Refer to the Figure. 1. If Name Two Congruent Angles. SOLUTION: Isosceles Triangle Theorem States That If Two Sides of T

Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 4-6 Isosceles and Equilateral Triangles 12

Refer to the figure. 4. m MRP

1. If name two congruent angles. SOLUTION: SOLUTION: Isosceles Triangle Theorem states that if two sides Since all the sides are congruent, is an of the triangle are congruent, then the angles opposite equilateral triangle. Each angle of an equilateral those sides are congruent. triangle measures 60°. Therefore, m MRP = 60°. Therefore, in triangle ABC, ANSWER: ANSWER: 60 BAC and BCA SENSE-MAKING Find the value of each variable. 2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, 5. in triangle EAC, SOLUTION: ANSWER: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem, Find each measure. 3. FH That is, .

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular 6. triangles are equilateral. Therefore, FH = GH = 12. SOLUTION: ANSWER: In the figure, Therefore, triangle WXY is 12 an isosceles triangle. By the Isosceles Triangle Theorem, . eSolutions4. m ManualMRP - Powered by Cognero Page 1

ANSWER: 16

SOLUTION: 7. PROOF Write a two-column proof. Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral Given: is isosceles; bisects ABC. triangle measures 60°. Therefore, m MRP = 60°. Prove:

ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that

ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the 16 distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

b. The Pythagorean Theorem tells us that: ANSWER:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. The Pythagorean Theorem tells us that b. If VR = 2.5 meters and QR = 2 meters, find the By CPCTC we know distance between and Explain your that VT = 1.5 m. The Seg. Add. Post. says QV +

reasoning. VT = QT. By substitution, we have 1.5 + 1.5 = QT.

So QT = 3 m. SOLUTION: Refer to the figure. a.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. b. The Pythagorean Theorem tells us that: SOLUTION:

By the Converse of Isosceles Triangle Theorem, in triangle ABF, By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. ANSWER: By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. 11. If name two congruent angles. ANSWER: a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

b. The Pythagorean Theorem tells us that 13. If BCF BFC, name two congruent segments. By CPCTC we know SOLUTION: that VT = 1.5 m. The Seg. Add. Post. says QV + By the Converse of Isosceles Triangle Theorem, in VT = QT. By substitution, we have 1.5 + 1.5 = QT.

So QT = 3 m. triangle BCF, ANSWER: Refer to the figure.

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, Here By Isosceles Triangle Theorem, . ANSWER: Apply the Triangle Sum Theorem.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER: 60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, SOLUTION: Given: ANSWER: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION:

AFH and AHF Since the triangle is isosceles, Therefore, Find each measure.

15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION: Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER:

60 SOLUTION: By the Converse of Isosceles Triangle Theorem, in 16. m SRT triangle ABC, That is, CB = 3.

ANSWER: 3

REGULARITY Find the value of each variable. SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, ANSWER: Solve for x. 65

17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

SOLUTION: Given: By the Isosceles Triangle Theorem, All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, And We know that, ANSWER: 4 18. CB ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in 21. triangle ABC, SOLUTION: That is, CB = 3. Given: ANSWER: By the Isosceles Triangle Theorem, 3 That is, REGULARITY Find the value of each variable.

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION:

Given: ANSWER: By the Isosceles Triangle Theorem, x = 11, y = 11

And We know that,

22. SOLUTION: ANSWER: Given: x = 13 By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y.

SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, ANSWER: x = 1.4, y = –1 Then, let Equation 1 be: . PROOF Write a paragraph proof.

Given: By the Triangle Sum Theorem, we can find Equation 23. is isosceles, and is 2: equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

Add the equations 1 and 2.

SOLUTION:

Proof: We are given that is an isosceles Substitute the value of x in one of the two equations to find the value of y. triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK,

MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent ANSWER: Supplements Theorem. By AAS, x = 11, y = 11 By CPCTC, JHK MHL. ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles 22. Triangle Theorem, we know that HJK HML. SOLUTION: Because is an equilateral triangle, we know Given: HLK LKH KHL and By the Triangle Angle-Sum JKH, HKL and HLK, Theorem, MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent So, the triangle is an equilateral triangle. Supplements Theorem. By AAS, Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y. Given: 24. W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

Segment Addition Postulate gives us XW + WY = XY SOLUTION: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY ANSWER: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and 25. BABYSITTING While babysitting her neighbor’s JKH, HKL and HLK, children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. MLH are supplementary, and HKL HLK, Using a jump rope to measure, Elisa is able to

we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is SOLUTION: equilateral. d. Proof: We are given W is the midpoint of If is isosceles, what is the minimum information needed to prove that and Q is the midpoint of Because W is the Explain your reasoning. midpoint of we know that Similarly, because Q is the midpoint of The SOLUTION: Segment Addition Postulate gives us XW + WY = XY a. Given: and XQ + QZ = XZ. Substitution gives XW + WY =

XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, .

If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem.

Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. b. ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. c. by the Reflexive Property. By SAS, So, by CPCTC.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since equilateral. you know that the triangle is isosceles, if one leg is d. If is isosceles, what is the minimum congruent to a leg of then you know that information needed to prove that both pairs of legs are congruent. Because the base Explain your reasoning. angles of an isosceles triangle are congruent, if you know that you know that SOLUTION: Therefore, with one pair of congruent corresponding sides and a. Given: one pair of congruent corresponding angles, the By Isosceles Triangle Theorem, . triangles can be proved congruent using either ASA Apply Triangle Sum Theorem. or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. b.

c. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that , you know that angles of an isosceles triangle are congruent, if you Therefore, know that you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by a. 65°; Because is isosceles, ABC bisects the angle formed by the sloped sides of ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65.

b. the roof, ABC.

SOLUTION:

ANSWER:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC. 27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

Sample answer: I constructed a pair of perpendicular ANSWER: segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles.

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Sample answer: I constructed a pair of perpendicular Proof: The base angles are congruent because it is segments and then used the same compass setting to an isosceles triangle. Let the measure of each acute mark points equidistant from their intersection. I angle be x. The acute angles of a right triangle are measured both legs for each triangle. Because AB = complementary, Solve for x AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles. ANSWER: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

ANSWER: Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to 44 mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = 30. m ACD AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a SOLUTION: protractor to confirm that A, D, and G are all We know right angles. that Therefore, . 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship ANSWER: between the base angles of an isosceles right 44 triangle. 31. m ACB SOLUTION: Conjecture: The measures of the base angles of an SOLUTION: isosceles right triangle are 45. The angles in a straight line add to 180°. Proof: The base angles are congruent because it is Therefore, an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are We know that complementary, Solve for x

ANSWER: ANSWER: Conjecture: The measures of the base angles of an 136 isosceles right triangle are 45. Proof: The base angles are congruent because it is 32. m ABC an isosceles triangle. Let the measure of each acute SOLUTION: angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles triangle. Therefore, REGULARITY Find each measure. By the Triangle Angle Sum Theorem,

ANSWER: 22 29. m∠CAD SOLUTION: 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids From the figure, shown. If each triangle is isosceles with vertex Therefore, is an isosceles triangle. angles G, H, and J, and G

H, and H J, show that the distance from B to F is three times the distance from D to F. By the Triangle Sum Theorem,

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike SOLUTION: to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. SOLUTION: 36. Corollary 4.4 By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 . 37. Theorem 4.11 SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 .

SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find Solve the equation for x.

Note that x can equal –8 here because

. ANSWER: ANSWER: 80 3 41. m LMP SOLUTION: Since the triangle LMP is isosceles, Angles at a point in a straight line 39. add up to 180 . SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

Therefore, The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. ANSWER: 80 ANSWER: 14 42. m JLK GAMES Use the diagram of a game timer SOLUTION: shown to find each measure. Vertical angles are congruent. Therefore,

First we need to find

We know that,

So, 40. m LPM ANSWER: SOLUTION: 20 Angles at a point in a straight line add up to 180 . 43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

In 41. m LMP

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the Substitute x = 45 in (3x – 55) to find interior angles of an isosceles triangle given the

measure of one exterior angle.

Therefore,

ANSWER: 80

42. m JLK a. Geometric Use a ruler and a protractor to draw SOLUTION: three different isosceles triangles, extending one of Vertical angles are congruent. Therefore, the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

b. Tabular Use a protractor to measure and record First we need to find m∠1 for each triangle. Use m 1 to calculate the We know that, measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same

measures. Organize your results in two tables.

c. Verbal Explain how you used m 1 to find the So, measures of 3, 4, and 5. Then explain how ANSWER: you used m 2 to find these same measures. 20 d. Algebraic If m 1 = x, write an expression for 43. m JKL the measures of 3, 4, and 5. Likewise, if m SOLUTION: 2 = x, write an expression for these same angle measures. Vertical angles are congruent. Therefore,

First we need to find SOLUTION: a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the c. 5 is supplementary to 1, so m 5 = 180 – measures of 3, 4, and 5. Then explain how m 1. 4 5, so m 4 = m 5. The sum of the you used m 2 to find these same measures. angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. Algebraic If m 1 = x, write an expression for m 3 = 180 – m 2. m 2 is twice as much as m the measures of 3, 4, and 5. Likewise, if m 4 and m 5, so m 4 = m 5 = 2 = x, write an expression for these same angle measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = c. 5 is supplementary to 1, so m 5 = 180 – 180 – m 4 – 5. 2 is supplementary to 3, so m 1. 4 5, so m 4 = m 5. The sum of the m 3 = 180 – m 2. m 2 is twice as much as m angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so 4 and m 5, so m 4 = m 5 = m 3 = 180 – m 2. m 2 is twice as much as m

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4 and m 5, so m 4 = m 5 =

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = ANSWER: a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

PRECISION Determine whether the following statements are sometimes, always, or never true. c. 5 is supplementary to 1, so m 5 = 180 – Explain. m 1. 4 5, so m 4 = m 5. The sum of the 46. If the measure of the vertex angle of an isosceles angle measures in a triangle must be 180, so m 3 = triangle is an integer, then the measure of each base 180 – m 4 – 5. 2 is supplementary to 3, so angle is an integer. m 3 = 180 – m 2. m 2 is twice as much as m SOLUTION:

4 and m 5, so m 4 = m 5 = The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 180; m 3 = 180 – x, m 4 = m 5 = ANSWER: 45. CHALLENGE In the figure, if is equilateral Sometimes; only if the measure of the vertex angle is even. and ZWP WJM JZL, prove that

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

measure of the vertex angle is even. For example, Neither; m G = or 55. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not Sometimes; only if the measure of the vertex angle is possible, explain why not. even. SOLUTION: It is not possible because a triangle cannot have more 47. If the measures of the base angles of an isosceles than one obtuse angle. triangle are integers, then the measure of its vertex angle is odd. ANSWER: It is not possible because a triangle cannot have more SOLUTION: than one obtuse angle. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 50. REASONING In isosceles m B = 90. angle) so if the base angles are integers, then 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even and 180 – 2 label each angle with its measure. (measure of the base angle) will be even. SOLUTION: ANSWER: The sum of the angle measures in a triangle must be Never; the measure of the vertex angle will be 180 – 180, 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will Since is isosceles, be even and 180 – 2(measure of the base angle) will And given that m B = 90. be even. ∠ Therefore,

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

than one obtuse angle. type of triangle in the proof. Doing this can save you ANSWER: steps when writing the proof. It is not possible because a triangle cannot have more 52. MULTI-STEP △AED and △BEC are congruent than one obtuse angle. and isosceles. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be

180, a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC B

C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION:

a. Of the statements given, only choice B is not ANSWER: necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ 51. WRITING IN MATH How can triangle △DEC by SAS, so AB ≅ EC. classifications help you prove triangle congruence? ANSWER: SOLUTION: a. B If a triangle is already classified, you can use the b. 40 previously proven properties of that type of triangle c. 80 in the proof. For example, if you know that a d. 60; Sample answer: All three angles in △AEB triangle is an equilateral triangle, you can use are 60° because base angles of an isosceles triangle Corollary 4.3 and 4.4 in the proof. Doing this can are congruent. So △AEB is equiangular and save you steps when writing the proof. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

ANSWER: 53. What is m∠FEG? Sample answer: If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B

C ∠CAD ≅ ∠DBC D Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two b. If m∠EAB = 40, what is the measure of ∠DCE? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is c. If m∠ADE = 50, what is the measure of ∠BEC? choice D.

d. What measure would ∠AEB need to be so that ANSWER: ? D SOLUTION: 54. A jeweler bends wire to make charms in the shape a. Of the statements given, only choice B is not of isosceles triangles. The base of the triangle necessarily true from the given information. measures 2 centimeters and another side measures b. The angles are congruent, so ∠DCE is 40°. 3.5 centimeters. How many charms can be made c. The sum of the measures of the angles of a from a piece of wire that is 63 centimeters long? triangle is 180, so ∠BEC must be 80°. SOLUTION: d. 60; All three angles in △AEB are 60° because Because we know that the shape of each charm is base angles of an isosceles triangle are congruent. So an isosceles triangle, then we can determine the side △AEB is equiangular and equilateral. △AEB ≅ lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, △DEC by SAS, so AB ≅ EC. the perimeter of each triangle is 9 cm.

ANSWER: To find how many charms can be made, divide 63 a. B cm by 9 cm. So, the jeweler can make 7 charms. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB 7 are 60° because base angles of an isosceles triangle 55. In the figure below, NDG LGD. are congruent. So △AEB is equiangular and

equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

a. What is m L? ∠ b. What is the value of x? A 34 c. What is m∠DGN? B 56 SOLUTION: C 60 a. In this figure, angles N and L are congruent so D 62 E 124 m∠L = 36°. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is SOLUTION: 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: a. 36 b. 54 c. 54 Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two 56. An equilateral triangle has a perimeter of 216 inches. angles would be 180 – 56 or 124°. What is the length of one side of the triangle? So, m∠FEG is , or 62. The correct answer is SOLUTION: choice D. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 ANSWER: to find the length of one side. D 216 ÷ 3 = 72 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle ANSWER: measures 2 centimeters and another side measures 72 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? Because we know that the shape of each charm is an isosceles triangle, then we can determine the side SOLUTION: lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, The perimeter can be found by adding all three side the perimeter of each triangle is 9 cm. lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or ANSWER: 16. 7 So, the perimeter of △JKL is 16. 55. In the figure below, NDG LGD. ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION: In an isosceles triangle, the base angles are a. What is m∠L? congruent. b. What is the value of x? c. What is m∠DGN? So, 5x + 5 = 7x – 19.

SOLUTION: Simplify. a. In this figure, angles N and L are congruent so m∠L = 36°. –2x = –24 b. Because we know two angle measures, and that x = 12 the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. ANSWER: c. ∠DGN is congruent to x°, so m∠DGN is 54°. 12

ANSWER: 59. QRV is an equilateral triangle. What additional a. 36 information would be enough to prove VRT is an b. 54 equilateral triangle? c. 54

56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72 A ANSWER: B m∠VRT = 60 72 C QRV VTU D VRT RST 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. Consider each choice. So, the perimeter of △JKL is 16. A does not provide enough information to ANSWER: determine how sides VT and TR are related. 16 B If we know that m∠VRT = 60, we still would not △ 58. ABC is an isosceles triangle with base angles have enough information to prove that the other two measuring (5x + 5)° and (7x – 19)°. Find the value of measures are 60 degrees. x. SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12 If , then m∠RVT = 60 because ANSWER: ∠QVR + ∠RVT + ∠TVU = 180. Since , 12 by the Isosceles Triangle Theorem. So, . 59. QRV is an equilateral triangle. What additional Therefore, is equilateral by Theorem 4.3. information would be enough to prove VRT is an equilateral triangle? D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

Refer to the figure. 4. m MRP

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular 6. triangles are equilateral. Therefore, FH = GH = 12. SOLUTION: ANSWER: 12 In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, . 4. m MRP

ANSWER: 16 SOLUTION: 7. PROOF Write a two-column proof. Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral Given: is isosceles; bisects ABC. triangle measures 60°. Therefore, m MRP = 60°. Prove:

ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 4-6 Isosceles12 and Equilateral Triangles

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that

ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the 16 distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: eSolutions8. ROLLERManual - COASTERSPowered by Cognero The roller coaster track a. Page 2 appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. The Pythagorean Theorem tells us that b. If VR = 2.5 meters and QR = 2 meters, find the By CPCTC we know distance between and Explain your that VT = 1.5 m. The Seg. Add. Post. says QV + reasoning. VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. SOLUTION: a. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. b. The Pythagorean Theorem tells us that: SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF, By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. ANSWER: By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 11. If name two congruent angles. a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

b. The Pythagorean Theorem tells us that 13. If BCF BFC, name two congruent segments. By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + SOLUTION: VT = QT. By substitution, we have 1.5 + 1.5 = QT. By the Converse of Isosceles Triangle Theorem, in So QT = 3 m. triangle BCF,

Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, Here By Isosceles Triangle Theorem, . ANSWER: Apply the Triangle Sum Theorem.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER:

60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR 14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION: AFH and AHF Since the triangle is isosceles, Therefore, Find each measure. 15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION: Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER:

60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3

REGULARITY Find the value of each variable. SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, ANSWER: Solve for x. 65 17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in 21. triangle ABC, That is, CB = 3. SOLUTION: Given: ANSWER: By the Isosceles Triangle Theorem, 3 REGULARITY Find the value of each variable. That is,

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11

And We know that,

22. SOLUTION: ANSWER: Given: x = 13 By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y. SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, ANSWER: x = 1.4, y = –1 Then, let Equation 1 be: . PROOF Write a paragraph proof. By the Triangle Sum Theorem, we can find Equation 23. Given: is isosceles, and is 2: equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

Add the equations 1 and 2.

SOLUTION:

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles 22. Triangle Theorem, we know that HJK HML. SOLUTION: Because is an equilateral triangle, we know Given: HLK LKH KHL and By the Triangle Angle-Sum JKH, HKL and HLK, Theorem, MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent So, the triangle is an equilateral triangle. Supplements Theorem. By AAS, Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y.

24. Given: W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary

and HLK and MLH are supplementary. Prove: JHK MHL SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY SOLUTION: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY ANSWER: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and 25. BABYSITTING While babysitting her neighbor’s JKH, HKL and HLK, children, Elisa observes that the supports on either MLH are supplementary, and HKL HLK, side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is SOLUTION: equilateral. Proof: We are given W is the midpoint of d. If is isosceles, what is the minimum information needed to prove that and Q is the midpoint of Because W is the Explain your reasoning. midpoint of we know that Similarly, because Q is the midpoint of The SOLUTION: Segment Addition Postulate gives us XW + WY = XY

and XQ + QZ = XZ. Substitution gives XW + WY = a. Given: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, . If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. b. ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The

Isosceles Triangle Theorem says XYZ XZY. c. by the Reflexive Property. By SAS, So, by CPCTC.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is d. One pair of congruent corresponding sides and equilateral. one pair of congruent corresponding angles; since d. If is isosceles, what is the minimum you know that the triangle is isosceles, if one leg is information needed to prove that congruent to a leg of then you know that Explain your reasoning. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that SOLUTION: Therefore, a. Given: with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the By Isosceles Triangle Theorem, . triangles can be proved congruent using either ASA Apply Triangle Sum Theorem. or SAS.

ANSWER: a. 65°; Because is isosceles, ABC

∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. b.

c. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that , you know that angles of an isosceles triangle are congruent, if you Therefore, know that you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. bisects the angle formed by the sloped sides of b. the roof, ABC.

SOLUTION:

ANSWER: d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

Sample answer: I constructed a pair of perpendicular isosceles right triangle are 45. segments and then used the same compass setting to Proof: The base angles are congruent because it is mark points equidistant from their intersection. I an isosceles triangle. Let the measure of each acute measured both legs for each triangle. Because AB = angle be x. The acute angles of a right triangle are AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = complementary, Solve for x 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles. ANSWER: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

Sample answer: I constructed a pair of perpendicular ANSWER: segments and then used the same compass setting to 44 mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = 30. m ACD AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a SOLUTION: protractor to confirm that A, D, and G are all We know right angles. that Therefore, . 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship ANSWER: between the base angles of an isosceles right 44 triangle. SOLUTION: 31. m ACB Conjecture: The measures of the base angles of an SOLUTION: isosceles right triangle are 45. The angles in a straight line add to 180°. Proof: The base angles are congruent because it is Therefore, an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are We know that complementary, Solve for x

ANSWER: ANSWER: Conjecture: The measures of the base angles of an 136 isosceles right triangle are 45. Proof: The base angles are congruent because it is 32. m ABC an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles triangle. Therefore, REGULARITY Find each measure.

By the Triangle Angle Sum Theorem,

H, and H J, show that the distance from B to F is three times the distance from D to F. By the Triangle Sum Theorem,

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

FITNESS 33. In the diagram, the rider will use his bike SOLUTION: to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. SOLUTION: 36. Corollary 4.4 By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

37. Theorem 4.11 The interior angles of a triangle add up to 180 .

SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 .

SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find

Solve the equation for x.

Note that x can equal –8 here because . ANSWER: ANSWER: 80 3 41. m LMP SOLUTION: Since the triangle LMP is isosceles, 39. Angles at a point in a straight line add up to 180 . SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

So, 40. m LPM SOLUTION: ANSWER: 20 Angles at a point in a straight line add up to 180 . 43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP In

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this Substitute x = 45 in (3x – 55) to find problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

Therefore,

ANSWER: 80

First we need to find b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the We know that, measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables.

So, c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how ANSWER: you used m 2 to find these same measures. 20 d. Algebraic If m 1 = x, write an expression for 43. m JKL the measures of 3, 4, and 5. Likewise, if m SOLUTION: 2 = x, write an expression for these same angle Vertical angles are congruent. Therefore, measures.

First we need to find SOLUTION: a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

b. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the c. 5 is supplementary to 1, so m 5 = 180 – measures of 3, 4, and 5. Then explain how m 1. 4 5, so m 4 = m 5. The sum of the you used m 2 to find these same measures. angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. Algebraic If m 1 = x, write an expression for m 3 = 180 – m 2. m 2 is twice as much as m the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle 4 and m 5, so m 4 = m 5 =

measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4 and m 5, so m 4 = m 5 =

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

ANSWER: 180; m 3 = 180 – x, m 4 = m 5 = a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

measure of the vertex angle is even. For example, Neither; m G = or 55. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not Sometimes; only if the measure of the vertex angle is possible, explain why not. even. SOLUTION: It is not possible because a triangle cannot have more 47. If the measures of the base angles of an isosceles than one obtuse angle. triangle are integers, then the measure of its vertex angle is odd. ANSWER: SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 50. REASONING In isosceles m B = 90. angle) so if the base angles are integers, then 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even and 180 – 2 label each angle with its measure. (measure of the base angle) will be even. SOLUTION: ANSWER: The sum of the angle measures in a triangle must be Never; the measure of the vertex angle will be 180 – 180, 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will

be even and 180 – 2(measure of the base angle) will Since is isosceles, be even. And given that m∠B = 90. Therefore,

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? SOLUTION: ANSWER: If a triangle is already classified, you can use the previously proven properties of that type of triangle Neither; m G = or 55. in the proof. For example, if you know that a triangle is an equilateral triangle, you can use 49. OPEN-ENDED If possible, draw an isosceles Corollary 4.3 and 4.4 in the proof. Doing this can triangle with base angles that are obtuse. If it is not save you steps when writing the proof. possible, explain why not. ANSWER: SOLUTION: Sample answer: If a triangle is already classified, It is not possible because a triangle cannot have more you can use the previously proven properties of that than one obtuse angle. type of triangle in the proof. Doing this can save you ANSWER: steps when writing the proof. It is not possible because a triangle cannot have more than one obtuse angle. 52. MULTI-STEP △AED and △BEC are congruent and isosceles. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be

180, a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC

B C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION:

52. MULTI-STEP △AED and △BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B

d. What measure would ∠AEB need to be so that ANSWER: ? D SOLUTION: 54. A jeweler bends wire to make charms in the shape a. Of the statements given, only choice B is not of isosceles triangles. The base of the triangle necessarily true from the given information. measures 2 centimeters and another side measures b. The angles are congruent, so ∠DCE is 40°. 3.5 centimeters. How many charms can be made c. The sum of the measures of the angles of a from a piece of wire that is 63 centimeters long? triangle is 180, so ∠BEC must be 80°. SOLUTION: △ d. 60; All three angles in AEB are 60° because Because we know that the shape of each charm is base angles of an isosceles triangle are congruent. So an isosceles triangle, then we can determine the side △AEB is equiangular and equilateral. △AEB ≅ lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, △DEC by SAS, so AB ≅ EC. the perimeter of each triangle is 9 cm.

ANSWER: To find how many charms can be made, divide 63 a. B cm by 9 cm. So, the jeweler can make 7 charms. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB 7 are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 55. In the figure below, NDG LGD.

equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

a. What is m∠L? b. What is the value of x? A 34 c. What is m∠DGN? B 56 SOLUTION: C 60 D 62 a. In this figure, angles N and L are congruent so E 124 m∠L = 36°. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is SOLUTION: 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: a. 36 b. 54 c. 54 Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two 56. An equilateral triangle has a perimeter of 216 inches. angles would be 180 – 56 or 124°. What is the length of one side of the triangle? So, m∠FEG is , or 62. The correct answer is SOLUTION: choice D. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 ANSWER: to find the length of one side. D 216 ÷ 3 = 72 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle ANSWER: measures 2 centimeters and another side measures 72 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of △ Because we know that the shape of each charm is JKL? an isosceles triangle, then we can determine the side SOLUTION: lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, The perimeter can be found by adding all three side the perimeter of each triangle is 9 cm. lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. Because the x-terms cancel each other out: 2x + 2x ANSWER: – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 7

△ 55. In the figure below, NDG LGD. So, the perimeter of JKL is 16. ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

In an isosceles triangle, the base angles are a. What is m L? ∠ congruent. b. What is the value of x? c. What is m∠DGN? So, 5x + 5 = 7x – 19. SOLUTION: Simplify. a. In this figure, angles N and L are congruent so

m∠L = 36°. –2x = –24 b. Because we know two angle measures, and that x = 12 the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. ANSWER: c. ∠DGN is congruent to x°, so m∠DGN is 54°. 12

ANSWER: 59. QRV is an equilateral triangle. What additional a. 36 information would be enough to prove VRT is an b. 54 equilateral triangle? c. 54 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72 A ANSWER: B m∠VRT = 60 72 C QRV VTU D VRT RST 57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of SOLUTION: △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. Consider each choice. So, the perimeter of △JKL is 16. A does not provide enough information to ANSWER: determine how sides VT and TR are related. 16 B If we know that m VRT = 60, we still would not 58. △ABC is an isosceles triangle with base angles ∠ have enough information to prove that the other two measuring (5x + 5)° and (7x – 19)°. Find the value of measures are 60 degrees. x.

SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: Refer to the figure. 12

4. m MRP

1. If name two congruent angles.

SOLUTION: Isosceles Triangle Theorem states that if two sides SOLUTION: of the triangle are congruent, then the angles opposite those sides are congruent. Since all the sides are congruent, is an Therefore, in triangle ABC, equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: ANSWER: BAC and BCA 60 2. If EAC ECA, name two congruent segments. SENSE-MAKING Find the value of each variable. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC, 5. ANSWER: SOLUTION: In the figure, . Therefore, triangle RST is Find each measure. an isosceles triangle. 3. FH By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: ANSWER: By the Triangle Angle-Sum Theorem, 12

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 6. 12 SOLUTION: In the figure, Therefore, triangle WXY is 4. m MRP an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: SOLUTION: 16 Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral 7. PROOF Write a two-column proof. triangle measures 60°. Therefore, m MRP = 60°. Given: is isosceles; bisects ABC. ANSWER: Prove: 60 SENSE-MAKING Find the value of each variable.

SOLUTION: 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown. 6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is ANSWER: isosceles with base , and prove that 16 b. If VR = 2.5 meters and QR = 2 meters, find the 7. PROOF Write a two-column proof. distance between and Explain your reasoning. Given: is isosceles; bisects ABC.

Prove: SOLUTION: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ROLLER COASTERS The 8. roller coaster track ANSWER: appears to be composed of congruent triangles. A portion of the track is shown. a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your b. The Pythagorean Theorem tells us that reasoning. By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + SOLUTION: VT = QT. By substitution, we have 1.5 + 1.5 = QT. a. So QT = 3 m. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER:

ABE and AEB b. The Pythagorean Theorem tells us that: 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By CPCTC we know that VT = 1.5 m. The By the Converse of Isosceles Triangle Theorem, in Segment Addition Postulate says QV + VT = QT. triangle ABF, By substitution, we have 1.5 + 1.5 = QT. So QT = ANSWER: 4-6 Isosceles3 m. and Equilateral Triangles

ANSWER: a. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, ANSWER: b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + 13. If BCF BFC, name two congruent segments. VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. By the Converse of Isosceles Triangle Theorem, in Refer to the figure. triangle BCF,

ANSWER:

14. If name two congruent angles. SOLUTION: 9. If name two congruent angles. By the Isosceles Triangle Theorem, in triangle AFH,

SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, ANSWER: AFH and AHF

ANSWER: Find each measure. ABE and AEB 15. m BAC

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ABF, SOLUTION: ANSWER: Here By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem. 11. If name two congruent angles. eSolutionsSOLUTION: Manual - Powered by Cognero Page 3 By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ANSWER: 60 ACD and ADC

12. If DAE DEA, name two congruent 16. m SRT segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: SOLUTION: Given: By Isosceles Triangle Theorem, . 13. If BCF BFC, name two congruent segments. SOLUTION: Apply Triangle Angle-Sum Theorem.

By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: 65 14. If name two congruent angles. 17. TR SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: AFH and AHF SOLUTION: Find each measure. Since the triangle is isosceles, 15. m BAC Therefore,

All the angles are congruent. Therefore, it is an SOLUTION: equiangular triangle. Since the equiangular triangle is an equilateral, Here By Isosceles Triangle Theorem, . ANSWER: Apply the Triangle Sum Theorem. 4

18. CB

ANSWER: 60

16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3 SOLUTION: Given: REGULARITY Find the value of each variable. By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

19. SOLUTION: Since all the angles are congruent, the sides are also ANSWER: congruent to each other. 65 Therefore, Solve for x. 17. TR

ANSWER: x = 5 SOLUTION: Since the triangle is isosceles, Therefore,

20.

SOLUTION: All the angles are congruent. Therefore, it is an Given: equiangular triangle. By the Isosceles Triangle Theorem, Since the equiangular triangle is an equilateral,

And ANSWER: We know 4 that, 18. CB

ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3. 21.

ANSWER: SOLUTION: 3 Given: By the Isosceles Triangle Theorem, REGULARITY Find the value of each variable. That is,

Then, let Equation 1 be: . 19. By the Triangle Sum Theorem, we can find Equation SOLUTION: 2: Since all the angles are congruent, the sides are also congruent to each other. Therefore, Solve for x. Add the equations 1 and 2.

ANSWER: x = 5 Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11 And We know that,

ANSWER: 22. x = 13 SOLUTION: Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. 21. Therefore,

SOLUTION: Set up two equations to solve for x and y. Given: By the Isosceles Triangle Theorem,

That is,

ANSWER: Then, let Equation 1 be: . x = 1.4, y = –1 By the Triangle Sum Theorem, we can find Equation PROOF Write a paragraph proof. 2: 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Add the equations 1 and 2. Prove: JHK MHL

JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK, x = 11, y = 11 we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH 22. and HKL are supplementary and HLK and SOLUTION: MLH are supplementary. From the Isosceles Given: Triangle Theorem, we know that HJK HML. By the Triangle Angle-Sum Because is an equilateral triangle, we know Theorem, HLK LKH KHL and JKH, HKL and HLK, So, the triangle is an equilateral triangle. MLH are supplementary, and HKL HLK, Therefore, we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, Set up two equations to solve for x and y. By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of ANSWER: Prove: x = 1.4, y = –1 PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, SOLUTION: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH MLH by the Congruent Proof: We are given W is the midpoint of Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, BABYSITTING MLH are supplementary, and HKL HLK, 25. While babysitting her neighbor’s children, Elisa observes that the supports on either we know JKH MLH by the Congruent ∠ side of a park swing set form two sets of triangles. Supplements Theorem. By AAS, Using a jump rope to measure, Elisa is able to By CPCTC, JHK MHL. determine that but

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. SOLUTION: c. If and show that is Proof: We are given W is the midpoint of equilateral. and Q is the midpoint of Because W is the d. If is isosceles, what is the minimum midpoint of we know that Similarly, information needed to prove that because Q is the midpoint of The Explain your reasoning. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = SOLUTION: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. a. Given: If we divide each side by 2, we have WY = QZ. The By Isosceles Triangle Theorem, . Isosceles Triangle Theorem says XYZ XZY. Apply Triangle Sum Theorem. by the Reflexive Property. By SAS, So, by CPCTC.

ANSWER: Proof: We are given W is the midpoint of b. and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. c.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is

equilateral. d. One pair of congruent corresponding sides and d. If is isosceles, what is the minimum one pair of congruent corresponding angles; since information needed to prove that you know that the triangle is isosceles, if one leg is Explain your reasoning. congruent to a leg of then you know that

both pairs of legs are congruent. Because the base SOLUTION: angles of an isosceles triangle are congruent, if you know that you know that a. Given: Therefore, By Isosceles Triangle Theorem, . with one pair of congruent corresponding sides and Apply Triangle Sum Theorem. one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: b. a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

c. c.

d. One pair of congruent corresponding sides and d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that congruent to a leg of then you know that both pairs of legs are congruent. Because the base both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you angles of an isosceles triangle are congruent, if you know that you know that know that , you know that Therefore, Therefore, with one pair of congruent corresponding sides and with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA triangles can be proved congruent using either ASA or SAS. or SAS.

ANSWER: 26. CHIMNEYS In the picture, and a. 65°; Because is isosceles, ABC is an isosceles triangle with base Show ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. that the chimney of the house, represented by b. bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since ANSWER: you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER: Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles.

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Sample answer: I constructed a pair of perpendicular Conjecture: The measures of the base angles of an segments and then used the same compass setting to isosceles right triangle are 45. mark points equidistant from their intersection. I Proof: The base angles are congruent because it is measured both legs for each triangle. Because AB = an isosceles triangle. Let the measure of each acute AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = angle be x. The acute angles of a right triangle are 2.3 cm, the triangles are isosceles. I used a complementary, Solve for x protractor to confirm that A, D, and G are all

right angles.

ANSWER: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I ANSWER: measured both legs for each triangle. Because AB = 44 AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 30. m ACD protractor to confirm that A, D, and G are all right angles. SOLUTION: We know 28. PROOF Based on your construction in Exercise 27, that make and prove a conjecture about the relationship Therefore, . between the base angles of an isosceles right triangle. ANSWER: SOLUTION: 44 Conjecture: The measures of the base angles of an 31. m ACB isosceles right triangle are 45. Proof: The base angles are congruent because it is SOLUTION: an isosceles triangle. Let the measure of each acute The angles in a straight line add to 180°. angle be x. The acute angles of a right triangle are Therefore, complementary, Solve for x We know that

ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. ANSWER: Proof: The base angles are congruent because it is 136 an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are 32. m ABC complementary, so x + x = 90 and x = 45. SOLUTION: REGULARITY Find each measure. From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

29. m∠CAD ANSWER: SOLUTION: 22

From the figure, 33. FITNESS In the diagram, the rider will use his bike Therefore, is an isosceles triangle. to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G By the Triangle Sum Theorem, H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD

SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that ANSWER:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, 34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex SOLUTION: angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION: ANSWER:

ANSWER: PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

36. Corollary 4.4 38. SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . 37. Theorem 4.11 SOLUTION: The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM 38. SOLUTION: SOLUTION: Angles at a point in a straight line add up to 180 . By the converse of Isosceles Triangle theorem,

Solve the equation for x. Substitute x = 45 in (3x – 55) to find

Note that x can equal –8 here because .

ANSWER: ANSWER: 3 80

41. m LMP SOLUTION: 39. Since the triangle LMP is isosceles, SOLUTION: Angles at a point in a straight line By the Isosceles Triangle Theorem, the third angle is add up to 180 . equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Therefore,

ANSWER: ANSWER: 14 80

GAMES Use the diagram of a game timer 42. m JLK shown to find each measure. SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM SOLUTION: So, Angles at a point in a straight line add up to 180 . ANSWER: 20

43. m JKL SOLUTION: Substitute x = 45 in (3x – 55) to find Vertical angles are congruent. Therefore,

First we need to find

We know that, ANSWER:

80 So, 41. m LMP SOLUTION: In Since the triangle LMP is isosceles, Angles at a point in a straight line Since the triangle JKL is isosceles, add up to 180 .

ANSWER: 80 Substitute x = 45 in (3x – 55) to find MULTIPLE REPRESENTATIONS 44. In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

Therefore,

ANSWER: 80

42. m JLK SOLUTION: Vertical angles are congruent. Therefore, a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown. First we need to find

b. Tabular Use a protractor to measure and record We know that, m 1 for each triangle. Use m 1 to calculate the ∠ measures of 3, 4, and 5. Then find and

record m 2 and use it to calculate these same measures. Organize your results in two tables. So, ANSWER: c. Verbal Explain how you used m 1 to find the 20 measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. 43. m JKL d. Algebraic If m 1 = x, write an expression for SOLUTION: the measures of 3, 4, and 5. Likewise, if m Vertical angles are congruent. Therefore, 2 = x, write an expression for these same angle measures. First we need to find

We know that, SOLUTION: a.

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw b. three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the d. Algebraic If m 1 = x, write an expression for angle measures in a triangle must be 180, so m 3 = the measures of 3, 4, and 5. Likewise, if m 180 – m 4 – 5. 2 is supplementary to 3, so 2 = x, write an expression for these same angle m 3 = 180 – m 2. m 2 is twice as much as m measures. 4 and m 5, so m 4 = m 5 = d. SOLUTION: m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

a. 180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so c. 5 is supplementary to 1, so m 5 = 180 – m 3 = 180 – m 2. m 2 is twice as much as m m 1. 4 5, so m 4 = m 5. The sum of the 4 and m 5, so m 4 = m 5 = angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – m 3 = 180 – m 2. m 2 is twice as much as m

180; m 3 = 180 – x, m 4 = m 5 = 4 and m 5, so m 4 = m 5 =

ANSWER: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

a. 180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

b. ANSWER:

c. 5 is supplementary to 1, so m 5 = 180 – PRECISION Determine whether the following m 1. 4 5, so m 4 = m 5. The sum of the statements are sometimes, always, or never true. angle measures in a triangle must be 180, so m 3 = Explain. 180 – m 4 – 5. 2 is supplementary to 3, so 46. If the measure of the vertex angle of an isosceles m 3 = 180 – m 2. m 2 is twice as much as m triangle is an integer, then the measure of each base angle is an integer. 4 and m 5, so m 4 = m 5 = SOLUTION: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, 180; m 3 = 180 – x, m 4 = m 5 = vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 45. CHALLENGE In the figure, if is equilateral ANSWER: and ZWP WJM JZL, prove that Sometimes; only if the measure of the vertex angle is even.

angle) so if the base angles are integers, then 2 SOLUTION: (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, ANSWER: vertex angle = 50, base angles = 65;

vertex angle = 55, base angles = 62.5. Neither; m G = or 55. ANSWER: Sometimes; only if the measure of the vertex angle is 49. OPEN-ENDED If possible, draw an isosceles even. triangle with base angles that are obtuse. If it is not possible, explain why not. 47. If the measures of the base angles of an isosceles SOLUTION: triangle are integers, then the measure of its vertex It is not possible because a triangle cannot have more angle is odd. than one obtuse angle.

SOLUTION: ANSWER: The statement is never true. The measure of the It is not possible because a triangle cannot have more vertex angle will be 180 – 2(measure of the base than one obtuse angle. angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 50. REASONING In isosceles m B = 90. (measure of the base angle) will be even. Draw the triangle. Indicate the congruent sides and label each angle with its measure. ANSWER: Never; the measure of the vertex angle will be 180 – SOLUTION: 2(measure of the base angle) so if the base angles The sum of the angle measures in a triangle must be are integers, then 2(measure of the base angle) will 180, be even and 180 – 2(measure of the base angle) will be even. Since is isosceles, 48. ERROR ANALYSIS Alexis and Miguela are And given that m∠B = 90. Therefore, finding m G in the figure shown. Alexis says that

m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. ANSWER:

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? ANSWER: SOLUTION:

Neither; m G = or 55. If a triangle is already classified, you can use the previously proven properties of that type of triangle 49. OPEN-ENDED If possible, draw an isosceles in the proof. For example, if you know that a triangle with base angles that are obtuse. If it is not triangle is an equilateral triangle, you can use possible, explain why not. Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. SOLUTION: It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Sample answer: If a triangle is already classified, ANSWER: you can use the previously proven properties of that type of triangle in the proof. Doing this can save you It is not possible because a triangle cannot have more than one obtuse angle. steps when writing the proof. △ △ 50. REASONING In isosceles m B = 90. 52. MULTI-STEP AED and BEC are congruent Draw the triangle. Indicate the congruent sides and and isosceles. label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, And given that m∠B = 90. a. Which statement is not necessarily true from the Therefore, information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC Construct the triangle with the angles measures 90, D 45, 45. b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? ANSWER: SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because 51. WRITING IN MATH How can triangle base angles of an isosceles triangle are congruent. So classifications help you prove triangle congruence? △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. SOLUTION: If a triangle is already classified, you can use the ANSWER: previously proven properties of that type of triangle a. B in the proof. For example, if you know that a b. 40 triangle is an equilateral triangle, you can use c. 80 Corollary 4.3 and 4.4 in the proof. Doing this can d. 60; Sample answer: All three angles in △AEB save you steps when writing the proof. are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and ANSWER: equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. Sample answer: If a triangle is already classified, you can use the previously proven properties of that 53. What is m∠FEG? type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

A 34 B 56 C 60 D 62

E 124 a. Which statement is not necessarily true from the information given? SOLUTION:

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two c. If m∠ADE = 50, what is the measure of ∠BEC? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is d. What measure would ∠AEB need to be so that choice D. ? ANSWER: SOLUTION: D a. Of the statements given, only choice B is not necessarily true from the given information. 54. A jeweler bends wire to make charms in the shape b. The angles are congruent, so ∠DCE is 40°. of isosceles triangles. The base of the triangle c. The sum of the measures of the angles of a measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made triangle is 180, so ∠BEC must be 80°. from a piece of wire that is 63 centimeters long? d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So SOLUTION: △AEB is equiangular and equilateral. △AEB ≅ Because we know that the shape of each charm is △DEC by SAS, so AB ≅ EC. an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, ANSWER: the perimeter of each triangle is 9 cm. a. B b. 40 To find how many charms can be made, divide 63 c. 80 cm by 9 cm. So, the jeweler can make 7 charms. d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle ANSWER: are congruent. So △AEB is equiangular and 7 equilateral. △AEB △DEC by SAS, so AB EC. ≅ ≅ 55. In the figure below, NDG LGD.

53. What is m∠FEG?

A 34 a. What is m∠L? B 56 b. What is the value of x? C 60 c. What is m∠DGN? D 62 SOLUTION: E 124 a. In this figure, angles N and L are congruent so

m∠L = 36°. SOLUTION: b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER:

a. 36 Consider triangle DEH. Since one angle is 56°, and b. 54 the triangle is isosceles, then the remaining two c. 54 angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? choice D. SOLUTION: ANSWER: Because the triangle is equilateral that means each D side has the same length, so divide the perimeter by 3 to find the length of one side. 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle 216 ÷ 3 = 72 measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made ANSWER: from a piece of wire that is 63 centimeters long? 72 SOLUTION: 57. △JKL is an isosceles triangle, and . JK = Because we know that the shape of each charm is 2x + 3, and KL = 10 – 4x. What is the perimeter of an isosceles triangle, then we can determine the side △JKL? lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. SOLUTION: The perimeter can be found by adding all three side To find how many charms can be made, divide 63 lengths. In this situation, since the triangle is cm by 9 cm. So, the jeweler can make 7 charms. isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

ANSWER: Because the x-terms cancel each other out: 2x + 2x 7 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 55. In the figure below, NDG LGD.

So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of

x. a. What is m∠L? SOLUTION: b. What is the value of x? In an isosceles triangle, the base angles are c. What is m∠DGN? congruent. SOLUTION: So, 5x + 5 = 7x – 19. a. In this figure, angles N and L are congruent so m∠L = 36°. Simplify. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is –2x = –24 180, then we can solve 36 + 90 + x = 180, or x = 54. x = 12 c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: 12 a. 36 b. 54 59. QRV is an equilateral triangle. What additional c. 54 information would be enough to prove VRT is an equilateral triangle? 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

ANSWER: A 72 B m∠VRT = 60 57. △JKL is an isosceles triangle, and . JK = C QRV VTU 2x + 3, and KL = 10 – 4x. What is the perimeter of D VRT RST △JKL? SOLUTION: SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. Consider each choice. ANSWER: 16 A does not provide enough information to determine how sides VT and TR are related. 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of B If we know that m∠VRT = 60, we still would not x. have enough information to prove that the other two SOLUTION: measures are 60 degrees. In an isosceles triangle, the base angles are congruent. C

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

ANSWER: 12 If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , 59. QRV is an equilateral triangle. What additional by the Isosceles Triangle Theorem. information would be enough to prove VRT is an So, . equilateral triangle? Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER:

C A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem, Refer to the figure.

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 1. If name two congruent angles. 12 SOLUTION: Isosceles Triangle Theorem states that if two sides 4. m MRP of the triangle are congruent, then the angles opposite those sides are congruent. Therefore, in triangle ABC,

ANSWER: BAC and BCA

2. If EAC ECA, name two congruent segments. SOLUTION: SOLUTION: Since all the sides are congruent, is an Converse of Isosceles Triangle Theorem states that equilateral triangle. Each angle of an equilateral if two angles of a triangle are congruent, then the triangle measures 60°. Therefore, m MRP = 60°. sides opposite those angles are congruent. Therefore, in triangle EAC, ANSWER: 60 ANSWER: SENSE-MAKING Find the value of each variable. Find each measure. 3. FH

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. SOLUTION: By the Converse of Isosceles Triangle Theorem, By the Triangle Angle-Sum Theorem, That is, .

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. ANSWER: ANSWER: 12 12

4. m MRP

6. SOLUTION: In the figure, Therefore, triangle WXY is SOLUTION: an isosceles triangle. Since all the sides are congruent, is an By the Isosceles Triangle Theorem, . equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: 60 ANSWER: SENSE-MAKING Find the value of each 16 variable. 7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove: 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . SOLUTION:

ANSWER: 12

ANSWER:

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, . 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

ANSWER: 16

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC.

Prove: a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: SOLUTION: a.

ANSWER:

8. ROLLER COASTERS The roller coaster track b. The Pythagorean Theorem tells us that: appears to be composed of congruent triangles. A portion of the track is shown.

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

b. The Pythagorean Theorem tells us that: 9. If name two congruent angles.

SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = ANSWER: 3 m. ABE and AEB

ANSWER: 10. If ∠ABF ∠AFB, name two congruent segments. a. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC b. The Pythagorean Theorem tells us that 12. If DAE DEA, name two congruent By CPCTC we know segments. that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. By the Converse of Isosceles Triangle Theorem, in triangle ADE, Refer to the figure. ANSWER:

13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

9. If name two congruent angles. triangle BCF,

SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: 14. If name two congruent angles. ABE and AEB SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, 10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in AFH and AHF triangle ABF, Find each measure. ANSWER: 15. m BAC

11. If name two congruent angles. SOLUTION:

By the Isosceles Triangle Theorem, in triangle ACD, SOLUTION: Here ANSWER: By Isosceles Triangle Theorem, . ACD and ADC Apply the Triangle Sum Theorem. 12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, ANSWER: 60 ANSWER:

16. m SRT

13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

14. If name two congruent angles. Apply Triangle Angle-Sum Theorem. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

4-6 IsoscelesANSWER: and Equilateral Triangles ANSWER: AFH and AHF 65 Find each measure. 17. TR 15. m BAC

SOLUTION: SOLUTION: Since the triangle is isosceles, Here Therefore, By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem.

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, ANSWER: 60 ANSWER: 4 16. m SRT 18. CB

SOLUTION: Given: SOLUTION: By Isosceles Triangle Theorem, . By the Converse of Isosceles Triangle Theorem, in

Apply Triangle Angle-Sum Theorem. triangle ABC, That is, CB = 3. ANSWER: 3 ANSWER: REGULARITY Find the value of each variable. 65 17. TR

19. SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. SOLUTION: Therefore,

Since the triangle is isosceles, Solve for x. Therefore, eSolutions Manual - Powered by Cognero Page 4

ANSWER: x = 5 All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: 20. 4 SOLUTION: 18. CB Given: By the Isosceles Triangle Theorem,

And We know

SOLUTION: that, By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3. ANSWER: ANSWER: x = 13 3 REGULARITY Find the value of each variable.

21. 19. SOLUTION: SOLUTION: Given: Since all the angles are congruent, the sides are also By the Isosceles Triangle Theorem, congruent to each other. Therefore, That is, Solve for x. Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 2: ANSWER: x = 5

Add the equations 1 and 2.

20. SOLUTION:

Given: Substitute the value of x in one of the two equations

By the Isosceles Triangle Theorem, to find the value of y.

And We know that,

ANSWER: ANSWER: x = 13 x = 11, y = 11

21. 22. SOLUTION: SOLUTION: Given: Given: By the Isosceles Triangle Theorem, By the Triangle Angle-Sum Theorem, That is, So, the triangle is an equilateral triangle. Then, let Equation 1 be: . Therefore,

By the Triangle Sum Theorem, we can find Equation Set up two equations to solve for x and y. 2:

Add the equations 1 and 2. ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is

equilateral. JKH and HKL are supplementary Substitute the value of x in one of the two equations and HLK and MLH are supplementary. to find the value of y. Prove: JHK MHL

SOLUTION: Proof: We are given that is an isosceles ANSWER: triangle and is an equilateral triangle, JKH

x = 11, y = 11 and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK,

22. MLH are supplementary, and HKL HLK, SOLUTION: we know JKH MLH by the Congruent Given: Supplements Theorem. By AAS, By the Triangle Angle-Sum By CPCTC, JHK MHL.

Theorem, ANSWER:

Proof: So, the triangle is an equilateral triangle. We are given that is an isosceles Therefore, triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and Set up two equations to solve for x and y. MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH ∠MLH by the Congruent x = 1.4, y = –1 Supplements Theorem. By AAS, By CPCTC, JHK MHL. PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary 24. Given: and HLK and MLH are supplementary. W is the midpoint of Prove: JHK MHL Q is the midpoint of

Prove:

SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and SOLUTION: MLH are supplementary. From the Isosceles Proof: We are given W is the midpoint of Triangle Theorem, we know that HJK HML. and Q is the midpoint of Because W is the Because is an equilateral triangle, we know midpoint of we know that Similarly, HLK LKH KHL and because Q is the midpoint of The JKH, HKL and HLK, Segment Addition Postulate gives us XW + WY = XY MLH are supplementary, and HKL HLK, and XQ + QZ = XZ. Substitution gives XW + WY = we know JKH MLH by the Congruent XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Supplements Theorem. By AAS, If we divide each side by 2, we have WY = QZ. The By CPCTC, JHK MHL. Isosceles Triangle Theorem says XYZ XZY. ANSWER: by the Reflexive Property. By SAS, Proof: We are given that is an isosceles So, by CPCTC. triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and ANSWER: MLH are supplementary. From the Isosceles Proof: We are given W is the midpoint of Triangle Theorem, we know that HJK HML. and Q is the midpoint of Because W is the Because is an equilateral triangle, we know midpoint of we know that Similarly, HLK LKH KHL and because Q is the midpoint of The JKH, HKL and HLK, Segment Addition Postulate gives us XW + WY = XY MLH are supplementary, and HKL HLK, and XQ + QZ = XZ. Substitution gives XW + WY = we know JKH ∠MLH by the Congruent XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Supplements Theorem. By AAS, If we divide each side by 2, we have WY = QZ. The By CPCTC, JHK MHL. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS,

24. Given: So, by CPCTC. W is the midpoint of Q is the midpoint of 25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either Prove: side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that but

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the

midpoint of we know that Similarly, because Q is the midpoint of The a. Elisa estimates m BAC to be 50. Based on this Segment Addition Postulate gives us XW + WY = XY estimate, what is m∠ABC? Explain.

and XQ + QZ = XZ. Substitution gives XW + WY = b. If show that is isosceles. XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The c. If and show that is Isosceles Triangle Theorem says XYZ XZY. equilateral. d. If is isosceles, what is the minimum by the Reflexive Property. By SAS, information needed to prove that

So, by CPCTC. Explain your reasoning.

ANSWER: Proof: We are given W is the midpoint of SOLUTION:

and Q is the midpoint of Because W is the a. Given: By Isosceles Triangle Theorem, . midpoint of we know that Similarly, Apply Triangle Sum Theorem. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The b. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC.

SOLUTION: a. Given:

By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem. d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base b. angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and d. One pair of congruent corresponding sides and one pair of congruent corresponding angles, the one pair of congruent corresponding angles; since triangles can be proved congruent using either ASA you know that the triangle is isosceles, if one leg is or SAS. congruent to a leg of then you know that ANSWER: both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you a. 65°; Because is isosceles, ABC know that , you know that ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. ∠ Therefore, b. with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC. c.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and

SOLUTION: mathematics. SOLUTION:

ANSWER:

ANSWER: 27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a Sample answer: I constructed a pair of perpendicular protractor to confirm that A, D, and G are all segments and then used the same compass setting to right angles. mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = 28. PROOF Based on your construction in Exercise 27, AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = make and prove a conjecture about the relationship 2.3 cm, the triangles are isosceles. I used a between the base angles of an isosceles right protractor to confirm that A, D, and G are all triangle. right angles. SOLUTION: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, Solve for x

ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD

Sample answer: I constructed a pair of perpendicular SOLUTION: segments and then used the same compass setting to From the figure, mark points equidistant from their intersection. I Therefore, is an isosceles triangle. measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a By the Triangle Sum Theorem, protractor to confirm that A, D, and G are all right angles.

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship ANSWER: between the base angles of an isosceles right 44 triangle.

SOLUTION: 30. m ACD Conjecture: The measures of the base angles of an isosceles right triangle are 45. SOLUTION: Proof: The base angles are congruent because it is We know an isosceles triangle. Let the measure of each acute that angle be x. The acute angles of a right triangle are Therefore, . complementary, Solve for x ANSWER: 44

ANSWER: 31. m ACB Conjecture: The measures of the base angles of an SOLUTION: isosceles right triangle are 45. The angles in a straight line add to 180°. Proof: The base angles are congruent because it is Therefore, an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are We know that complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

ANSWER: 136

32. m ABC SOLUTION: 29. m∠CAD From the figure, So, is an isosceles SOLUTION: triangle. Therefore, From the figure, Therefore, is an isosceles triangle. By the Triangle Angle Sum Theorem,

By the Triangle Sum Theorem, ANSWER: 22

ANSWER: 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids 44 shown. If each triangle is isosceles with vertex angles G, H, and J, and G 30. m ACD H, and H J, show that the distance SOLUTION: from B to F is three times the distance from D to F. We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

ANSWER: 136

32. m ABC ANSWER: SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex 34. Given: is isosceles; Prove: X and YZV are complementary. angles G, H, and J, and G

H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: ANSWER:

37. Theorem 4.11 SOLUTION: ANSWER:

ANSWER:

36. Corollary 4.4 SOLUTION:

Find the value of each variable.

38. SOLUTION: By the converse of Isosceles Triangle theorem,

ANSWER:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 37. Theorem 4.11 SOLUTION:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

ANSWER:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 Find the value of each variable. GAMES Use the diagram of a game timer shown to find each measure.

38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x. 40. m LPM SOLUTION: Note that x can equal –8 here because Angles at a point in a straight line add up to 180 .

. ANSWER: 3

Substitute x = 45 in (3x – 55) to find

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is ANSWER: equal to (2y – 5) . 80

The interior angles of a triangle add up to 180 . 41. m LMP

SOLUTION: Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 Substitute x = 45 in (3x – 55) to find GAMES Use the diagram of a game timer shown to find each measure.

Therefore,

ANSWER: 80

42. m JLK SOLUTION: 40. m LPM Vertical angles are congruent. Therefore,

SOLUTION:

Angles at a point in a straight line add up to 180 . First we need to find

We know that,

Substitute x = 45 in (3x – 55) to find So, ANSWER: 20

ANSWER: 43. m JKL 80 SOLUTION: Vertical angles are congruent. Therefore, 41. m LMP SOLUTION: First we need to find Since the triangle LMP is isosceles, Angles at a point in a straight line We know that,

add up to 180 .

So,

In Substitute x = 45 in (3x – 55) to find Since the triangle JKL is isosceles,

Therefore, ANSWER: ANSWER: 80 80 44. MULTIPLE REPRESENTATIONS In this 42. m JLK problem, you will explore possible measures of the SOLUTION: interior angles of an isosceles triangle given the

Vertical angles are congruent. Therefore, measure of one exterior angle.

First we need to find

We know that,

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of

So, the sides adjacent to the vertex angle and to one of ANSWER: the base angles, and labeling as shown.

20 b. Tabular Use a protractor to measure and record 43. m JKL m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and SOLUTION: record m 2 and use it to calculate these same Vertical angles are congruent. Therefore, measures. Organize your results in two tables.

First we need to find c. Verbal Explain how you used m 1 to find the

measures of 3, 4, and 5. Then explain how We know that, you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m So, 2 = x, write an expression for these same angle

measures. In

Since the triangle JKL is isosceles, SOLUTION: a.

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same b. measures. Organize your results in two tables.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: c. 5 is supplementary to 1, so m 5 = 180 – a. m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

b. c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: c. 5 is supplementary to 1, so m 5 = 180 – a. m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER:

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – PRECISION Determine whether the following 180; m 3 = 180 – x, m 4 = m 5 = statements are sometimes, always, or never true. Explain. 45. CHALLENGE In the figure, if is equilateral 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base and ZWP WJM JZL, prove that angle is an integer.

SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is even. SOLUTION: 47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even.

ANSWER: ANSWER: Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will be even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles

triangle is an integer, then the measure of each base angle is an integer. SOLUTION: Neither of them is correct. This is an isosceles SOLUTION: triangle with a vertex angle of 70. Since this is an The statement is sometimes true. It is true only if the isosceles triangle, the base angles are congruent. measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65;

vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex ANSWER: angle is odd. Neither; m G = or 55. SOLUTION: The statement is never true. The measure of the 49. OPEN-ENDED If possible, draw an isosceles vertex angle will be 180 – 2(measure of the base triangle with base angles that are obtuse. If it is not angle) so if the base angles are integers, then 2 possible, explain why not. (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even. SOLUTION: It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles ANSWER: are integers, then 2(measure of the base angle) will It is not possible because a triangle cannot have more be even and 180 – 2(measure of the base angle) will than one obtuse angle. be even. 50. REASONING In isosceles m B = 90. 48. ERROR ANALYSIS Alexis and Miguela are Draw the triangle. Indicate the congruent sides and finding m G in the figure shown. Alexis says that label each angle with its measure. m G = 35, while Miguela says that m G = 60. Is SOLUTION: either of them correct? Explain your reasoning. The sum of the angle measures in a triangle must be 180,

Since is isosceles, And given that m∠B = 90. Therefore,

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an Construct the triangle with the angles measures 90, isosceles triangle, the base angles are congruent. 45, 45.

ANSWER: ANSWER:

Neither; m G = or 55.

49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? SOLUTION: It is not possible because a triangle cannot have more SOLUTION: than one obtuse angle. If a triangle is already classified, you can use the ANSWER: previously proven properties of that type of triangle in the proof. For example, if you know that a It is not possible because a triangle cannot have more than one obtuse angle. triangle is an equilateral triangle, you can use Corollary 4.3 and 4.4 in the proof. Doing this can 50. REASONING In isosceles m B = 90. save you steps when writing the proof. Draw the triangle. Indicate the congruent sides and label each angle with its measure. ANSWER: Sample answer: If a triangle is already classified, SOLUTION: you can use the previously proven properties of that The sum of the angle measures in a triangle must be type of triangle in the proof. Doing this can save you 180, steps when writing the proof.

Since is isosceles, 52. MULTI-STEP △AED and △BEC are congruent And given that m∠B = 90. and isosceles. Therefore,

Construct the triangle with the angles measures 90, 45, 45. a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D ANSWER: b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? SOLUTION: a. Of the statements given, only choice B is not SOLUTION: necessarily true from the given information. If a triangle is already classified, you can use the b. The angles are congruent, so ∠DCE is 40°. previously proven properties of that type of triangle c. The sum of the measures of the angles of a in the proof. For example, if you know that a triangle is 180, so ∠BEC must be 80°. triangle is an equilateral triangle, you can use d. 60; All three angles in △AEB are 60° because Corollary 4.3 and 4.4 in the proof. Doing this can base angles of an isosceles triangle are congruent. So save you steps when writing the proof. △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. ANSWER: Sample answer: If a triangle is already classified, ANSWER: you can use the previously proven properties of that a. B type of triangle in the proof. Doing this can save you b. 40 steps when writing the proof. c. 80 d. 60; Sample answer: All three angles in △AEB 52. MULTI-STEP △AED and △BEC are congruent are 60° because base angles of an isosceles triangle and isosceles. are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC A 34 B B 56 C ∠CAD ≅ ∠DBC C 60 D D 62 E 124

b. If m∠EAB = 40, what is the measure of ∠DCE? SOLUTION: c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION:

a. Of the statements given, only choice B is not Consider triangle DEH. Since one angle is 56°, and necessarily true from the given information. the triangle is isosceles, then the remaining two b. The angles are congruent, so ∠DCE is 40°. angles would be 180 – 56 or 124°. c. The sum of the measures of the angles of a So, m∠FEG is , or 62. The correct answer is triangle is 180, so ∠BEC must be 80°. choice D. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So ANSWER: △AEB is equiangular and equilateral. △AEB ≅ D △DEC by SAS, so AB ≅ EC. 54. A jeweler bends wire to make charms in the shape ANSWER: of isosceles triangles. The base of the triangle a. B measures 2 centimeters and another side measures b. 40 3.5 centimeters. How many charms can be made c. 80 from a piece of wire that is 63 centimeters long? d. 60; Sample answer: All three angles in △AEB SOLUTION: are 60° because base angles of an isosceles triangle Because we know that the shape of each charm is are congruent. So △AEB is equiangular and an isosceles triangle, then we can determine the side △ △ equilateral. AEB ≅ DEC by SAS, so AB ≅ EC. lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. 53. What is m∠FEG? To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms.

ANSWER: 7

55. In the figure below, NDG LGD. A 34 B 56 C 60 D 62 E 124

SOLUTION:

a. What is m∠L? b. What is the value of x? c. What is m∠DGN?

SOLUTION: a. In this figure, angles N and L are congruent so Consider triangle DEH. Since one angle is 56°, and m∠L = 36°. the triangle is isosceles, then the remaining two b. Because we know two angle measures, and that angles would be 180 – 56 or 124°. the sum of the measures of the angles of a triangle is So, m∠FEG is , or 62. The correct answer is 180, then we can solve 36 + 90 + x = 180, or x = 54. choice D. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: D a. 36 b. 54 54. A jeweler bends wire to make charms in the shape c. 54 of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures 56. An equilateral triangle has a perimeter of 216 inches. 3.5 centimeters. How many charms can be made What is the length of one side of the triangle? from a piece of wire that is 63 centimeters long? SOLUTION: SOLUTION: Because the triangle is equilateral that means each Because we know that the shape of each charm is side has the same length, so divide the perimeter by 3 an isosceles triangle, then we can determine the side to find the length of one side. lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. 216 ÷ 3 = 72

To find how many charms can be made, divide 63 ANSWER: cm by 9 cm. So, the jeweler can make 7 charms. 72

ANSWER: 57. △JKL is an isosceles triangle, and . JK = 7 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? 55. In the figure below, NDG LGD. SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or

16.

a. What is m L? ∠ So, the perimeter of △JKL is 16. b. What is the value of x? c. What is m∠DGN? ANSWER: SOLUTION: 16 a. In this figure, angles N and L are congruent so 58. △ABC is an isosceles triangle with base angles m∠L = 36°. measuring (5x + 5)° and (7x – 19)°. Find the value of b. Because we know two angle measures, and that x. the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. SOLUTION: c. ∠DGN is congruent to x°, so m∠DGN is 54°. In an isosceles triangle, the base angles are congruent. ANSWER: a. 36 So, 5x + 5 = 7x – 19. b. 54 c. 54 Simplify.

56. An equilateral triangle has a perimeter of 216 inches. –2x = –24 What is the length of one side of the triangle? x = 12 SOLUTION: ANSWER: Because the triangle is equilateral that means each 12 side has the same length, so divide the perimeter by 3 to find the length of one side. 59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an 216 ÷ 3 = 72 equilateral triangle?

ANSWER: 72

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION:

The perimeter can be found by adding all three side lengths. In this situation, since the triangle is A isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. B m∠VRT = 60

Because the x-terms cancel each other out: 2x + 2x C QRV VTU – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or D VRT RST 16. SOLUTION:

So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x.

SOLUTION: In an isosceles triangle, the base angles are Consider each choice. congruent. A does not provide enough information to So, 5x + 5 = 7x – 19. determine how sides VT and TR are related.

Simplify. B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two –2x = –24 measures are 60 degrees. x = 12 C ANSWER: 12

59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an equilateral triangle?

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

A D Since we do not have any measures to indicate B m∠VRT = 60 that either or is equilateral, knowing C QRV VTU they are congruent would not help prove is D VRT RST equilateral.

SOLUTION: The correct answer is choice C. ANSWER: C

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: Refer to the figure. By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. 1. If name two congruent angles. ANSWER: SOLUTION: 12 Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite 4. m MRP those sides are congruent.

Therefore, in triangle ABC,

ANSWER: BAC and BCA

2. If EAC ECA, name two congruent segments. SOLUTION: SOLUTION: Converse of Isosceles Triangle Theorem states that Since all the sides are congruent, is an if two angles of a triangle are congruent, then the equilateral triangle. Each angle of an equilateral sides opposite those angles are congruent. Therefore, triangle measures 60°. Therefore, m MRP = 60°. in triangle EAC, ANSWER: ANSWER: 60

SENSE-MAKING Find the value of each Find each measure. variable. 3. FH

5. SOLUTION:

In the figure, . Therefore, triangle RST is SOLUTION: an isosceles triangle. By the Triangle Angle-Sum Theorem, By the Converse of Isosceles Triangle Theorem,

That is, .

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: ANSWER: 12 12

4. m MRP

6. SOLUTION: SOLUTION: In the figure, Therefore, triangle WXY is Since all the sides are congruent, is an an isosceles triangle. equilateral triangle. Each angle of an equilateral By the Isosceles Triangle Theorem, . triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: 60 SENSE-MAKING Find the value of each ANSWER: variable. 16

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. 5. Prove:

SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION:

ANSWER: 12

ANSWER:

ANSWER: 16

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove:

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your

reasoning. SOLUTION: SOLUTION: a.

ANSWER:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A b. The Pythagorean Theorem tells us that: portion of the track is shown.

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

b. The Pythagorean Theorem tells us that: 9. If name two congruent angles. SOLUTION: By CPCTC we know that VT = 1.5 m. The By the Isosceles Triangle Theorem, in triangle ABE, Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ANSWER: ABE and AEB ANSWER: a. 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: b. The Pythagorean Theorem tells us that ACD and ADC By CPCTC we know 12. If DAE DEA, name two congruent that VT = 1.5 m. The Seg. Add. Post. says QV + segments. VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. SOLUTION: By the Converse of Isosceles Triangle Theorem, in Refer to the figure. triangle ADE,

ANSWER:

13. If BCF BFC, name two congruent segments. SOLUTION: 9. If name two congruent angles. By the Converse of Isosceles Triangle Theorem, in

SOLUTION: triangle BCF, By the Isosceles Triangle Theorem, in triangle ABE, ANSWER:

ANSWER: ABE and AEB 14. If name two congruent angles. SOLUTION: 10. If ∠ABF ∠AFB, name two congruent segments. By the Isosceles Triangle Theorem, in triangle AFH, SOLUTION: By the Converse of Isosceles Triangle Theorem, in ANSWER:

triangle ABF, AFH and AHF

ANSWER: Find each measure. 15. m BAC

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

SOLUTION: ANSWER: Here ACD and ADC By Isosceles Triangle Theorem, .

12. If DAE DEA, name two congruent Apply the Triangle Sum Theorem.

segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, ANSWER: ANSWER: 60

16. m SRT 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: SOLUTION:

Given: By Isosceles Triangle Theorem, . 14. If name two congruent angles. SOLUTION: Apply Triangle Angle-Sum Theorem. By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: AFH and AHF ANSWER: 65 Find each measure. 15. m BAC 17. TR

SOLUTION: SOLUTION: Here Since the triangle is isosceles, By Isosceles Triangle Theorem, . Therefore,

Apply the Triangle Sum Theorem.

All the angles are congruent. Therefore, it is an equiangular triangle. ANSWER: Since the equiangular triangle is an equilateral, 60

ANSWER: 16. m SRT 4 18. CB

SOLUTION: Given: By Isosceles Triangle Theorem, . SOLUTION:

Apply Triangle Angle-Sum Theorem. By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3 ANSWER: 65 REGULARITY Find the value of each variable. 17. TR

19. SOLUTION: Since all the angles are congruent, the sides are also SOLUTION: congruent to each other. Since the triangle is isosceles, Therefore, Therefore, Solve for x.

ANSWER: All the angles are congruent. Therefore, it is an x = 5 equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: 4 20. 18. CB SOLUTION: Given: By the Isosceles Triangle Theorem,

And SOLUTION: We know

By the Converse of Isosceles Triangle Theorem, in that, triangle ABC, That is, CB = 3.

4-6 IsoscelesANSWER: and Equilateral Triangles ANSWER: 3 x = 13 REGULARITY Find the value of each variable.

19. 21. SOLUTION: SOLUTION: Since all the angles are congruent, the sides are also Given: congruent to each other. By the Isosceles Triangle Theorem, Therefore, Solve for x. That is,

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation ANSWER: 2: x = 5

Add the equations 1 and 2.

20. SOLUTION: Given:

By the Isosceles Triangle Theorem, Substitute the value of x in one of the two equations to find the value of y.

And We know that,

ANSWER: x = 13 ANSWER: x = 11, y = 11

21. SOLUTION: 22. Given: SOLUTION: By the Isosceles Triangle Theorem, Given: By the Triangle Angle-Sum That is, eSolutions Manual - Powered by Cognero Theorem, Page 5

Then, let Equation 1 be: . So, the triangle is an equilateral triangle. Therefore, By the Triangle Sum Theorem, we can find Equation 2: Set up two equations to solve for x and y.

Add the equations 1 and 2.

ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof.

23. Given: is isosceles, and is Substitute the value of x in one of the two equations equilateral. JKH and HKL are supplementary to find the value of y. and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: ANSWER: Proof: We are given that is an isosceles x = 11, y = 11 triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and 22. JKH, HKL and HLK, SOLUTION: MLH are supplementary, and HKL HLK, Given: we know JKH MLH by the Congruent By the Triangle Angle-Sum Supplements Theorem. By AAS, Theorem, By CPCTC, JHK MHL.

ANSWER: So, the triangle is an equilateral triangle. Therefore, Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH Set up two equations to solve for x and y. and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK,

x = 1.4, y = –1 we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, PROOF Write a paragraph proof. By CPCTC, JHK MHL. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. 24. Given: Prove: JHK MHL W is the midpoint of Q is the midpoint of Prove:

SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH

and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles SOLUTION: Triangle Theorem, we know that HJK HML. Proof: We are given W is the midpoint of Because is an equilateral triangle, we know and Q is the midpoint of Because W is the HLK LKH KHL and midpoint of we know that Similarly, JKH, HKL and HLK, because Q is the midpoint of The MLH are supplementary, and HKL HLK, Segment Addition Postulate gives us XW + WY = XY we know JKH MLH by the Congruent

Supplements Theorem. By AAS, and XQ + QZ = XZ. Substitution gives XW + WY =

By CPCTC, JHK MHL. XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The ANSWER: Isosceles Triangle Theorem says XYZ XZY. Proof: We are given that is an isosceles by the Reflexive Property. By SAS, triangle and is an equilateral triangle, JKH So, by CPCTC. and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles ANSWER: Triangle Theorem, we know that HJK HML. Proof: We are given W is the midpoint of Because is an equilateral triangle, we know and Q is the midpoint of Because W is the HLK LKH KHL and midpoint of we know that Similarly, JKH, HKL and HLK, because Q is the midpoint of The MLH are supplementary, and HKL HLK, Segment Addition Postulate gives us XW + WY = XY we know JKH ∠MLH by the Congruent and XQ + QZ = XZ. Substitution gives XW + WY = Supplements Theorem. By AAS, XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ.

By CPCTC, JHK MHL. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. 24. Given: by the Reflexive Property. By SAS, W is the midpoint of So, by CPCTC. Q is the midpoint of BABYSITTING Prove: 25. While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that but

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly,

because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY a. Elisa estimates m BAC to be 50. Based on this and XQ + QZ = XZ. Substitution gives XW + WY = estimate, what is m∠ABC? Explain. XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. b. If show that is isosceles. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. c. If and show that is equilateral. by the Reflexive Property. By SAS, d. If is isosceles, what is the minimum So, by CPCTC. information needed to prove that

ANSWER: Explain your reasoning.

Proof: We are given W is the midpoint of SOLUTION: and Q is the midpoint of Because W is the a. Given: midpoint of we know that Similarly, By Isosceles Triangle Theorem, . because Q is the midpoint of The Apply Triangle Sum Theorem. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. b. by the Reflexive Property. By SAS, So, by CPCTC.

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem. d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that b. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. c. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the d. One pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles; since or SAS. you know that the triangle is isosceles, if one leg is ANSWER: congruent to a leg of then you know that a. 65°; Because is isosceles, ABC both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. know that , you know that b. Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of c. the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then SOLUTION: verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different ANSWER: isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = Sample answer: I constructed a pair of perpendicular 2.3 cm, the triangles are isosceles. I used a segments and then used the same compass setting to mark points equidistant from their intersection. I protractor to confirm that A, D, and G are all right angles. measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 28. PROOF Based on your construction in Exercise 27, 2.3 cm, the triangles are isosceles. I used a make and prove a conjecture about the relationship protractor to confirm that A, D, and G are all between the base angles of an isosceles right right angles. triangle. ANSWER: SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, Solve for x

REGULARITY Find each measure.

29. m∠CAD Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to SOLUTION: mark points equidistant from their intersection. I From the figure, measured both legs for each triangle. Because AB = Therefore, is an isosceles triangle. AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a

protractor to confirm that A, D, and G are all By the Triangle Sum Theorem, right angles.

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right ANSWER: triangle. 44 SOLUTION: Conjecture: The measures of the base angles of an 30. m ACD isosceles right triangle are 45. Proof: The base angles are congruent because it is SOLUTION: an isosceles triangle. Let the measure of each acute We know angle be x. The acute angles of a right triangle are that complementary, Solve for x Therefore, .

ANSWER: 44 ANSWER: Conjecture: The measures of the base angles of an 31. m ACB isosceles right triangle are 45. SOLUTION: Proof: The base angles are congruent because it is The angles in a straight line add to 180°. an isosceles triangle. Let the measure of each acute Therefore, angle be x. The acute angles of a right triangle are

complementary, so x + x = 90 and x = 45. We know that REGULARITY Find each measure.

ANSWER: 136

32. m ABC 29. m∠CAD SOLUTION: SOLUTION: From the figure, So, is an isosceles From the figure, triangle. Therefore, Therefore, is an isosceles triangle. By the Triangle Angle Sum Theorem,

By the Triangle Sum Theorem,

ANSWER: 22 ANSWER: 44 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex 30. m ACD angles G, H, and J, and G SOLUTION: H, and H J, show that the distance We know from B to F is three times the distance from D to F. that Therefore, .

ANSWER: 44

31. m ACB

SOLUTION: SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

ANSWER: 136

32. m ABC SOLUTION: ANSWER: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex 34. Given: is isosceles; angles G, H, and J, and G Prove: X and YZV are complementary. H, and H J, show that the distance

from B to F is three times the distance from D to F.

SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: ANSWER:

37. Theorem 4.11 ANSWER: SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

Find the value of each variable.

38. SOLUTION: By the converse of Isosceles Triangle theorem, ANSWER:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 37. Theorem 4.11 3 SOLUTION:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

ANSWER: The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: Find the value of each variable. 14 GAMES Use the diagram of a game timer shown to find each measure.

38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

40. m LPM Note that x can equal –8 here because SOLUTION:

. Angles at a point in a straight line add up to 180 . ANSWER: 3

Substitute x = 45 in (3x – 55) to find

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . ANSWER: 80 The interior angles of a triangle add up to 180 . 41. m LMP SOLUTION: Since the triangle LMP is isosceles, Angles at a point in a straight line The measure of an angle cannot be negative, and 2(– add up to 180 . 18) – 5 = –41, so y = 14.

ANSWER: 14

GAMES Use the diagram of a game timer Substitute x = 45 in (3x – 55) to find shown to find each measure.

Therefore,

ANSWER: 80

42. m JLK

40. m LPM SOLUTION: SOLUTION: Vertical angles are congruent. Therefore,

Angles at a point in a straight line add up to 180 .

First we need to find

We know that,

Substitute x = 45 in (3x – 55) to find

So,

ANSWER: 20 ANSWER: 80 43. m JKL SOLUTION:

41. m LMP Vertical angles are congruent. Therefore, SOLUTION: Since the triangle LMP is isosceles, First we need to find Angles at a point in a straight line add up to 180 . We know that,

So,

Substitute x = 45 in (3x – 55) to find In

Since the triangle JKL is isosceles,

Therefore,

ANSWER: ANSWER: 80 80 42. m JLK 44. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will explore possible measures of the Vertical angles are congruent. Therefore, interior angles of an isosceles triangle given the measure of one exterior angle.

First we need to find

We know that,

a. Geometric So, Use a ruler and a protractor to draw three different isosceles triangles, extending one of ANSWER: the sides adjacent to the vertex angle and to one of 20 the base angles, and labeling as shown.

43. m JKL b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the SOLUTION: measures of 3, 4, and 5. Then find and Vertical angles are congruent. Therefore, record m 2 and use it to calculate these same measures. Organize your results in two tables. First we need to find

c. Verbal Explain how you used m 1 to find the We know that, measures of 3, 4, and 5. Then explain how

you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for So, the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle In measures.

Since the triangle JKL is isosceles, SOLUTION: a.

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables. b. c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION:

a. c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – b. m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a. c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER:

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = PRECISION Determine whether the following statements are sometimes, always, or never true. 45. CHALLENGE In the figure, if is equilateral Explain. and ZWP WJM JZL, prove that 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer.

SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is SOLUTION: even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: SOLUTION: Neither of them is correct. This is an isosceles The statement is sometimes true. It is true only if the triangle with a vertex angle of 70. Since this is an measure of the vertex angle is even. For example, isosceles triangle, the base angles are congruent. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. ANSWER:

SOLUTION: Neither; m G = or 55. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 49. OPEN-ENDED If possible, draw an isosceles angle) so if the base angles are integers, then 2 triangle with base angles that are obtuse. If it is not (measure of the base angle) will be even and 180 – 2 possible, explain why not. (measure of the base angle) will be even. SOLUTION: ANSWER: It is not possible because a triangle cannot have more Never; the measure of the vertex angle will be 180 – than one obtuse angle. 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will ANSWER: be even and 180 – 2(measure of the base angle) will It is not possible because a triangle cannot have more be even. than one obtuse angle.

48. ERROR ANALYSIS Alexis and Miguela are 50. REASONING In isosceles m B = 90. finding m G in the figure shown. Alexis says that Draw the triangle. Indicate the congruent sides and m G = 35, while Miguela says that m G = 60. Is label each angle with its measure. either of them correct? Explain your reasoning. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, And given that m∠B = 90.

Therefore,

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. Construct the triangle with the angles measures 90,

45, 45.

ANSWER: ANSWER:

Neither; m G = or 55.

49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. SOLUTION: 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? It is not possible because a triangle cannot have more than one obtuse angle. SOLUTION: ANSWER: If a triangle is already classified, you can use the It is not possible because a triangle cannot have more previously proven properties of that type of triangle than one obtuse angle. in the proof. For example, if you know that a triangle is an equilateral triangle, you can use 50. REASONING In isosceles m B = 90. Corollary 4.3 and 4.4 in the proof. Doing this can Draw the triangle. Indicate the congruent sides and save you steps when writing the proof. label each angle with its measure. ANSWER: SOLUTION: Sample answer: If a triangle is already classified, The sum of the angle measures in a triangle must be you can use the previously proven properties of that

180, type of triangle in the proof. Doing this can save you

steps when writing the proof. Since is isosceles, And given that m∠B = 90. 52. MULTI-STEP △AED and △BEC are congruent Therefore, and isosceles.

Construct the triangle with the angles measures 90, 45, 45.

a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC ANSWER: D

b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that 51. WRITING IN MATH How can triangle ? classifications help you prove triangle congruence? SOLUTION: SOLUTION: a. Of the statements given, only choice B is not If a triangle is already classified, you can use the necessarily true from the given information. previously proven properties of that type of triangle b. The angles are congruent, so ∠DCE is 40°. in the proof. For example, if you know that a c. The sum of the measures of the angles of a triangle is an equilateral triangle, you can use triangle is 180, so ∠BEC must be 80°. Corollary 4.3 and 4.4 in the proof. Doing this can d. 60; All three angles in △AEB are 60° because save you steps when writing the proof. base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ ANSWER: △DEC by SAS, so AB ≅ EC. Sample answer: If a triangle is already classified, you can use the previously proven properties of that ANSWER: type of triangle in the proof. Doing this can save you a. B steps when writing the proof. b. 40 c. 80 52. MULTI-STEP △AED and △BEC are congruent d. 60; Sample answer: All three angles in △AEB and isosceles. are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B A 34 C ∠CAD ≅ ∠DBC B 56 D C 60 D 62 b. If m∠EAB = 40, what is the measure of ∠DCE? E 124

c. If m∠ADE = 50, what is the measure of ∠BEC? SOLUTION:

d. What measure would ∠AEB need to be so that ? SOLUTION: a. Of the statements given, only choice B is not

necessarily true from the given information. Consider triangle DEH. Since one angle is 56°, and b. The angles are congruent, so ∠DCE is 40°. the triangle is isosceles, then the remaining two c. The sum of the measures of the angles of a angles would be 180 – 56 or 124°. triangle is 180, so ∠BEC must be 80°. So, m∠FEG is , or 62. The correct answer is d. 60; All three angles in △AEB are 60° because choice D. base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ ANSWER: △DEC by SAS, so AB ≅ EC. D

ANSWER: 54. A jeweler bends wire to make charms in the shape a. B of isosceles triangles. The base of the triangle b. 40 measures 2 centimeters and another side measures c. 80 3.5 centimeters. How many charms can be made d. 60; Sample answer: All three angles in △AEB from a piece of wire that is 63 centimeters long? are 60° because base angles of an isosceles triangle SOLUTION: are congruent. So △AEB is equiangular and Because we know that the shape of each charm is equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, 53. What is m∠FEG? the perimeter of each triangle is 9 cm.

To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms.

ANSWER: 7

A 34 55. In the figure below, NDG LGD. B 56 C 60 D 62 E 124

SOLUTION:

a. What is m∠L? b. What is the value of x? c. What is m∠DGN?

SOLUTION: Consider triangle DEH. Since one angle is 56°, and a. In this figure, angles N and L are congruent so the triangle is isosceles, then the remaining two m L = 36°. angles would be 180 – 56 or 124°. ∠ b. Because we know two angle measures, and that So, m∠FEG is , or 62. The correct answer is the sum of the measures of the angles of a triangle is choice D. 180, then we can solve 36 + 90 + x = 180, or x = 54. c. DGN is congruent to x°, so m DGN is 54°. ANSWER: ∠ ∠ D ANSWER: a. 36 54. A jeweler bends wire to make charms in the shape b. of isosceles triangles. The base of the triangle 54 measures 2 centimeters and another side measures c. 54 3.5 centimeters. How many charms can be made 56. An equilateral triangle has a perimeter of 216 inches. from a piece of wire that is 63 centimeters long? What is the length of one side of the triangle? SOLUTION: SOLUTION: Because we know that the shape of each charm is Because the triangle is equilateral that means each an isosceles triangle, then we can determine the side side has the same length, so divide the perimeter by 3 lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, to find the length of one side. the perimeter of each triangle is 9 cm.

216 ÷ 3 = 72 To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. ANSWER: ANSWER: 72 7 57. △JKL is an isosceles triangle, and . JK = 55. In the figure below, NDG LGD. 2x + 3, and KL = 10 – 4x. What is the perimeter of △ JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x

– 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or a. What is m∠L? 16. b. What is the value of x? So, the perimeter of △JKL is 16. c. What is m∠DGN? SOLUTION: ANSWER: a. In this figure, angles N and L are congruent so 16 m∠L = 36°. 58. △ABC is an isosceles triangle with base angles b. Because we know two angle measures, and that measuring (5x + 5)° and (7x – 19)°. Find the value of the sum of the measures of the angles of a triangle is x. 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. SOLUTION: In an isosceles triangle, the base angles are ANSWER: congruent. a. 36 b. 54 So, 5x + 5 = 7x – 19. c. 54 Simplify. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? –2x = –24 x = 12 SOLUTION: Because the triangle is equilateral that means each ANSWER: side has the same length, so divide the perimeter by 3 12 to find the length of one side. 59. QRV is an equilateral triangle. What additional 216 ÷ 3 = 72 information would be enough to prove VRT is an equilateral triangle? ANSWER:

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side

lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. A

Because the x-terms cancel each other out: 2x + 2x B m∠VRT = 60 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or C QRV VTU 16. D VRT RST

So, the perimeter of △JKL is 16. SOLUTION:

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

In an isosceles triangle, the base angles are congruent. Consider each choice.

So, 5x + 5 = 7x – 19. A does not provide enough information to determine how sides VT and TR are related. Simplify.

B If we know that m∠VRT = 60, we still would not –2x = –24 have enough information to prove that the other two x = 12 measures are 60 degrees. ANSWER: C 12

59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an equilateral triangle?

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3. A B m∠VRT = 60 D Since we do not have any measures to indicate C QRV VTU that either or is equilateral, knowing D VRT RST they are congruent would not help prove is equilateral. SOLUTION: The correct answer is choice C.

ANSWER: C

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

Find each measure. 3. FH 1. If name two congruent angles. SOLUTION: Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite those sides are congruent. Therefore, in triangle ABC,

ANSWER: SOLUTION: By the Triangle Angle-Sum Theorem, BAC and BCA

2. If EAC ECA, name two congruent segments.

SOLUTION: Since the measures of all the three angles are 60°; Converse of Isosceles Triangle Theorem states that the triangle must be equiangular. All the equiangular if two angles of a triangle are congruent, then the triangles are equilateral. Therefore, FH = GH = 12. sides opposite those angles are congruent. Therefore, ANSWER: in triangle EAC, 12 ANSWER: 4. m MRP

Find each measure. 3. FH

SOLUTION: Since all the sides are congruent, is an SOLUTION: equilateral triangle. Each angle of an equilateral By the Triangle Angle-Sum Theorem, triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: 60

Since the measures of all the three angles are 60°; SENSE-MAKING Find the value of each the triangle must be equiangular. All the equiangular variable. triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

5. 4. m MRP SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: ANSWER: 12 60 SENSE-MAKING Find the value of each variable.

5. SOLUTION: In the figure, Therefore, triangle WXY is SOLUTION: an isosceles triangle. In the figure, . Therefore, triangle RST is By the Isosceles Triangle Theorem, . an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER: 16

7. PROOF Write a two-column proof. ANSWER: 12 Given: is isosceles; bisects ABC. Prove:

6. SOLUTION: In the figure, Therefore, triangle WXY is SOLUTION: an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: 16 ANSWER:

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

SOLUTION:

a. If and are perpendicular to is ANSWER: isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a. 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the b. The Pythagorean Theorem tells us that: distance between and Explain your reasoning. By CPCTC we know that VT = 1.5 m. The SOLUTION: Segment Addition Postulate says QV + VT = QT. a. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: a.

b. The Pythagorean Theorem tells us that:

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The By CPCTC we know Segment Addition Postulate says QV + VT = QT. that VT = 1.5 m. The Seg. Add. Post. says QV + By substitution, we have 1.5 + 1.5 = QT. So QT = VT = QT. By substitution, we have 1.5 + 1.5 = QT. 3 m. So QT = 3 m.

ANSWER: Refer to the figure. a.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

b. The Pythagorean Theorem tells us that 10. If ∠ABF ∠AFB, name two congruent segments. By CPCTC we know SOLUTION: that VT = 1.5 m. The Seg. Add. Post. says QV + By the Converse of Isosceles Triangle Theorem, in VT = QT. By substitution, we have 1.5 + 1.5 = QT. triangle ABF, So QT = 3 m. ANSWER: Refer to the figure.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

9. If name two congruent angles. ANSWER: SOLUTION: ACD and ADC By the Isosceles Triangle Theorem, in triangle ABE, 12. If DAE DEA, name two congruent

segments. ANSWER: SOLUTION: ABE and AEB By the Converse of Isosceles Triangle Theorem, in triangle ADE, 10. If ∠ABF ∠AFB, name two congruent segments. ANSWER: SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF, 13. If BCF BFC, name two congruent segments. ANSWER: SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, 11. If name two congruent angles. ANSWER: SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, 14. If name two congruent angles. ANSWER: SOLUTION: ACD and ADC By the Isosceles Triangle Theorem, in triangle AFH,

12. If DAE DEA, name two congruent segments. ANSWER: SOLUTION: AFH and AHF By the Converse of Isosceles Triangle Theorem, in Find each measure. triangle ADE, 15. m BAC ANSWER:

13. If BCF BFC, name two congruent segments. SOLUTION: SOLUTION: By the Converse of Isosceles Triangle Theorem, in Here

triangle BCF, By Isosceles Triangle Theorem, .

ANSWER: Apply the Triangle Sum Theorem.

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, ANSWER: 60

ANSWER: 16. m SRT AFH and AHF

Find each measure. 15. m BAC

SOLUTION: Given: By Isosceles Triangle Theorem, . SOLUTION: Apply Triangle Angle-Sum Theorem. Here

By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem.

ANSWER: 65 17. TR ANSWER: 60

16. m SRT

SOLUTION: Since the triangle is isosceles, Therefore,

SOLUTION: Given: By Isosceles Triangle Theorem, . All the angles are congruent. Therefore, it is an Apply Triangle Angle-Sum Theorem. equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: ANSWER: 4 65 18. CB 17. TR

SOLUTION: SOLUTION: By the Converse of Isosceles Triangle Theorem, in Since the triangle is isosceles, triangle ABC, Therefore, That is, CB = 3.

ANSWER: 3 REGULARITY Find the value of each variable. All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral,

19. ANSWER: 4 SOLUTION: Since all the angles are congruent, the sides are also 18. CB congruent to each other. Therefore, Solve for x.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in x = 5 triangle ABC, That is, CB = 3.

ANSWER: 3 20. REGULARITY Find the value of each variable. SOLUTION: Given: By the Isosceles Triangle Theorem,

19. SOLUTION: And Since all the angles are congruent, the sides are also We know congruent to each other. that, Therefore, Solve for x.

ANSWER: x = 13

ANSWER: x = 5

21. SOLUTION: 20. Given: SOLUTION: By the Isosceles Triangle Theorem,

Given: By the Isosceles Triangle Theorem, That is,

Then, let Equation 1 be: . And We know By the Triangle Sum Theorem, we can find Equation 2: that,

Add the equations 1 and 2. ANSWER: x = 13

Substitute the value of x in one of the two equations 21. to find the value of y. SOLUTION: Given: By the Isosceles Triangle Theorem,

That is,

Then, let Equation 1 be: . ANSWER: By the Triangle Sum Theorem, we can find Equation x = 11, y = 11 2:

Add the equations 1 and 2.

22. SOLUTION: Given: By the Triangle Angle-Sum Theorem, Substitute the value of x in one of the two equations to find the value of y. So, the triangle is an equilateral triangle. Therefore,

Set up two equations to solve for x and y.

4-6 IsoscelesANSWER: and Equilateral Triangles ANSWER: x = 11, y = 11 x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

22. SOLUTION: Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. SOLUTION: Therefore, Proof: We are given that is an isosceles

triangle and is an equilateral triangle, JKH Set up two equations to solve for x and y. and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, x = 1.4, y = –1 MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent PROOF Write a paragraph proof. Supplements Theorem. By AAS, 23. Given: is isosceles, and is By CPCTC, JHK MHL. equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. ANSWER: Prove: JHK MHL Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and SOLUTION: JKH, HKL and HLK, Proof: We are given that is an isosceles MLH are supplementary, and HKL HLK, triangle and is an equilateral triangle, JKH we know JKH ∠MLH by the Congruent and HKL are supplementary and HLK and Supplements Theorem. By AAS, MLH are supplementary. From the Isosceles By CPCTC, JHK MHL. Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know 24. Given: HLK LKH KHL and W is the midpoint of JKH, HKL and HLK, Q is the midpoint of MLH are supplementary, and HKL HLK, Prove: we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL. eSolutionsANSWER: Manual - Powered by Cognero Page 6 Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles SOLUTION: Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know Proof: We are given W is the midpoint of HLK LKH KHL and and Q is the midpoint of Because W is the JKH, HKL and HLK, midpoint of we know that Similarly, MLH are supplementary, and HKL HLK, because Q is the midpoint of The

we know JKH ∠MLH by the Congruent Segment Addition Postulate gives us XW + WY = XY Supplements Theorem. By AAS, and XQ + QZ = XZ. Substitution gives XW + WY = By CPCTC, JHK MHL. XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The 24. Given: Isosceles Triangle Theorem says XYZ XZY. W is the midpoint of by the Reflexive Property. By SAS, Q is the midpoint of So, by CPCTC. Prove: ANSWER:

Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. SOLUTION: If we divide each side by 2, we have WY = QZ. The Proof: We are given W is the midpoint of Isosceles Triangle Theorem says XYZ XZY. and Q is the midpoint of Because W is the by the Reflexive Property. By SAS, midpoint of we know that Similarly, So, by CPCTC. because Q is the midpoint of The BABYSITTING Segment Addition Postulate gives us XW + WY = XY 25. While babysitting her neighbor’s children, Elisa observes that the supports on either and XQ + QZ = XZ. Substitution gives XW + WY = side of a park swing set form two sets of triangles. XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Using a jump rope to measure, Elisa is able to If we divide each side by 2, we have WY = QZ. The determine that but Isosceles Triangle Theorem says XYZ XZY.

by the Reflexive Property. By SAS, So, by CPCTC.

ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = a. Elisa estimates m BAC to be 50. Based on this XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. estimate, what is m∠ABC? Explain. If we divide each side by 2, we have WY = QZ. The b. If show that is isosceles. Isosceles Triangle Theorem says XYZ XZY. c. If and show that is by the Reflexive Property. By SAS, equilateral. So, by CPCTC. d. If is isosceles, what is the minimum information needed to prove that 25. BABYSITTING While babysitting her neighbor’s Explain your reasoning. children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to SOLUTION:

determine that but a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

c. d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is c. congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and c. one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, SOLUTION: with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

ANSWER:

SOLUTION:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION: ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

ANSWER: Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to Conjecture: The measures of the base angles of an mark points equidistant from their intersection. I isosceles right triangle are 45. measured both legs for each triangle. Because AB = Proof: The base angles are congruent because it is AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = an isosceles triangle. Let the measure of each acute 2.3 cm, the triangles are isosceles. I used a angle be x. The acute angles of a right triangle are protractor to confirm that A, D, and G are all complementary, so x + x = 90 and x = 45. right angles. REGULARITY Find each measure.

By the Triangle Sum Theorem, ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is ANSWER: an isosceles triangle. Let the measure of each acute 44 angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. 30. m ACD REGULARITY Find each measure. SOLUTION: We know that Therefore, .

ANSWER: 44 29. m∠CAD

SOLUTION: 31. m ACB From the figure, SOLUTION: Therefore, is an isosceles triangle. The angles in a straight line add to 180°. Therefore,

By the Triangle Sum Theorem, We know that

ANSWER: ANSWER: 44 136

32. m ABC 30. m ACD SOLUTION: SOLUTION: We know From the figure, So, is an isosceles that triangle. Therefore, Therefore, . By the Triangle Angle Sum Theorem, ANSWER: 44

31. m ACB ANSWER: SOLUTION: 22 The angles in a straight line add to 180°. Therefore, 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex We know that angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, SOLUTION:

By the Triangle Angle Sum Theorem,

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance ANSWER: from B to F is three times the distance from D to F.

SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION: ANSWER:

ANSWER: 34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

36. Corollary 4.4 37. Theorem 4.11 SOLUTION: SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

37. Theorem 4.11 SOLUTION: 38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

ANSWER: Note that x can equal –8 here because .

ANSWER: 3

Find the value of each variable. 39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . 38. SOLUTION: The interior angles of a triangle add up to 180 .

By the converse of Isosceles Triangle theorem,

Solve the equation for x. The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Note that x can equal –8 here because ANSWER: . 14 ANSWER: GAMES Use the diagram of a game timer 3 shown to find each measure.

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

40. m LPM The interior angles of a triangle add up to 180 . SOLUTION: Angles at a point in a straight line add up to 180 .

The measure of an angle cannot be negative, and 2(–

18) – 5 = –41, so y = 14. Substitute x = 45 in (3x – 55) to find ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure. ANSWER: 80

41. m LMP SOLUTION: Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 . 40. m LPM SOLUTION: Angles at a point in a straight line add up to 180 .

Substitute x = 45 in (3x – 55) to find

Substitute x = 45 in (3x – 55) to find Therefore, ANSWER: 80

ANSWER: 42. m JLK 80 SOLUTION: Vertical angles are congruent. Therefore, 41. m LMP SOLUTION: Since the triangle LMP is isosceles, First we need to find Angles at a point in a straight line add up to 180 . We know that,

So,

Substitute x = 45 in (3x – 55) to find ANSWER: 20 43. m JKL SOLUTION: Therefore, Vertical angles are congruent. Therefore, ANSWER: 80 First we need to find

42. m JLK We know that,

SOLUTION:

Vertical angles are congruent. Therefore,

So,

First we need to find In

We know that,

Since the triangle JKL is isosceles,

So, ANSWER: ANSWER: 80 20 44. MULTIPLE REPRESENTATIONS In this 43. m JKL problem, you will explore possible measures of the SOLUTION: interior angles of an isosceles triangle given the measure of one exterior angle. Vertical angles are congruent. Therefore,

First we need to find

We know that,

a. Geometric So, Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of In the base angles, and labeling as shown.

Since the triangle JKL is isosceles, b. Tabular Use a protractor to measure and record

m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same ANSWER: measures. Organize your results in two tables.

80 c. Verbal Explain how you used m 1 to find the 44. MULTIPLE REPRESENTATIONS In this measures of 3, 4, and 5. Then explain how problem, you will explore possible measures of the you used m 2 to find these same measures. interior angles of an isosceles triangle given the measure of one exterior angle. d. Algebraic If m 1 = x, write an expression for

the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: a. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

b. ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = b. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

c. 5 is supplementary to 1, so m 5 = 180 – SOLUTION: m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that ANSWER:

SOLUTION: PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. ANSWER: ANSWER: Sometimes; only if the measure of the vertex angle is even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 PRECISION Determine whether the following (measure of the base angle) will be even and 180 – 2 statements are sometimes, always, or never true. (measure of the base angle) will be even. Explain. 46. If the measure of the vertex angle of an isosceles ANSWER: triangle is an integer, then the measure of each base Never; the measure of the vertex angle will be 180 – angle is an integer. 2(measure of the base angle) so if the base angles SOLUTION: are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will The statement is sometimes true. It is true only if the be even. measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; 48. ERROR ANALYSIS Alexis and Miguela are vertex angle = 55, base angles = 62.5. finding m G in the figure shown. Alexis says that ANSWER: m G = 35, while Miguela says that m G = 60. Is Sometimes; only if the measure of the vertex angle is either of them correct? Explain your reasoning. even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd.

SOLUTION: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base SOLUTION: angle) so if the base angles are integers, then 2 Neither of them is correct. This is an isosceles (measure of the base angle) will be even and 180 – 2 triangle with a vertex angle of 70. Since this is an (measure of the base angle) will be even. isosceles triangle, the base angles are congruent.

48. ERROR ANALYSIS Alexis and Miguela are ANSWER: finding m G in the figure shown. Alexis says that

m G = 35, while Miguela says that m G = 60. Is Neither; m G = or 55. either of them correct? Explain your reasoning.

49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle. SOLUTION: ANSWER: Neither of them is correct. This is an isosceles It is not possible because a triangle cannot have more triangle with a vertex angle of 70. Since this is an than one obtuse angle. isosceles triangle, the base angles are congruent. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, ANSWER: And given that m∠B = 90.

Neither; m G = or 55. Therefore,

49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. Construct the triangle with the angles measures 90, SOLUTION: 45, 45. It is not possible because a triangle cannot have more than one obtuse angle.

ANSWER: It is not possible because a triangle cannot have more than one obtuse angle.

50. REASONING In isosceles m B = 90. ANSWER: Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, 51. WRITING IN MATH How can triangle And given that m∠B = 90. classifications help you prove triangle congruence? Therefore, SOLUTION:

If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. For example, if you know that a triangle is an equilateral triangle, you can use Construct the triangle with the angles measures 90, Corollary 4.3 and 4.4 in the proof. Doing this can 45, 45. save you steps when writing the proof. ANSWER: Sample answer: If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof. ANSWER: 52. MULTI-STEP △AED and △BEC are congruent and isosceles.

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence?

SOLUTION: a. Which statement is not necessarily true from the If a triangle is already classified, you can use the information given? previously proven properties of that type of triangle in the proof. For example, if you know that a A △AEB ≅ △DEC triangle is an equilateral triangle, you can use B Corollary 4.3 and 4.4 in the proof. Doing this can C ∠CAD ≅ ∠DBC save you steps when writing the proof. D

ANSWER: b. If m∠EAB = 40, what is the measure of ∠DCE? Sample answer: If a triangle is already classified, you can use the previously proven properties of that c. If m∠ADE = 50, what is the measure of ∠BEC? type of triangle in the proof. Doing this can save you steps when writing the proof. d. What measure would ∠AEB need to be so that ? 52. MULTI-STEP △AED and △BEC are congruent and isosceles. SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because

base angles of an isosceles triangle are congruent. So a. Which statement is not necessarily true from the △AEB is equiangular and equilateral. △AEB ≅ information given? △DEC by SAS, so AB ≅ EC.

ANSWER: A △AEB ≅ △DEC a. B B b. 40 C ∠CAD ≅ ∠DBC c. 80 D d. 60; Sample answer: All three angles in △AEB

are 60° because base angles of an isosceles triangle b. If m∠EAB = 40, what is the measure of ∠DCE? are congruent. So △AEB is equiangular and

equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. c. If m∠ADE = 50, what is the measure of ∠BEC? 53. What is m∠FEG? d. What measure would ∠AEB need to be so that ? SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so DCE is 40°. ∠ c. The sum of the measures of the angles of a A 34 triangle is 180, so ∠BEC must be 80°. B 56 d. △ 60; All three angles in AEB are 60° because C 60 base angles of an isosceles triangle are congruent. So D 62 △ △ AEB is equiangular and equilateral. AEB ≅ E 124 △DEC by SAS, so AB ≅ EC. ANSWER: SOLUTION: a. B b. 40 c. 80 d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two 53. What is m∠FEG? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is choice D.

ANSWER: D

54. A jeweler bends wire to make charms in the shape A 34 of isosceles triangles. The base of the triangle B 56 measures 2 centimeters and another side measures C 60 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? D 62 E 124 SOLUTION: Because we know that the shape of each charm is SOLUTION: an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm.

To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms.

ANSWER: Consider triangle DEH. Since one angle is 56°, and 7 the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. 55. In the figure below, NDG LGD. So, m∠FEG is , or 62. The correct answer is choice D.

ANSWER: D

54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle

measures 2 centimeters and another side measures a. 3.5 centimeters. How many charms can be made What is m∠L? from a piece of wire that is 63 centimeters long? b. What is the value of x? c. What is m∠DGN? SOLUTION: Because we know that the shape of each charm is SOLUTION: an isosceles triangle, then we can determine the side a. In this figure, angles N and L are congruent so lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, m∠L = 36°. the perimeter of each triangle is 9 cm. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is To find how many charms can be made, divide 63 180, then we can solve 36 + 90 + x = 180, or x = 54. cm by 9 cm. So, the jeweler can make 7 charms. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: 7 a. 36 b. 54 55. In the figure below, NDG LGD. c. 54

a. What is m∠L? 216 ÷ 3 = 72 b. What is the value of x? c. What is m∠DGN? ANSWER: 72 SOLUTION: a. In this figure, angles N and L are congruent so 57. △JKL is an isosceles triangle, and . JK = m∠L = 36°. 2x + 3, and KL = 10 – 4x. What is the perimeter of b. Because we know two angle measures, and that △JKL? the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. SOLUTION: c. ∠DGN is congruent to x°, so m∠DGN is 54°. The perimeter can be found by adding all three side lengths. In this situation, since the triangle is ANSWER: isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. a. 36 b. 54 Because the x-terms cancel each other out: 2x + 2x c. 54 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? So, the perimeter of △JKL is 16. SOLUTION: ANSWER: Because the triangle is equilateral that means each 16 side has the same length, so divide the perimeter by 3 to find the length of one side. 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of 216 ÷ 3 = 72 x.

ANSWER: SOLUTION: 72 In an isosceles triangle, the base angles are congruent. 57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of So, 5x + 5 = 7x – 19. △JKL? Simplify. SOLUTION: The perimeter can be found by adding all three side –2x = –24 lengths. In this situation, since the triangle is x = 12 isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. ANSWER: Because the x-terms cancel each other out: 2x + 2x 12 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an So, the perimeter of △JKL is 16. equilateral triangle?

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

In an isosceles triangle, the base angles are congruent. A

B m∠VRT = 60 So, 5x + 5 = 7x – 19. C QRV VTU Simplify. D VRT RST SOLUTION: –2x = –24 x = 12

ANSWER: 12

59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an equilateral triangle?

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees. A B m∠VRT = 60 C C QRV VTU D VRT RST SOLUTION:

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3. Consider each choice. D Since we do not have any measures to indicate A does not provide enough information to that either or is equilateral, knowing determine how sides VT and TR are related. they are congruent would not help prove is equilateral. B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two The correct answer is choice C. measures are 60 degrees. ANSWER: C C

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12 Refer to the figure.

4. m MRP

1. If name two congruent angles. SOLUTION: Isosceles Triangle Theorem states that if two sides SOLUTION: of the triangle are congruent, then the angles opposite Since all the sides are congruent, is an

those sides are congruent. equilateral triangle. Each angle of an equilateral

Therefore, in triangle ABC, triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: ANSWER: BAC and BCA 60

In the figure, . Therefore, triangle RST is an isosceles triangle. Find each measure. By the Converse of Isosceles Triangle Theorem, 3. FH

That is, .

SOLUTION: ANSWER: By the Triangle Angle-Sum Theorem, 12

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. 6. ANSWER: 12 SOLUTION: In the figure, Therefore, triangle WXY is

4. m MRP an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: SOLUTION: 16 Since all the sides are congruent, is an 7. PROOF Write a two-column proof. equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. Given: is isosceles; bisects ABC. ANSWER: Prove: 60 SENSE-MAKING Find the value of each variable.

SOLUTION: 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

8. ROLLER COASTERS The roller coaster track ANSWER: appears to be composed of congruent triangles. A a. portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the b. The Pythagorean Theorem tells us that distance between and Explain your reasoning. By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. a. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

b. The Pythagorean Theorem tells us that: 10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: By CPCTC we know that VT = 1.5 m. The By the Converse of Isosceles Triangle Theorem, in

Segment Addition Postulate says QV + VT = QT. triangle ABF, By substitution, we have 1.5 + 1.5 = QT. So QT = ANSWER: 3 m. ANSWER: a. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: b. The Pythagorean Theorem tells us that By CPCTC we know 13. If BCF BFC, name two congruent segments. that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. By the Converse of Isosceles Triangle Theorem, in triangle BCF, Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, 9. If name two congruent angles.

SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle ABE, AFH and AHF

ANSWER: Find each measure.

ABE and AEB 15. m BAC

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF, SOLUTION:

ANSWER: Here By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ANSWER: 60 ACD and ADC 16. m SRT 12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, ANSWER: SOLUTION: Given: By Isosceles Triangle Theorem, . 13. If BCF BFC, name two congruent segments. Apply Triangle Angle-Sum Theorem. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: 65

14. If name two congruent angles. 17. TR SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: AFH and AHF SOLUTION: Since the triangle is isosceles, Find each measure. Therefore, 15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. SOLUTION: Since the equiangular triangle is an equilateral, Here By Isosceles Triangle Theorem, . ANSWER:

Apply the Triangle Sum Theorem. 4

18. CB

ANSWER: 60

16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER:

3 SOLUTION: Given: REGULARITY Find the value of each variable. By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

19. SOLUTION: Since all the angles are congruent, the sides are also ANSWER: congruent to each other. 65 Therefore, Solve for x. 17. TR

ANSWER: x = 5 SOLUTION: Since the triangle is isosceles, Therefore,

20. SOLUTION:

Given: All the angles are congruent. Therefore, it is an By the Isosceles Triangle Theorem, equiangular triangle. Since the equiangular triangle is an equilateral,

And ANSWER: We know 4 that, 18. CB

ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC,

That is, CB = 3. 21. SOLUTION: ANSWER:

3 Given: By the Isosceles Triangle Theorem, REGULARITY Find the value of each variable. That is,

Then, let Equation 1 be: .

19. By the Triangle Sum Theorem, we can find Equation SOLUTION: 2: Since all the angles are congruent, the sides are also congruent to each other.

Therefore, Add the equations 1 and 2. Solve for x.

ANSWER: x = 5 Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION:

Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11 And We know that,

22. ANSWER: x = 13 SOLUTION: Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore, 21.

SOLUTION: Set up two equations to solve for x and y. Given: By the Isosceles Triangle Theorem,

That is, ANSWER:

Then, let Equation 1 be: . x = 1.4, y = –1

By the Triangle Sum Theorem, we can find Equation PROOF Write a paragraph proof. 2: 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary.

Prove: Add the equations 1 and 2. JHK MHL

Substitute the value of x in one of the two equations SOLUTION: to find the value of y. Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH MLH by the Congruent x = 11, y = 11 Supplements Theorem. By AAS, By CPCTC, JHK MHL.

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH 22. and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles SOLUTION: Triangle Theorem, we know that HJK HML. Given: Because is an equilateral triangle, we know By the Triangle Angle-Sum HLK LKH KHL and Theorem, JKH, HKL and HLK, So, the triangle is an equilateral triangle. MLH are supplementary, and HKL HLK, Therefore, we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS,

By CPCTC, JHK MHL. Set up two equations to solve for x and y.

24. Given: W is the midpoint of Q is the midpoint of ANSWER: Prove: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, SOLUTION: because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY Proof: We are given that is an isosceles and XQ + QZ = XZ. Substitution gives XW + WY = triangle and is an equilateral triangle, JKH XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. and HKL are supplementary and HLK and If we divide each side by 2, we have WY = QZ. The MLH are supplementary. From the Isosceles Isosceles Triangle Theorem says XYZ XZY. Triangle Theorem, we know that HJK HML. by the Reflexive Property. By SAS, Because is an equilateral triangle, we know

HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK, Proof: We are given W is the midpoint of we know JKH MLH by the Congruent Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY Proof: We are given that is an isosceles and XQ + QZ = XZ. Substitution gives XW + WY = triangle and is an equilateral triangle, JKH XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. and HKL are supplementary and HLK and If we divide each side by 2, we have WY = QZ. The MLH are supplementary. From the Isosceles Isosceles Triangle Theorem says XYZ XZY. Triangle Theorem, we know that HJK HML. by the Reflexive Property. By SAS, Because is an equilateral triangle, we know So, by CPCTC. HLK LKH KHL and JKH, HKL and HLK, 25. BABYSITTING While babysitting her neighbor’s MLH are supplementary, and HKL HLK, children, Elisa observes that the supports on either we know JKH ∠MLH by the Congruent side of a park swing set form two sets of triangles. Supplements Theorem. By AAS, Using a jump rope to measure, Elisa is able to 4-6 Isosceles and Equilateral Triangles By CPCTC, JHK MHL. determine that but

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain.

b. If show that is isosceles. SOLUTION: c. If and show that is Proof: We are given W is the midpoint of equilateral. and Q is the midpoint of Because W is the d. If is isosceles, what is the minimum information needed to prove that midpoint of we know that Similarly, Explain your reasoning. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = SOLUTION: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. a. Given: If we divide each side by 2, we have WY = QZ. The By Isosceles Triangle Theorem, . Isosceles Triangle Theorem says XYZ XZY. Apply Triangle Sum Theorem. by the Reflexive Property. By SAS, So, by CPCTC.

ANSWER: b. Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, c. So, by CPCTC.

eSolutions Manual - Powered by Cognero Page 7

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles.

c. If and show that is equilateral. d. One pair of congruent corresponding sides and d. If is isosceles, what is the minimum one pair of congruent corresponding angles; since information needed to prove that you know that the triangle is isosceles, if one leg is Explain your reasoning. congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you SOLUTION: know that you know that a. Given: Therefore, By Isosceles Triangle Theorem, . with one pair of congruent corresponding sides and Apply Triangle Sum Theorem. one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: b. a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

c. c.

d. d. One pair of congruent corresponding sides and One pair of congruent corresponding sides and one pair of congruent corresponding angles; since one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that congruent to a leg of then you know that both pairs of legs are congruent. Because the base both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you angles of an isosceles triangle are congruent, if you know that you know that know that , you know that Therefore, Therefore, with one pair of congruent corresponding sides and with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA triangles can be proved congruent using either ASA

or SAS. or SAS.

SOLUTION:

d. One pair of congruent corresponding sides and ANSWER: one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle.

SOLUTION: Sample answer: I constructed a pair of perpendicular Conjecture: The measures of the base angles of an segments and then used the same compass setting to isosceles right triangle are 45. mark points equidistant from their intersection. I Proof: The base angles are congruent because it is measured both legs for each triangle. Because AB = an isosceles triangle. Let the measure of each acute AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = angle be x. The acute angles of a right triangle are 2.3 cm, the triangles are isosceles. I used a complementary, Solve for x protractor to confirm that A, D, and G are all right angles.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to ANSWER: mark points equidistant from their intersection. I 44 measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 30. m ACD protractor to confirm that A, D, and G are all SOLUTION: right angles. We know 28. PROOF Based on your construction in Exercise 27, that make and prove a conjecture about the relationship Therefore, . between the base angles of an isosceles right triangle. ANSWER: 44 SOLUTION: Conjecture: The measures of the base angles of an 31. m ACB isosceles right triangle are 45. SOLUTION: Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute The angles in a straight line add to 180°. angle be x. The acute angles of a right triangle are Therefore, complementary, Solve for x We know that

ANSWER: Conjecture: The measures of the base angles of an ANSWER: isosceles right triangle are 45. 136 Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute 32. m ABC angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. SOLUTION: REGULARITY Find each measure. From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

29. m∠CAD ANSWER: 22 SOLUTION: From the figure, 33. FITNESS In the diagram, the rider will use his bike Therefore, is an isosceles triangle. to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G By the Triangle Sum Theorem, H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that ANSWER:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, 34. Given: is isosceles;

Prove: X and YZV are complementary.

ANSWER: 22 FITNESS 33. In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex SOLUTION: angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION: ANSWER:

ANSWER: PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. 36. Corollary 4.4 SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . 37. Theorem 4.11 The interior angles of a triangle add up to 180 . SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM 38. SOLUTION: SOLUTION: Angles at a point in a straight line add up to 180 . By the converse of Isosceles Triangle theorem,

Solve the equation for x. Substitute x = 45 in (3x – 55) to find

Note that x can equal –8 here because .

ANSWER: ANSWER: 3 80

41. m LMP SOLUTION: 39. Since the triangle LMP is isosceles, Angles at a point in a straight line SOLUTION: add up to 180 . By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Therefore,

ANSWER: ANSWER: 14 80 GAMES Use the diagram of a game timer 42. m JLK shown to find each measure. SOLUTION:

Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM So, SOLUTION: Angles at a point in a straight line add up to 180 . ANSWER: 20

43. m JKL SOLUTION:

Vertical angles are congruent. Therefore, Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

ANSWER: 80 So, 41. m LMP In SOLUTION:

Since the triangle LMP is isosceles, Since the triangle JKL is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

Substitute x = 45 in (3x – 55) to find 44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

Therefore,

ANSWER: 80

42. m JLK

SOLUTION: a. Geometric Use a ruler and a protractor to draw Vertical angles are congruent. Therefore, three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown. First we need to find b. Tabular Use a protractor to measure and record We know that, m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables. So, c. Verbal Explain how you used m 1 to find the ANSWER: measures of 3, 4, and 5. Then explain how 20 you used m 2 to find these same measures.

43. m JKL d. Algebraic If m 1 = x, write an expression for SOLUTION: the measures of 3, 4, and 5. Likewise, if m Vertical angles are congruent. Therefore, 2 = x, write an expression for these same angle measures. First we need to find

We know that, SOLUTION: a.

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw b. three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how c. 5 is supplementary to 1, so m 5 = 180 – you used m 2 to find these same measures. m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = d. Algebraic If m 1 = x, write an expression for 180 – m 4 – 5. 2 is supplementary to 3, so the measures of 3, 4, and 5. Likewise, if m m 3 = 180 – m 2. m 2 is twice as much as m 2 = x, write an expression for these same angle

measures. 4 and m 5, so m 4 = m 5 =

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – SOLUTION: a. 180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – m 3 = 180 – m 2. m 2 is twice as much as m

180; m 3 = 180 – x, m 4 = m 5 = 4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – ANSWER:

a. 180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

b. ANSWER:

c. 5 is supplementary to 1, so m 5 = 180 – PRECISION Determine whether the following statements are sometimes, always, or never true. m 1. 4 5, so m 4 = m 5. The sum of the Explain. angle measures in a triangle must be 180, so m 3 = 46. If the measure of the vertex angle of an isosceles 180 – m 4 – 5. 2 is supplementary to 3, so triangle is an integer, then the measure of each base m 3 = 180 – m 2. m 2 is twice as much as m angle is an integer.

4 and m 5, so m 4 = m 5 = SOLUTION: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – The statement is sometimes true. It is true only if the

measure of the vertex angle is even. For example, 180; m 3 = 180 – x, m 4 = m 5 = vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

45. CHALLENGE In the figure, if is equilateral ANSWER: and ZWP WJM JZL, prove that Sometimes; only if the measure of the vertex angle is even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning.

ANSWER: 49. OPEN-ENDED If possible, draw an isosceles Sometimes; only if the measure of the vertex angle is triangle with base angles that are obtuse. If it is not even. possible, explain why not. SOLUTION: 47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex It is not possible because a triangle cannot have more angle is odd. than one obtuse angle. SOLUTION: ANSWER: The statement is never true. The measure of the It is not possible because a triangle cannot have more vertex angle will be 180 – 2(measure of the base than one obtuse angle. angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 50. REASONING In isosceles m B = 90. (measure of the base angle) will be even. Draw the triangle. Indicate the congruent sides and label each angle with its measure. ANSWER: SOLUTION: Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles The sum of the angle measures in a triangle must be are integers, then 2(measure of the base angle) will 180, be even and 180 – 2(measure of the base angle) will be even. Since is isosceles, And given that m∠B = 90. 48. ERROR ANALYSIS Alexis and Miguela are Therefore, finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an ANSWER: isosceles triangle, the base angles are congruent.

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? ANSWER: SOLUTION: If a triangle is already classified, you can use the Neither; m G = or 55. previously proven properties of that type of triangle in the proof. For example, if you know that a 49. OPEN-ENDED If possible, draw an isosceles triangle is an equilateral triangle, you can use triangle with base angles that are obtuse. If it is not Corollary 4.3 and 4.4 in the proof. Doing this can possible, explain why not. save you steps when writing the proof. SOLUTION: It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Sample answer: If a triangle is already classified, you can use the previously proven properties of that ANSWER: type of triangle in the proof. Doing this can save you It is not possible because a triangle cannot have more steps when writing the proof. than one obtuse angle. 52. MULTI-STEP △AED and △BEC are congruent 50. REASONING In isosceles m B = 90. and isosceles. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, a. Which statement is not necessarily true from the And given that m∠B = 90. information given? Therefore,

A △AEB ≅ △DEC B

C ∠CAD ≅ ∠DBC

D Construct the triangle with the angles measures 90,

45, 45. b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ?

SOLUTION: ANSWER: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So 51. WRITING IN MATH How can triangle △ △ classifications help you prove triangle congruence? AEB is equiangular and equilateral. AEB ≅ △DEC by SAS, so AB ≅ EC. SOLUTION: If a triangle is already classified, you can use the ANSWER: previously proven properties of that type of triangle a. B in the proof. For example, if you know that a b. 40 c. triangle is an equilateral triangle, you can use 80 d. 60; Sample answer: All three angles in △AEB Corollary 4.3 and 4.4 in the proof. Doing this can are 60° because base angles of an isosceles triangle save you steps when writing the proof. are congruent. So △AEB is equiangular and △ △ ANSWER: equilateral. AEB ≅ DEC by SAS, so AB ≅ EC. Sample answer: If a triangle is already classified, 53. What is m∠FEG? you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

A 34 B 56 C 60 D 62 E 124

a. Which statement is not necessarily true from the information given? SOLUTION:

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

Consider triangle DEH. Since one angle is 56°, and b. If m EAB = 40, what is the measure of DCE? ∠ ∠ the triangle is isosceles, then the remaining two

angles would be 180 – 56 or 124°. c. If m∠ADE = 50, what is the measure of ∠BEC? So, m FEG is , or 62. The correct answer is ∠ d. What measure would ∠AEB need to be so that choice D. ? ANSWER: SOLUTION: D a. Of the statements given, only choice B is not 54. A jeweler bends wire to make charms in the shape necessarily true from the given information. of isosceles triangles. The base of the triangle b. The angles are congruent, so ∠DCE is 40°. measures 2 centimeters and another side measures c. The sum of the measures of the angles of a 3.5 centimeters. How many charms can be made triangle is 180, so ∠BEC must be 80°. from a piece of wire that is 63 centimeters long? d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So SOLUTION: △AEB is equiangular and equilateral. △AEB ≅ Because we know that the shape of each charm is an isosceles triangle, then we can determine the side △DEC by SAS, so AB ≅ EC. lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, ANSWER: the perimeter of each triangle is 9 cm. a. B To find how many charms can be made, divide 63 b. 40 cm by 9 cm. So, the jeweler can make 7 charms. c. 80 d. 60; Sample answer: All three angles in △AEB ANSWER: are 60° because base angles of an isosceles triangle 7 are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. 55. In the figure below, NDG LGD.

53. What is m∠FEG?

a. What is m∠L? A 34 b. What is the value of x? B 56 c. What is m∠DGN? C 60 D 62 SOLUTION: E 124 a. In this figure, angles N and L are congruent so m∠L = 36°. SOLUTION: b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: a. 36

b. 54 Consider triangle DEH. Since one angle is 56°, and c. the triangle is isosceles, then the remaining two 54 angles would be 180 – 56 or 124°. 56. An equilateral triangle has a perimeter of 216 inches. So, m∠FEG is , or 62. The correct answer is What is the length of one side of the triangle? choice D. SOLUTION: ANSWER: Because the triangle is equilateral that means each D side has the same length, so divide the perimeter by 3 to find the length of one side. 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle 216 ÷ 3 = 72 measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made ANSWER: from a piece of wire that is 63 centimeters long? 72

SOLUTION: 57. △JKL is an isosceles triangle, and . JK = Because we know that the shape of each charm is 2x + 3, and KL = 10 – 4x. What is the perimeter of an isosceles triangle, then we can determine the side △JKL? lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. SOLUTION: The perimeter can be found by adding all three side To find how many charms can be made, divide 63 lengths. In this situation, since the triangle is cm by 9 cm. So, the jeweler can make 7 charms. isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

ANSWER: Because the x-terms cancel each other out: 2x + 2x 7 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 55. In the figure below, NDG LGD. So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x.

a. What is m∠L? SOLUTION: b. What is the value of x? In an isosceles triangle, the base angles are c. What is m∠DGN? congruent.

SOLUTION: So, 5x + 5 = 7x – 19. a. In this figure, angles N and L are congruent so m∠L = 36°. Simplify. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is –2x = –24 180, then we can solve 36 + 90 + x = 180, or x = 54. x = 12 c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: 12 a. 36 59. QRV is an equilateral triangle. What additional b. 54 c. 54 information would be enough to prove VRT is an equilateral triangle? 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

ANSWER: A 72 B m∠VRT = 60 C QRV VTU 57. △JKL is an isosceles triangle, and . JK = D VRT RST 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. Consider each choice. ANSWER: 16 A does not provide enough information to determine how sides VT and TR are related. 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of B If we know that m∠VRT = 60, we still would not x. have enough information to prove that the other two measures are 60 degrees. SOLUTION: In an isosceles triangle, the base angles are C congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

ANSWER: If , then m∠RVT = 60 because 12 ∠QVR + ∠RVT + ∠TVU = 180. Since , 59. QRV is an equilateral triangle. What additional by the Isosceles Triangle Theorem. information would be enough to prove VRT is an So, . equilateral triangle? Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12 Refer to the figure. 4. m MRP

1. If name two congruent angles.

SOLUTION: SOLUTION: Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite Since all the sides are congruent, is an those sides are congruent. equilateral triangle. Each angle of an equilateral Therefore, in triangle ABC, triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: ANSWER: 60 BAC and BCA SENSE-MAKING Find the value of each 2. If EAC ECA, name two congruent segments. variable. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC, 5.

ANSWER: SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. Find each measure. By the Converse of Isosceles Triangle Theorem, 3. FH That is, .

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. 6. ANSWER: SOLUTION: 12 In the figure, Therefore, triangle WXY is an isosceles triangle. 4. m MRP By the Isosceles Triangle Theorem, .

ANSWER: 16 SOLUTION: 7. PROOF Write a two-column proof. Since all the sides are congruent, is an

equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. Given: is isosceles; bisects ABC. Prove: ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that ANSWER: 16 b. If VR = 2.5 meters and QR = 2 meters, find the 7. PROOF Write a two-column proof. distance between and Explain your reasoning.

Given: is isosceles; bisects ABC. Prove: SOLUTION: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the b. The Pythagorean Theorem tells us that distance between and Explain your By CPCTC we know reasoning. that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. a. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

b. The Pythagorean Theorem tells us that: 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in By CPCTC we know that VT = 1.5 m. The triangle ABF, Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = ANSWER: 3 m.

ANSWER: 11. If name two congruent angles. a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

b. The Pythagorean Theorem tells us that By CPCTC we know 13. If BCF BFC, name two congruent segments. that VT = 1.5 m. The Seg. Add. Post. says QV + SOLUTION: VT = QT. By substitution, we have 1.5 + 1.5 = QT. By the Converse of Isosceles Triangle Theorem, in So QT = 3 m. triangle BCF,

Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, 9. If name two congruent angles. SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle ABE, AFH and AHF

Find each measure. ANSWER: 15. m BAC ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, Here ANSWER: By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER: 60 ANSWER: ACD and ADC 16. m SRT 12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ADE, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

13. If BCF BFC, name two congruent segments. Apply Triangle Angle-Sum Theorem.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: 65 17. TR 14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION: AFH and AHF Since the triangle is isosceles, Find each measure. Therefore,

15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION:

Here By Isosceles Triangle Theorem, . ANSWER: 4 Apply the Triangle Sum Theorem. 18. CB

ANSWER: 60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3 SOLUTION: REGULARITY Find the value of each variable. Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19. SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. ANSWER: Therefore, 65 Solve for x.

17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore,

20. SOLUTION: Given: All the angles are congruent. Therefore, it is an By the Isosceles Triangle Theorem, equiangular triangle. Since the equiangular triangle is an equilateral, And We know ANSWER: that, 4 18. CB ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, 21.

That is, CB = 3. SOLUTION: ANSWER: Given: 3 By the Isosceles Triangle Theorem,

REGULARITY Find the value of each variable. That is,

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2: SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, Add the equations 1 and 2. Solve for x.

ANSWER: x = 5 Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION:

Given: By the Isosceles Triangle Theorem, ANSWER:

x = 11, y = 11

And We know that,

22. ANSWER: SOLUTION: x = 13 Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore, 21. Set up two equations to solve for x and y. SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, ANSWER:

Then, let Equation 1 be: . x = 1.4, y = –1 PROOF Write a paragraph proof. By the Triangle Sum Theorem, we can find Equation 23. Given: is isosceles, and is 2: equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL Add the equations 1 and 2.

SOLUTION: Substitute the value of x in one of the two equations Proof: We are given that is an isosceles to find the value of y. triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK,

MLH are supplementary, and HKL HLK, ANSWER: we know JKH MLH by the Congruent x = 11, y = 11 Supplements Theorem. By AAS, By CPCTC, JHK MHL.

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and 22. MLH are supplementary. From the Isosceles SOLUTION: Triangle Theorem, we know that HJK HML. Given: Because is an equilateral triangle, we know By the Triangle Angle-Sum HLK LKH KHL and Theorem, JKH, HKL and HLK, MLH are supplementary, and HKL HLK, So, the triangle is an equilateral triangle. we know JKH ∠MLH by the Congruent Therefore, Supplements Theorem. By AAS, By CPCTC, JHK MHL. Set up two equations to solve for x and y. 24. Given: W is the midpoint of Q is the midpoint of Prove: ANSWER:

x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The SOLUTION: Segment Addition Postulate gives us XW + WY = XY Proof: We are given that is an isosceles and XQ + QZ = XZ. Substitution gives XW + WY = triangle and is an equilateral triangle, JKH XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. and HKL are supplementary and HLK and If we divide each side by 2, we have WY = QZ. The MLH are supplementary. From the Isosceles Isosceles Triangle Theorem says XYZ XZY. Triangle Theorem, we know that HJK HML. by the Reflexive Property. By SAS, Because is an equilateral triangle, we know So, by CPCTC. HLK LKH KHL and ANSWER: JKH, HKL and HLK, MLH are supplementary, and HKL HLK, Proof: We are given W is the midpoint of we know JKH MLH by the Congruent and Q is the midpoint of Because W is the Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The ANSWER: Segment Addition Postulate gives us XW + WY = XY Proof: We are given that is an isosceles and XQ + QZ = XZ. Substitution gives XW + WY = triangle and is an equilateral triangle, JKH XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. by the Reflexive Property. By SAS, Because is an equilateral triangle, we know So, by CPCTC. HLK LKH KHL and 25. BABYSITTING While babysitting her neighbor’s JKH, HKL and HLK, children, Elisa observes that the supports on either MLH are supplementary, and HKL HLK, side of a park swing set form two sets of triangles. we know JKH ∠MLH by the Congruent Using a jump rope to measure, Elisa is able to Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. SOLUTION: c. If and show that is equilateral. Proof: We are given W is the midpoint of d. If is isosceles, what is the minimum and Q is the midpoint of Because W is the information needed to prove that midpoint of we know that Similarly, Explain your reasoning. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY SOLUTION: and XQ + QZ = XZ. Substitution gives XW + WY = a. Given: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, . If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles.

c. If and show that is d. One pair of congruent corresponding sides and

equilateral. one pair of congruent corresponding angles; since d. If is isosceles, what is the minimum you know that the triangle is isosceles, if one leg is information needed to prove that congruent to a leg of then you know that Explain your reasoning. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you SOLUTION: know that you know that Therefore, a. Given: with one pair of congruent corresponding sides and By Isosceles Triangle Theorem, . one pair of congruent corresponding angles, the Apply Triangle Sum Theorem. triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC b. ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

4-6 Isosceles and Equilateral Triangles

c. c.

d. One pair of congruent corresponding sides and d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that congruent to a leg of then you know that both pairs of legs are congruent. Because the base both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you angles of an isosceles triangle are congruent, if you know that , you know that know that you know that Therefore, Therefore, with one pair of congruent corresponding sides and with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA triangles can be proved congruent using either ASA or SAS. or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show a. 65°; Because is isosceles, ABC that the chimney of the house, represented by ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. bisects the angle formed by the sloped sides of b. the roof, ABC.

c. eSolutions Manual - Powered by Cognero Page 8 SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an Sample answer: I constructed a pair of perpendicular isosceles right triangle are 45. segments and then used the same compass setting to Proof: The base angles are congruent because it is mark points equidistant from their intersection. I an isosceles triangle. Let the measure of each acute measured both legs for each triangle. Because AB = angle be x. The acute angles of a right triangle are AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = complementary, Solve for x 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

Sample answer: I constructed a pair of perpendicular ANSWER: segments and then used the same compass setting to mark points equidistant from their intersection. I 44 measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 30. m ACD 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all SOLUTION: right angles. We know that 28. PROOF Based on your construction in Exercise 27, Therefore, . make and prove a conjecture about the relationship between the base angles of an isosceles right ANSWER: triangle. 44

SOLUTION: 31. m ACB Conjecture: The measures of the base angles of an isosceles right triangle are 45. SOLUTION: Proof: The base angles are congruent because it is The angles in a straight line add to 180°. an isosceles triangle. Let the measure of each acute Therefore, angle be x. The acute angles of a right triangle are complementary, Solve for x We know that

ANSWER: Conjecture: The measures of the base angles of an ANSWER: isosceles right triangle are 45. 136 Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute 32. m ABC angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles REGULARITY Find each measure. triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER: 29. m∠CAD 22 SOLUTION: 33. FITNESS In the diagram, the rider will use his bike From the figure, to hop across the tops of each of the concrete solids Therefore, is an isosceles triangle. shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance By the Triangle Sum Theorem, from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, 34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids SOLUTION: shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION: ANSWER:

ANSWER: PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. 36. Corollary 4.4 SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

37. Theorem 4.11 The interior angles of a triangle add up to 180 . SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 . SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x. Substitute x = 45 in (3x – 55) to find

Note that x can equal –8 here because . ANSWER: ANSWER: 80 3

41. m LMP SOLUTION: Since the triangle LMP is isosceles, 39. Angles at a point in a straight line SOLUTION: add up to 180 . By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– Therefore, 18) – 5 = –41, so y = 14. ANSWER: ANSWER: 80 14 42. m JLK GAMES Use the diagram of a game timer shown to find each measure. SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM So,

SOLUTION: ANSWER: Angles at a point in a straight line add up to 180 . 20

43. m JKL SOLUTION: Vertical angles are congruent. Therefore, Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP In SOLUTION: Since the triangle LMP is isosceles, Since the triangle JKL is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this

Substitute x = 45 in (3x – 55) to find problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

Therefore,

ANSWER: 80

42. m JLK SOLUTION: a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of Vertical angles are congruent. Therefore, the sides adjacent to the vertex angle and to one of

the base angles, and labeling as shown.

First we need to find b. Tabular Use a protractor to measure and record

m∠1 for each triangle. Use m 1 to calculate the We know that, measures of 3, 4, and 5. Then find and

record m 2 and use it to calculate these same measures. Organize your results in two tables.

So, c. Verbal Explain how you used m 1 to find the ANSWER: measures of 3, 4, and 5. Then explain how 20 you used m 2 to find these same measures.

43. m JKL d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m SOLUTION: 2 = x, write an expression for these same angle Vertical angles are congruent. Therefore, measures.

First we need to find SOLUTION: We know that, a.

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

b. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

2 = x, write an expression for these same angle 4 and m 5, so m 4 = m 5 = measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

SOLUTION: 180; m 3 = 180 – x, m 4 = m 5 = a. ANSWER: a.

4 and m 5, so m 4 = m 5 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4 and m 5, so m 4 = m 5 = 180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

ANSWER: 180; m 3 = 180 – x, m 4 = m 5 = a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

b. ANSWER:

PRECISION Determine whether the following c. 5 is supplementary to 1, so m 5 = 180 – statements are sometimes, always, or never true. m 1. 4 5, so m 4 = m 5. The sum of the Explain. angle measures in a triangle must be 180, so m 3 = 46. If the measure of the vertex angle of an isosceles 180 – m 4 – 5. 2 is supplementary to 3, so triangle is an integer, then the measure of each base angle is an integer. m 3 = 180 – m 2. m 2 is twice as much as m SOLUTION: 4 and m 5, so m 4 = m 5 = The statement is sometimes true. It is true only if the d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; 180; m 3 = 180 – x, m 4 = m 5 = vertex angle = 55, base angles = 62.5.

ANSWER: 45. CHALLENGE In the figure, if is equilateral Sometimes; only if the measure of the vertex angle is and ZWP WJM JZL, prove that even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

PRECISION Determine whether the following

statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the ANSWER: measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; Neither; m G = or 55. vertex angle = 55, base angles = 62.5.

ANSWER: 49. OPEN-ENDED If possible, draw an isosceles Sometimes; only if the measure of the vertex angle is triangle with base angles that are obtuse. If it is not

even. possible, explain why not. SOLUTION: 47. If the measures of the base angles of an isosceles It is not possible because a triangle cannot have more triangle are integers, then the measure of its vertex than one obtuse angle. angle is odd. ANSWER: SOLUTION: It is not possible because a triangle cannot have more The statement is never true. The measure of the than one obtuse angle. vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 50. REASONING In isosceles m B = 90. (measure of the base angle) will be even and 180 – 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even. label each angle with its measure. ANSWER: SOLUTION: Never; the measure of the vertex angle will be 180 – The sum of the angle measures in a triangle must be 2(measure of the base angle) so if the base angles 180, are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will Since is isosceles, be even. And given that m∠B = 90. Therefore, 48. ERROR ANALYSIS Alexis and Miguela are

finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an ANSWER: isosceles triangle, the base angles are congruent.

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? SOLUTION: ANSWER: If a triangle is already classified, you can use the

Neither; m G = or 55. previously proven properties of that type of triangle in the proof. For example, if you know that a triangle is an equilateral triangle, you can use 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not Corollary 4.3 and 4.4 in the proof. Doing this can possible, explain why not. save you steps when writing the proof. SOLUTION: ANSWER: It is not possible because a triangle cannot have more Sample answer: If a triangle is already classified, than one obtuse angle. you can use the previously proven properties of that type of triangle in the proof. Doing this can save you ANSWER: steps when writing the proof. It is not possible because a triangle cannot have more

than one obtuse angle. 52. MULTI-STEP △AED and △BEC are congruent and isosceles. 50. REASONING In isosceles m B = 90.

Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, a. Which statement is not necessarily true from the And given that m∠B = 90. information given? Therefore, A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D Construct the triangle with the angles measures 90, 45, 45. b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION: ANSWER: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because

base angles of an isosceles triangle are congruent. So 51. WRITING IN MATH How can triangle △AEB is equiangular and equilateral. △AEB ≅ classifications help you prove triangle congruence? △DEC by SAS, so AB ≅ EC. SOLUTION: ANSWER: If a triangle is already classified, you can use the a. B previously proven properties of that type of triangle b. 40 in the proof. For example, if you know that a c. 80 triangle is an equilateral triangle, you can use d. 60; Sample answer: All three angles in △AEB Corollary 4.3 and 4.4 in the proof. Doing this can are 60° because base angles of an isosceles triangle △ save you steps when writing the proof. are congruent. So AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. ANSWER: Sample answer: If a triangle is already classified, 53. What is m∠FEG?

you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

A 34 B 56 C 60 D 62 E 124

a. Which statement is not necessarily true from the SOLUTION: information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D Consider triangle DEH. Since one angle is 56°, and b. If m∠EAB = 40, what is the measure of ∠DCE? the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. c. If m∠ADE = 50, what is the measure of ∠BEC? So, m∠FEG is , or 62. The correct answer is choice D. d. What measure would ∠AEB need to be so that ANSWER: ? D SOLUTION: a. Of the statements given, only choice B is not 54. A jeweler bends wire to make charms in the shape necessarily true from the given information. of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures b. The angles are congruent, so ∠DCE is 40°. 3.5 centimeters. How many charms can be made c. The sum of the measures of the angles of a from a piece of wire that is 63 centimeters long? triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because SOLUTION: base angles of an isosceles triangle are congruent. So Because we know that the shape of each charm is △AEB is equiangular and equilateral. △AEB ≅ an isosceles triangle, then we can determine the side △DEC by SAS, so AB ≅ EC. lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. ANSWER: a. B To find how many charms can be made, divide 63 b. 40 cm by 9 cm. So, the jeweler can make 7 charms. c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle 7 △ are congruent. So AEB is equiangular and 55. In the figure below, NDG LGD. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. 53. What is m∠FEG?

a. What is m∠L? A 34 b. What is the value of x? B 56 c. What is m∠DGN? C 60 SOLUTION: D 62 a. In this figure, angles N and L are congruent so E 124 m∠L = 36°.

b. Because we know two angle measures, and that SOLUTION: the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: a. 36

b. 54 Consider triangle DEH. Since one angle is 56°, and c. 54 the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? So, m∠FEG is , or 62. The correct answer is choice D. SOLUTION: Because the triangle is equilateral that means each ANSWER: side has the same length, so divide the perimeter by 3 D to find the length of one side.

54. A jeweler bends wire to make charms in the shape 216 ÷ 3 = 72 of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures ANSWER: 3.5 centimeters. How many charms can be made 72 from a piece of wire that is 63 centimeters long? 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of Because we know that the shape of each charm is △JKL? an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, SOLUTION: the perimeter of each triangle is 9 cm. The perimeter can be found by adding all three side lengths. In this situation, since the triangle is To find how many charms can be made, divide 63 isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. cm by 9 cm. So, the jeweler can make 7 charms. Because the x-terms cancel each other out: 2x + 2x ANSWER: – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 7 16.

55. In the figure below, NDG LGD. So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x.

SOLUTION: a. What is m∠L? In an isosceles triangle, the base angles are b. What is the value of x? congruent. c. What is m∠DGN? So, 5x + 5 = 7x – 19. SOLUTION: a. In this figure, angles N and L are congruent so Simplify. m∠L = 36°. b. Because we know two angle measures, and that –2x = –24 the sum of the measures of the angles of a triangle is x = 12 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: 12 ANSWER: a. 36 59. QRV is an equilateral triangle. What additional b. 54 information would be enough to prove VRT is an c. 54 equilateral triangle?

216 ÷ 3 = 72 A ANSWER: B m∠VRT = 60 72 C QRV VTU 57. △JKL is an isosceles triangle, and . JK = D VRT RST 2x + 3, and KL = 10 – 4x. What is the perimeter of SOLUTION: △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. Consider each choice.

ANSWER: A does not provide enough information to 16 determine how sides VT and TR are related.

58. △ABC is an isosceles triangle with base angles B If we know that m∠VRT = 60, we still would not measuring (5x + 5)° and (7x – 19)°. Find the value of have enough information to prove that the other two x. measures are 60 degrees. SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

ANSWER: If , then m∠RVT = 60 because 12 ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. 59. QRV is an equilateral triangle. What additional So, . information would be enough to prove VRT is an Therefore, is equilateral by Theorem 4.3. equilateral triangle? D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

Refer to the figure. 4. m MRP

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. 6. SOLUTION: ANSWER: 12 In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, . 4. m MRP

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that ANSWER: 16 b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. The Pythagorean Theorem tells us that b. If VR = 2.5 meters and QR = 2 meters, find the By CPCTC we know distance between and Explain your reasoning. that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. SOLUTION: a. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

ANSWER: 11. If name two congruent angles. a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, Here By Isosceles Triangle Theorem, . ANSWER:

Apply the Triangle Sum Theorem.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER:

60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ADE, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR 14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION: AFH and AHF Since the triangle is isosceles, Therefore, Find each measure. 15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION:

Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER: 60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3

SOLUTION: REGULARITY Find the value of each variable. Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. ANSWER: Therefore, Solve for x. 65

17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

SOLUTION: Given: By the Isosceles Triangle Theorem, All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral,

And We know ANSWER: that, 4 18. CB ANSWER: x = 13

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11

And We know that,

22.

ANSWER: SOLUTION:

x = 13 Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y. SOLUTION: Given: By the Isosceles Triangle Theorem,

Add the equations 1 and 2.

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and 22. MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. SOLUTION: Because is an equilateral triangle, we know Given: HLK LKH KHL and By the Triangle Angle-Sum JKH, HKL and HLK,

Theorem, MLH are supplementary, and HKL HLK,

we know JKH MLH by the Congruent ∠ So, the triangle is an equilateral triangle. Supplements Theorem. By AAS,

Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y. 24. Given: W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary.

Prove: JHK MHL SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY SOLUTION: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY ANSWER: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and 25. BABYSITTING While babysitting her neighbor’s JKH, HKL and HLK, children, Elisa observes that the supports on either MLH are supplementary, and HKL HLK, side of a park swing set form two sets of triangles. we know JKH ∠MLH by the Congruent Using a jump rope to measure, Elisa is able to Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles.

So, by CPCTC. b. ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The

Isosceles Triangle Theorem says XYZ XZY. c. by the Reflexive Property. By SAS, So, by CPCTC.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m ABC? Explain. ∠ b. If show that is isosceles. c. If and show that is d. One pair of congruent corresponding sides and equilateral. one pair of congruent corresponding angles; since d. If is isosceles, what is the minimum you know that the triangle is isosceles, if one leg is information needed to prove that congruent to a leg of then you know that Explain your reasoning. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that SOLUTION: Therefore, a. Given: with one pair of congruent corresponding sides and By Isosceles Triangle Theorem, . one pair of congruent corresponding angles, the Apply Triangle Sum Theorem. triangles can be proved congruent using either ASA

or SAS. ANSWER: a. 65°; Because is isosceles, ABC ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. ∠ b.

c. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that , you know that angles of an isosceles triangle are congruent, if you Therefore, know that you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show a. 65°; Because is isosceles, ABC that the chimney of the house, represented by ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. bisects the angle formed by the sloped sides of b. the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of 4-6 Isosceles and Equilateral Triangles the roof, ABC. 27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then eSolutionsverifyManual your- constructionsPowered by Cognero using measurement and Page 9 mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an Sample answer: I constructed a pair of perpendicular isosceles right triangle are 45. segments and then used the same compass setting to Proof: The base angles are congruent because it is mark points equidistant from their intersection. I an isosceles triangle. Let the measure of each acute measured both legs for each triangle. Because AB = angle be x. The acute angles of a right triangle are AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = complementary, Solve for x 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles. ANSWER: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

ANSWER: ANSWER: Conjecture: The measures of the base angles of an 136 isosceles right triangle are 45. Proof: The base angles are congruent because it is 32. m ABC an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles triangle. Therefore, REGULARITY Find each measure.

By the Triangle Angle Sum Theorem,

ANSWER:

29. m∠CAD 22 SOLUTION: 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids From the figure, shown. If each triangle is isosceles with vertex Therefore, is an isosceles triangle. angles G, H, and J, and G H, and H J, show that the distance

By the Triangle Sum Theorem, from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38.

36. Corollary 4.4 SOLUTION: By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

37. Theorem 4.11 The interior angles of a triangle add up to 180 .

SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 . SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find Solve the equation for x.

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

40. m LPM So, SOLUTION: ANSWER: Angles at a point in a straight line add up to 180 . 20

43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP In

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

Therefore,

ANSWER: 80

First we need to find SOLUTION:

a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

b. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the c. 5 is supplementary to 1, so m 5 = 180 – measures of 3, 4, and 5. Then explain how m 1. 4 5, so m 4 = m 5. The sum of the you used m 2 to find these same measures. angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. Algebraic If m 1 = x, write an expression for m 3 = 180 – m 2. m 2 is twice as much as m the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle 4 and m 5, so m 4 = m 5 = measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

ANSWER: 180; m 3 = 180 – x, m 4 = m 5 = a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the ANSWER: measure of the vertex angle is even. For example, Neither; m G = or 55. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not Sometimes; only if the measure of the vertex angle is possible, explain why not. even. SOLUTION: It is not possible because a triangle cannot have more 47. If the measures of the base angles of an isosceles than one obtuse angle. triangle are integers, then the measure of its vertex angle is odd. ANSWER: SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 50. REASONING In isosceles m B = 90. angle) so if the base angles are integers, then 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even and 180 – 2 label each angle with its measure. (measure of the base angle) will be even. SOLUTION: ANSWER: The sum of the angle measures in a triangle must be Never; the measure of the vertex angle will be 180 – 180, 2(measure of the base angle) so if the base angles

are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will Since is isosceles, be even. And given that m∠B = 90. Therefore, 48. ERROR ANALYSIS Alexis and Miguela are

finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? SOLUTION: ANSWER: If a triangle is already classified, you can use the previously proven properties of that type of triangle Neither; m G = or 55. in the proof. For example, if you know that a triangle is an equilateral triangle, you can use 49. OPEN-ENDED If possible, draw an isosceles Corollary 4.3 and 4.4 in the proof. Doing this can triangle with base angles that are obtuse. If it is not save you steps when writing the proof. possible, explain why not. ANSWER: SOLUTION: Sample answer: If a triangle is already classified, It is not possible because a triangle cannot have more than one obtuse angle. you can use the previously proven properties of that type of triangle in the proof. Doing this can save you ANSWER: steps when writing the proof. It is not possible because a triangle cannot have more than one obtuse angle. 52. MULTI-STEP △AED and △BEC are congruent and isosceles. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC

B C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ?

SOLUTION: a. Of the statements given, only choice B is not ANSWER: necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ WRITING IN MATH How can triangle 51. △DEC by SAS, so AB ≅ EC. classifications help you prove triangle congruence? ANSWER: SOLUTION: a. B If a triangle is already classified, you can use the b. 40 previously proven properties of that type of triangle c. 80 in the proof. For example, if you know that a d. 60; Sample answer: All three angles in △AEB triangle is an equilateral triangle, you can use are 60° because base angles of an isosceles triangle Corollary 4.3 and 4.4 in the proof. Doing this can are congruent. So △AEB is equiangular and save you steps when writing the proof. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

△ △ 52. MULTI-STEP AED and BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two b. If m∠EAB = 40, what is the measure of ∠DCE? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is c. If m∠ADE = 50, what is the measure of ∠BEC? choice D.

d. What measure would ∠AEB need to be so that ANSWER: ? D

SOLUTION: 54. A jeweler bends wire to make charms in the shape a. Of the statements given, only choice B is not of isosceles triangles. The base of the triangle necessarily true from the given information. measures 2 centimeters and another side measures b. The angles are congruent, so ∠DCE is 40°. 3.5 centimeters. How many charms can be made c. The sum of the measures of the angles of a from a piece of wire that is 63 centimeters long? triangle is 180, so ∠BEC must be 80°. SOLUTION: d. 60; All three angles in △AEB are 60° because Because we know that the shape of each charm is base angles of an isosceles triangle are congruent. So an isosceles triangle, then we can determine the side △ △ AEB is equiangular and equilateral. AEB ≅ lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, △DEC by SAS, so AB ≅ EC. the perimeter of each triangle is 9 cm.

ANSWER: To find how many charms can be made, divide 63 a. B cm by 9 cm. So, the jeweler can make 7 charms. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB 7 are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 55. In the figure below, NDG LGD. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

54. A jeweler bends wire to make charms in the shape 216 ÷ 3 = 72 of isosceles triangles. The base of the triangle ANSWER: measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made 72 from a piece of wire that is 63 centimeters long? 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of Because we know that the shape of each charm is △JKL? an isosceles triangle, then we can determine the side SOLUTION: lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, The perimeter can be found by adding all three side the perimeter of each triangle is 9 cm. lengths. In this situation, since the triangle is

isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. To find how many charms can be made, divide 63

cm by 9 cm. So, the jeweler can make 7 charms. Because the x-terms cancel each other out: 2x + 2x ANSWER: – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 7 16.

55. In the figure below, NDG LGD. So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

In an isosceles triangle, the base angles are a. What is m∠L? congruent. b. What is the value of x?

c. What is m∠DGN? So, 5x + 5 = 7x – 19. SOLUTION: a. In this figure, angles N and L are congruent so Simplify.

m L = 36°. ∠ –2x = –24 b. Because we know two angle measures, and that x = 12 the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. ANSWER: c. ∠DGN is congruent to x°, so m∠DGN is 54°. 12 ANSWER: 59. QRV is an equilateral triangle. What additional a. 36 information would be enough to prove VRT is an b. 54 equilateral triangle? c. 54 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. Consider each choice. So, the perimeter of △JKL is 16. A does not provide enough information to ANSWER: determine how sides VT and TR are related. 16 58. △ABC is an isosceles triangle with base angles B If we know that m∠VRT = 60, we still would not measuring (5x + 5)° and (7x – 19)°. Find the value of have enough information to prove that the other two x. measures are 60 degrees.

SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12 If , then m∠RVT = 60 because ANSWER: ∠QVR + ∠RVT + ∠TVU = 180. Since , 12 by the Isosceles Triangle Theorem. 59. QRV is an equilateral triangle. What additional So, . Therefore, is equilateral by Theorem 4.3. information would be enough to prove VRT is an

equilateral triangle? D Since we do not have any measures to indicate

that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

Refer to the figure.

SOLUTION: By the Triangle Angle-Sum Theorem,

1. If name two congruent angles. SOLUTION: Isosceles Triangle Theorem states that if two sides Since the measures of all the three angles are 60°; of the triangle are congruent, then the angles opposite the triangle must be equiangular. All the equiangular those sides are congruent. triangles are equilateral. Therefore, FH = GH = 12. Therefore, in triangle ABC, ANSWER: ANSWER: 12 BAC and BCA 4. m MRP 2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC,

ANSWER: SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral Find each measure. triangle measures 60°. Therefore, m MRP = 60°. 3. FH ANSWER:

60 SENSE-MAKING Find the value of each variable.

SOLUTION: By the Triangle Angle-Sum Theorem, 5. SOLUTION: In the figure, . Therefore, triangle RST is Since the measures of all the three angles are 60°; an isosceles triangle. the triangle must be equiangular. All the equiangular By the Converse of Isosceles Triangle Theorem, triangles are equilateral. Therefore, FH = GH = 12. ANSWER: That is, . 12

4. m MRP

ANSWER: 12

SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral 6. triangle measures 60°. Therefore, m MRP = 60°. SOLUTION: ANSWER: In the figure, Therefore, triangle WXY is 60 an isosceles triangle. SENSE-MAKING Find the value of each By the Isosceles Triangle Theorem, . variable.

ANSWER: 5. 16 SOLUTION: 7. PROOF Write a two-column proof. In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem, Given: is isosceles; bisects ABC. Prove: That is, .

ANSWER: 12 SOLUTION:

6. SOLUTION: ANSWER: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: 8. ROLLER COASTERS The roller coaster track 16 appears to be composed of congruent triangles. A portion of the track is shown. 7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove:

a. If and are perpendicular to is SOLUTION: isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a. ANSWER:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The

Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = a. If and are perpendicular to is 3 m. isosceles with base , and prove that ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the a. distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure. b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. 9. If name two congruent angles.

ANSWER: SOLUTION: a. By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. b. The Pythagorean Theorem tells us that SOLUTION: By CPCTC we know By the Isosceles Triangle Theorem, in triangle ACD, that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ANSWER: ACD and ADC Refer to the figure. 12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

9. If name two congruent angles. ANSWER:

SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, 13. If BCF BFC, name two congruent segments.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in ABE and AEB triangle BCF,

ANSWER: 10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in 14. If name two congruent angles. triangle ABF, SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: 11. If name two congruent angles. AFH and AHF SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, Find each measure. 15. m BAC

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: SOLUTION: Here By the Converse of Isosceles Triangle Theorem, in By Isosceles Triangle Theorem, . triangle ADE, Apply the Triangle Sum Theorem. ANSWER:

13. If BCF BFC, name two congruent segments. SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in 60 triangle BCF, 16. m SRT ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, SOLUTION: Given: By Isosceles Triangle Theorem, . ANSWER:

AFH and AHF Apply Triangle Angle-Sum Theorem.

Find each measure. 15. m BAC

ANSWER: 65 17. TR SOLUTION: Here By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem. SOLUTION: Since the triangle is isosceles, Therefore,

ANSWER: 60

16. m SRT All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER:

4 SOLUTION: Given: 18. CB By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in 65 triangle ABC, That is, CB = 3. 17. TR ANSWER: 3 REGULARITY Find the value of each variable.

SOLUTION: Since the triangle is isosceles, Therefore, 19. SOLUTION:

Since all the angles are congruent, the sides are also congruent to each other. Therefore,

All the angles are congruent. Therefore, it is an Solve for x. equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: ANSWER: 4 x = 5 18. CB

20. SOLUTION: SOLUTION: Given: By the Converse of Isosceles Triangle Theorem, in By the Isosceles Triangle Theorem, triangle ABC, That is, CB = 3. And ANSWER: We know 3 that, REGULARITY Find the value of each variable.

ANSWER: x = 13 19. SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, Solve for x. 21. SOLUTION:

Given: By the Isosceles Triangle Theorem, ANSWER: That is, x = 5

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 2: 20. SOLUTION: Add the equations 1 and 2. Given: By the Isosceles Triangle Theorem,

And We know that, Substitute the value of x in one of the two equations to find the value of y.

ANSWER: x = 13

ANSWER: 21. x = 11, y = 11 SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, 22. Then, let Equation 1 be: . SOLUTION:

Given: By the Triangle Sum Theorem, we can find Equation By the Triangle Angle-Sum 2: Theorem,

So, the triangle is an equilateral triangle. Add the equations 1 and 2. Therefore,

Set up two equations to solve for x and y.

Substitute the value of x in one of the two equations ANSWER:

to find the value of y. x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

ANSWER: x = 11, y = 11

SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH 22. and HKL are supplementary and HLK and SOLUTION: MLH are supplementary. From the Isosceles Given: Triangle Theorem, we know that HJK HML. By the Triangle Angle-Sum Because is an equilateral triangle, we know Theorem, HLK LKH KHL and JKH, HKL and HLK, So, the triangle is an equilateral triangle. MLH are supplementary, and HKL HLK, Therefore, we know JKH MLH by the Congruent Supplements Theorem. By AAS, Set up two equations to solve for x and y. By CPCTC, JHK MHL.

ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and ANSWER: MLH are supplementary. From the Isosceles x = 1.4, y = –1 Triangle Theorem, we know that HJK HML. PROOF Write a paragraph proof. Because is an equilateral triangle, we know 23. Given: is isosceles, and is HLK LKH KHL and equilateral. JKH and HKL are supplementary JKH, HKL and HLK, and HLK and MLH are supplementary. MLH are supplementary, and HKL HLK, Prove: JHK MHL we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

24. Given: W is the midpoint of

Q is the midpoint of SOLUTION: Prove: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and

JKH, HKL and HLK, MLH are supplementary, and HKL HLK, SOLUTION: we know JKH MLH by the Congruent Proof: We are given W is the midpoint of Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH ∠MLH by the Congruent Proof: We are given W is the midpoint of Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, because Q is the midpoint of The 24. Given: Segment Addition Postulate gives us XW + WY = XY W is the midpoint of and XQ + QZ = XZ. Substitution gives XW + WY = Q is the midpoint of XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Prove: If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC.

by the Reflexive Property. By SAS, a. Elisa estimates m BAC to be 50. Based on this

So, by CPCTC. estimate, what is m∠ABC? Explain.

ANSWER: b. If show that is isosceles. c. If and show that is Proof: We are given W is the midpoint of equilateral. and Q is the midpoint of Because W is the d. If is isosceles, what is the minimum midpoint of we know that Similarly, information needed to prove that because Q is the midpoint of The Explain your reasoning. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = SOLUTION: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The a. Given: Isosceles Triangle Theorem says XYZ XZY. By Isosceles Triangle Theorem, .

by the Reflexive Property. By SAS, Apply Triangle Sum Theorem.

So, by CPCTC.

25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either b. side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that but

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

b. d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the c. triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. d. One pair of congruent corresponding sides and b. one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA

c. or SAS. 26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by ANSWER: bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I 27. CONSTRUCTION Construct three different measured both legs for each triangle. Because AB = isosceles right triangles. Explain your method. Then AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = verify your constructions using measurement and 2.3 cm, the triangles are isosceles. I used a mathematics. protractor to confirm that A, D, and G are all right angles. SOLUTION: ANSWER:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to Sample answer: I constructed a pair of perpendicular mark points equidistant from their intersection. I segments and then used the same compass setting to measured both legs for each triangle. Because AB = mark points equidistant from their intersection. I AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = measured both legs for each triangle. Because AB = 2.3 cm, the triangles are isosceles. I used a AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = protractor to confirm that A, D, and G are all 2.3 cm, the triangles are isosceles. I used a 4-6 Isoscelesright angles. and Equilateral Triangles protractor to confirm that A, D, and G are all right angles. ANSWER: 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, Solve for x

REGULARITY Find each measure.

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 29. m∠CAD protractor to confirm that A, D, and G are all SOLUTION: right angles. From the figure, 28. PROOF Based on your construction in Exercise 27, Therefore, is an isosceles triangle. make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. By the Triangle Sum Theorem, SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is ANSWER: an isosceles triangle. Let the measure of each acute 44 angle be x. The acute angles of a right triangle are complementary, Solve for x 30. m ACD SOLUTION: We know ANSWER: that Conjecture: The measures of the base angles of an Therefore, . eSolutionsisoscelesManual right- Powered triangleby Cognero are 45. Page 10 Proof: The base angles are congruent because it is ANSWER: an isosceles triangle. Let the measure of each acute 44 angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. 31. m ACB REGULARITY Find each measure. SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

29. m∠CAD SOLUTION: ANSWER: From the figure, 136 Therefore, is an isosceles triangle. 32. m ABC

SOLUTION: By the Triangle Sum Theorem, From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, ANSWER: 44

30. m ACD ANSWER: 22 SOLUTION: We know 33. FITNESS In the diagram, the rider will use his bike that to hop across the tops of each of the concrete solids Therefore, . shown. If each triangle is isosceles with vertex angles G, H, and J, and G ANSWER: H, and H J, show that the distance 44 from B to F is three times the distance from D to F. 31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that SOLUTION:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER:

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION:

ANSWER:

34. Given: is isosceles; Prove: X and YZV are complementary. PROOF Write a two-column proof of each

corollary or theorem. 35. Corollary 4.3 SOLUTION:

SOLUTION:

ANSWER:

36. Corollary 4.4 PROOF Write a two-column proof of each SOLUTION: corollary or theorem. 35. Corollary 4.3 SOLUTION:

ANSWER:

37. Theorem 4.11 SOLUTION:

36. Corollary 4.4 ANSWER: SOLUTION:

Find the value of each variable.

ANSWER:

38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

37. Theorem 4.11 SOLUTION: Note that x can equal –8 here because .

ANSWER: 3

39. ANSWER: SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Find the value of each variable.

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

38. ANSWER: SOLUTION: 14 By the converse of Isosceles Triangle theorem, GAMES Use the diagram of a game timer shown to find each measure.

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER:

3 40. m LPM SOLUTION: Angles at a point in a straight line add up to 180 .

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . Substitute x = 45 in (3x – 55) to find

The interior angles of a triangle add up to 180 .

ANSWER: 80

The measure of an angle cannot be negative, and 2(– 41. m LMP 18) – 5 = –41, so y = 14. SOLUTION: ANSWER: Since the triangle LMP is isosceles, 14 Angles at a point in a straight line GAMES Use the diagram of a game timer add up to 180 . shown to find each measure.

Substitute x = 45 in (3x – 55) to find

Therefore, 40. m LPM ANSWER: SOLUTION: 80 Angles at a point in a straight line add up to 180 . 42. m JLK SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP ANSWER: SOLUTION: 20 Since the triangle LMP is isosceles, 43. m JKL Angles at a point in a straight line add up to 180 . SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that, Substitute x = 45 in (3x – 55) to find

So, Therefore, In ANSWER: 80 Since the triangle JKL is isosceles,

42. m JLK SOLUTION: Vertical angles are congruent. Therefore, ANSWER:

First we need to find 44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the We know that, interior angles of an isosceles triangle given the measure of one exterior angle.

So,

ANSWER: 20

First we need to find b. Tabular Use a protractor to measure and record We know that, m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables. So, c. Verbal Explain how you used m 1 to find the In measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. Since the triangle JKL is isosceles, d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures. ANSWER: 80 SOLUTION: 44. MULTIPLE REPRESENTATIONS In this a. problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for b. the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = b. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = ANSWER: 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

ANSWER: Sometimes; only if the measure of the vertex angle is even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: ANSWER: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even.

ERROR ANALYSIS PRECISION Determine whether the following 48. Alexis and Miguela are statements are sometimes, always, or never true. finding m G in the figure shown. Alexis says that Explain. m G = 35, while Miguela says that m G = 60. Is 46. If the measure of the vertex angle of an isosceles either of them correct? Explain your reasoning. triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. SOLUTION: ANSWER: Neither of them is correct. This is an isosceles Sometimes; only if the measure of the vertex angle is triangle with a vertex angle of 70. Since this is an even. isosceles triangle, the base angles are congruent.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 ANSWER: (measure of the base angle) will be even. Neither; m G = or 55. ANSWER: Never; the measure of the vertex angle will be 180 – 49. OPEN-ENDED If possible, draw an isosceles 2(measure of the base angle) so if the base angles triangle with base angles that are obtuse. If it is not are integers, then 2(measure of the base angle) will possible, explain why not. be even and 180 – 2(measure of the base angle) will be even. SOLUTION: It is not possible because a triangle cannot have more 48. ERROR ANALYSIS Alexis and Miguela are than one obtuse angle. finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning. It is not possible because a triangle cannot have more than one obtuse angle. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180, SOLUTION:

Neither of them is correct. This is an isosceles

triangle with a vertex angle of 70. Since this is an Since is isosceles, isosceles triangle, the base angles are congruent. And given that m∠B = 90. Therefore,

Construct the triangle with the angles measures 90, 45, 45.

ANSWER:

Neither; m G = or 55.

49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not ANSWER: possible, explain why not. SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle.

ANSWER: It is not possible because a triangle cannot have more 51. WRITING IN MATH How can triangle than one obtuse angle. classifications help you prove triangle congruence? 50. REASONING In isosceles m B = 90. SOLUTION: Draw the triangle. Indicate the congruent sides and If a triangle is already classified, you can use the label each angle with its measure. previously proven properties of that type of triangle SOLUTION: in the proof. For example, if you know that a The sum of the angle measures in a triangle must be triangle is an equilateral triangle, you can use 180, Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. Since is isosceles, ANSWER: And given that m∠B = 90. Therefore, Sample answer: If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof. 52. MULTI-STEP △AED and △BEC are congruent Construct the triangle with the angles measures 90, and isosceles. 45, 45.

ANSWER: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC

D 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? b. If m∠EAB = 40, what is the measure of ∠DCE?

SOLUTION: c. If m∠ADE = 50, what is the measure of ∠BEC? If a triangle is already classified, you can use the previously proven properties of that type of triangle d. What measure would ∠AEB need to be so that in the proof. For example, if you know that a ? triangle is an equilateral triangle, you can use SOLUTION: Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. a. Of the statements given, only choice B is not necessarily true from the given information. ANSWER: b. The angles are congruent, so ∠DCE is 40°. Sample answer: If a triangle is already classified, c. The sum of the measures of the angles of a you can use the previously proven properties of that triangle is 180, so ∠BEC must be 80°. type of triangle in the proof. Doing this can save you d. 60; All three angles in △AEB are 60° because steps when writing the proof. base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ 52. MULTI-STEP △AED and △BEC are congruent △DEC by SAS, so AB ≅ EC. and isosceles. ANSWER: a. B b. 40 c. 80 d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and

equilateral. △AEB △DEC by SAS, so AB EC. a. Which statement is not necessarily true from the ≅ ≅ information given? 53. What is m∠FEG?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? A 34 c. If m∠ADE = 50, what is the measure of ∠BEC? B 56 C 60 d. What measure would ∠AEB need to be so that D 62 ? E 124

SOLUTION: a. Of the statements given, only choice B is not SOLUTION: necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ Consider triangle DEH. Since one angle is 56°, and △DEC by SAS, so AB ≅ EC. the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. ANSWER: So, m∠FEG is , or 62. The correct answer is a. B choice D. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB D are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made 53. What is m∠FEG? from a piece of wire that is 63 centimeters long? SOLUTION: Because we know that the shape of each charm is an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm.

A 34 To find how many charms can be made, divide 63 B 56 cm by 9 cm. So, the jeweler can make 7 charms. C 60 ANSWER: D 62 E 124 7 55. In the figure below, NDG LGD. SOLUTION:

Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two a. What is m∠L? angles would be 180 – 56 or 124°. b. What is the value of x? So, m∠FEG is , or 62. The correct answer is c. What is m∠DGN? choice D. SOLUTION: ANSWER: a. In this figure, angles N and L are congruent so D m∠L = 36°. b. Because we know two angle measures, and that 54. A jeweler bends wire to make charms in the shape the sum of the measures of the angles of a triangle is of isosceles triangles. The base of the triangle 180, then we can solve 36 + 90 + x = 180, or x = 54. measures 2 centimeters and another side measures c. ∠DGN is congruent to x°, so m∠DGN is 54°. 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? ANSWER: SOLUTION: a. 36 b. Because we know that the shape of each charm is 54 an isosceles triangle, then we can determine the side c. 54 lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, 56. An equilateral triangle has a perimeter of 216 inches. the perimeter of each triangle is 9 cm. What is the length of one side of the triangle?

To find how many charms can be made, divide 63 SOLUTION: cm by 9 cm. So, the jeweler can make 7 charms. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 ANSWER: to find the length of one side. 7 216 ÷ 3 = 72 55. In the figure below, NDG LGD. ANSWER: 72

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION:

The perimeter can be found by adding all three side a. What is m∠L? lengths. In this situation, since the triangle is b. What is the value of x? isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. c. What is m∠DGN? SOLUTION: Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or a. In this figure, angles N and L are congruent so 16. m∠L = 36°. b. Because we know two angle measures, and that So, the perimeter of △JKL is 16. the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. ANSWER: c. ∠DGN is congruent to x°, so m∠DGN is 54°. 16

ANSWER: 58. △ABC is an isosceles triangle with base angles a. 36 measuring (5x + 5)° and (7x – 19)°. Find the value of b. 54 x. c. 54 SOLUTION: 56. An equilateral triangle has a perimeter of 216 inches. In an isosceles triangle, the base angles are What is the length of one side of the triangle? congruent.

SOLUTION: So, 5x + 5 = 7x – 19. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 Simplify. to find the length of one side. –2x = –24 216 ÷ 3 = 72 x = 12

ANSWER: ANSWER: 72 12

57. △JKL is an isosceles triangle, and . JK = 59. QRV is an equilateral triangle. What additional 2x + 3, and KL = 10 – 4x. What is the perimeter of information would be enough to prove VRT is an △JKL? equilateral triangle? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

△ So, the perimeter of JKL is 16. A ANSWER: B m∠VRT = 60 16 C QRV VTU D VRT RST 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of SOLUTION: x. SOLUTION: In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 Consider each choice. x = 12 A does not provide enough information to ANSWER: determine how sides VT and TR are related. 12 B If we know that m∠VRT = 60, we still would not 59. QRV is an equilateral triangle. What additional have enough information to prove that the other two information would be enough to prove VRT is an measures are 60 degrees. equilateral triangle? C

A If , then m∠RVT = 60 because B m VRT = 60 ∠ ∠QVR + ∠RVT + ∠TVU = 180. Since , C QRV VTU by the Isosceles Triangle Theorem. D VRT RST So, . SOLUTION: Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER:

C Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

Refer to the figure. SOLUTION: By the Triangle Angle-Sum Theorem,

1. If name two congruent angles. Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular SOLUTION: triangles are equilateral. Therefore, FH = GH = 12. Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite ANSWER: those sides are congruent. 12 Therefore, in triangle ABC, 4. m MRP ANSWER: BAC and BCA

2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, SOLUTION: in triangle EAC, Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral ANSWER: triangle measures 60°. Therefore, m MRP = 60°.

ANSWER: Find each measure. 60 3. FH SENSE-MAKING Find the value of each

variable.

5. SOLUTION: By the Triangle Angle-Sum Theorem, SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

Since the measures of all the three angles are 60°;

the triangle must be equiangular. All the equiangular That is, . triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

4. m MRP ANSWER: 12

SOLUTION: 6. Since all the sides are congruent, is an SOLUTION: equilateral triangle. Each angle of an equilateral In the figure, Therefore, triangle WXY is triangle measures 60°. Therefore, m MRP = 60°. an isosceles triangle. ANSWER: By the Isosceles Triangle Theorem, .

SENSE-MAKING Find the value of each variable.

ANSWER: 16

7. PROOF Write a two-column proof. 5. SOLUTION: Given: is isosceles; bisects ABC. In the figure, . Therefore, triangle RST is Prove: an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION:

ANSWER: 12

ANSWER: 6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A

ANSWER: portion of the track is shown.

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove:

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the SOLUTION: distance between and Explain your reasoning.

SOLUTION: a.

ANSWER:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: a. If and are perpendicular to is a. isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The 9. If name two congruent angles. Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = SOLUTION: 3 m. By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ANSWER: a. ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, b. The Pythagorean Theorem tells us that By CPCTC we know ANSWER: that VT = 1.5 m. The Seg. Add. Post. says QV + ACD and ADC VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. 12. If DAE DEA, name two congruent segments. Refer to the figure. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

9. If name two congruent angles. 13. If BCF BFC, name two congruent segments. SOLUTION: SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. 14. If name two congruent angles. SOLUTION: SOLUTION: By the Converse of Isosceles Triangle Theorem, in By the Isosceles Triangle Theorem, in triangle AFH, triangle ABF,

ANSWER: ANSWER: AFH and AHF

Find each measure. 11. If name two congruent angles. 15. m BAC SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC SOLUTION: 12. If DAE DEA, name two congruent Here segments. By Isosceles Triangle Theorem, .

SOLUTION: Apply the Triangle Sum Theorem. By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

ANSWER: 13. If BCF BFC, name two congruent segments. 60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER:

14. If name two congruent angles. SOLUTION: Given: SOLUTION: By Isosceles Triangle Theorem, . By the Isosceles Triangle Theorem, in triangle AFH, Apply Triangle Angle-Sum Theorem. ANSWER: AFH and AHF

Find each measure. 15. m BAC ANSWER: 65 17. TR

SOLUTION: Here By Isosceles Triangle Theorem, . SOLUTION:

Apply the Triangle Sum Theorem. Since the triangle is isosceles, Therefore,

ANSWER: 60 All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, 16. m SRT

ANSWER: 4 18. CB

SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3. ANSWER: 65 ANSWER: 3 17. TR REGULARITY Find the value of each variable.

19. SOLUTION: SOLUTION:

Since the triangle is isosceles, Since all the angles are congruent, the sides are also

Therefore, congruent to each other.

Therefore, Solve for x.

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, ANSWER: x = 5 ANSWER: 4 18. CB 20. SOLUTION: Given: By the Isosceles Triangle Theorem,

SOLUTION: By the Converse of Isosceles Triangle Theorem, in And triangle ABC, We know That is, CB = 3. that, ANSWER: 3 REGULARITY Find the value of each variable. ANSWER: x = 13

19. SOLUTION: Since all the angles are congruent, the sides are also 21. congruent to each other. SOLUTION: Therefore, Solve for x. Given: By the Isosceles Triangle Theorem,

That is,

ANSWER: Then, let Equation 1 be: . x = 5 By the Triangle Sum Theorem, we can find Equation 2:

Add the equations 1 and 2. 20. SOLUTION: Given: By the Isosceles Triangle Theorem,

And Substitute the value of x in one of the two equations We know to find the value of y. that,

ANSWER: x = 13

ANSWER: x = 11, y = 11

21. SOLUTION: Given: By the Isosceles Triangle Theorem, 22.

SOLUTION: That is, Given: By the Triangle Angle-Sum Then, let Equation 1 be: . Theorem,

By the Triangle Sum Theorem, we can find Equation 2: So, the triangle is an equilateral triangle. Therefore,

Set up two equations to solve for x and y. Add the equations 1 and 2.

ANSWER: x = 1.4, y = –1

Substitute the value of x in one of the two equations PROOF Write a paragraph proof. to find the value of y. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

ANSWER:

x = 11, y = 11 SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles 22. Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know SOLUTION: HLK LKH KHL and Given: JKH, HKL and HLK, By the Triangle Angle-Sum MLH are supplementary, and HKL HLK, Theorem, we know JKH MLH by the Congruent Supplements Theorem. By AAS, So, the triangle is an equilateral triangle. By CPCTC, JHK MHL. Therefore, ANSWER: Set up two equations to solve for x and y. Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. ANSWER: Because is an equilateral triangle, we know x = 1.4, y = –1 HLK LKH KHL and JKH, HKL and HLK, PROOF Write a paragraph proof. MLH are supplementary, and HKL HLK, 23. Given: is isosceles, and is we know JKH ∠MLH by the Congruent equilateral. JKH and HKL are supplementary Supplements Theorem. By AAS, and HLK and MLH are supplementary. By CPCTC, JHK MHL. Prove: JHK MHL

24. Given: W is the midpoint of Q is the midpoint of Prove:

SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know SOLUTION: HLK LKH KHL and Proof: We are given W is the midpoint of JKH, HKL and HLK, and Q is the midpoint of Because W is the MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent midpoint of we know that Similarly, Supplements Theorem. By AAS, because Q is the midpoint of The By CPCTC, JHK MHL. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = ANSWER: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Proof: We are given that is an isosceles If we divide each side by 2, we have WY = QZ. The triangle and is an equilateral triangle, JKH Isosceles Triangle Theorem says XYZ XZY. and HKL are supplementary and HLK and by the Reflexive Property. By SAS, MLH are supplementary. From the Isosceles So, by CPCTC. Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know ANSWER: HLK LKH KHL and Proof: We are given W is the midpoint of JKH, HKL and HLK, MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH ∠MLH by the Congruent midpoint of we know that Similarly, Supplements Theorem. By AAS, because Q is the midpoint of The By CPCTC, JHK MHL. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. 24. Given: If we divide each side by 2, we have WY = QZ. The

W is the midpoint of Isosceles Triangle Theorem says XYZ XZY.

Q is the midpoint of by the Reflexive Property. By SAS,

Prove: So, by CPCTC.

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The a. Elisa estimates m BAC to be 50. Based on this Isosceles Triangle Theorem says XYZ XZY. estimate, what is m∠ABC? Explain. by the Reflexive Property. By SAS, b. If show that is isosceles. So, by CPCTC. c. If and show that is equilateral. ANSWER: d. If is isosceles, what is the minimum Proof: We are given W is the midpoint of information needed to prove that Explain your reasoning. and Q is the midpoint of Because W is the

midpoint of we know that Similarly, SOLUTION: because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY a. Given: and XQ + QZ = XZ. Substitution gives XW + WY = By Isosceles Triangle Theorem, . XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Apply Triangle Sum Theorem. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS,

So, by CPCTC. b. 25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that but

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since b. you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

c. ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS. d. One pair of congruent corresponding sides and ANSWER: one pair of congruent corresponding angles; since a. 65°; Because is isosceles, ABC you know that the triangle is isosceles, if one leg is ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. congruent to a leg of then you know that b. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

CHIMNEYS 26. In the picture, and c. is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION: 27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to Sample answer: I constructed a pair of perpendicular mark points equidistant from their intersection. I segments and then used the same compass setting to measured both legs for each triangle. Because AB = mark points equidistant from their intersection. I AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = measured both legs for each triangle. Because AB = 2.3 cm, the triangles are isosceles. I used a AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = protractor to confirm that A, D, and G are all 2.3 cm, the triangles are isosceles. I used a right angles. protractor to confirm that A, D, and G are all right angles. 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship ANSWER: between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, Solve for x

REGULARITY Find each measure.

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I 29. m∠CAD measured both legs for each triangle. Because AB = SOLUTION: AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a From the figure, protractor to confirm that A, D, and G are all Therefore, is an isosceles triangle. right angles.

28. PROOF Based on your construction in Exercise 27, By the Triangle Sum Theorem, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: ANSWER: Conjecture: The measures of the base angles of an 44 isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute 30. m ACD angle be x. The acute angles of a right triangle are SOLUTION: complementary, Solve for x We know that Therefore, . ANSWER: ANSWER: Conjecture: The measures of the base angles of an 44 isosceles right triangle are 45. Proof: The base angles are congruent because it is 31. m ACB an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. The angles in a straight line add to 180°. Therefore, REGULARITY Find each measure.

We know that

ANSWER:

29. m∠CAD 136 SOLUTION: 32. m ABC

From the figure, SOLUTION: Therefore, is an isosceles triangle. From the figure, So, is an isosceles

triangle. Therefore,

By the Triangle Sum Theorem,

By the Triangle Angle Sum Theorem,

ANSWER: ANSWER: 4-6 Isosceles44 and Equilateral Triangles 22

30. m ACD 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids SOLUTION: shown. If each triangle is isosceles with vertex We know angles G, H, and J, and G that H, and H J, show that the distance

Therefore, . from B to F is three times the distance from D to F. ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°.

Therefore, SOLUTION: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, ANSWER: By the Triangle Angle Sum Theorem,

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F. 34. Given: is isosceles; Prove: X and YZV are complementary.

eSolutionsSOLUTION: Manual - Powered by Cognero SOLUTION: Page 11

ANSWER: ANSWER:

PROOF Write a two-column proof of each 34. Given: is isosceles; corollary or theorem. 35. Corollary 4.3 Prove: X and YZV are complementary. SOLUTION:

SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

ANSWER:

ANSWER: 37. Theorem 4.11 SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

Find the value of each variable.

38. ANSWER: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because . 37. Theorem 4.11 SOLUTION: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is ANSWER: equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– Find the value of each variable. 18) – 5 = –41, so y = 14. ANSWER: 14 38. GAMES Use the diagram of a game timer shown to find each measure. SOLUTION:

By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because

. 40. m LPM ANSWER: SOLUTION: 3 Angles at a point in a straight line add up to 180 .

39. Substitute x = 45 in (3x – 55) to find SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 . ANSWER:

41. m LMP SOLUTION: The measure of an angle cannot be negative, and 2(– Since the triangle LMP is isosceles, 18) – 5 = –41, so y = 14. Angles at a point in a straight line ANSWER: add up to 180 . 14 GAMES Use the diagram of a game timer shown to find each measure.

Substitute x = 45 in (3x – 55) to find

Therefore,

ANSWER: 80 40. m LPM SOLUTION: 42. m JLK Angles at a point in a straight line add up to 180 . SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

Substitute x = 45 in (3x – 55) to find We know that,

So, ANSWER: 80 ANSWER: 20

41. m LMP 43. m JKL SOLUTION: SOLUTION: Since the triangle LMP is isosceles, Vertical angles are congruent. Therefore, Angles at a point in a straight line

add up to 180 . First we need to find

We know that,

Substitute x = 45 in (3x – 55) to find So,

Therefore, Since the triangle JKL is isosceles,

ANSWER: 80

42. m JLK ANSWER: SOLUTION: 80 Vertical angles are congruent. Therefore, 44. MULTIPLE REPRESENTATIONS In this

problem, you will explore possible measures of the interior angles of an isosceles triangle given the First we need to find measure of one exterior angle.

We know that,

So,

ANSWER: a. Geometric Use a ruler and a protractor to draw 20 three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of 43. m JKL the base angles, and labeling as shown. SOLUTION: Vertical angles are congruent. Therefore, b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the First we need to find measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same We know that, measures. Organize your results in two tables.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how So, you used m 2 to find these same measures.

In d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m Since the triangle JKL is isosceles, 2 = x, write an expression for these same angle measures.

SOLUTION: ANSWER: a. 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how b. you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = b. 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

b. SOLUTION:

c. 5 is supplementary to 1, so m 5 = 180 – ANSWER: m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral PRECISION Determine whether the following and ZWP WJM JZL, prove that statements are sometimes, always, or never true.

Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. SOLUTION: ANSWER: Sometimes; only if the measure of the vertex angle is even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is PRECISION Determine whether the following either of them correct? Explain your reasoning. statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, SOLUTION: vertex angle = 50, base angles = 65; Neither of them is correct. This is an isosceles vertex angle = 55, base angles = 62.5. triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. ANSWER:

Sometimes; only if the measure of the vertex angle is

even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: The statement is never true. The measure of the ANSWER: vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 Neither; m G = or 55. (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even. 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not ANSWER: possible, explain why not. Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles SOLUTION: are integers, then 2(measure of the base angle) will It is not possible because a triangle cannot have more be even and 180 – 2(measure of the base angle) will than one obtuse angle. be even. ANSWER: 48. ERROR ANALYSIS Alexis and Miguela are It is not possible because a triangle cannot have more finding m G in the figure shown. Alexis says that than one obtuse angle. m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, SOLUTION: And given that m∠B = 90. Neither of them is correct. This is an isosceles Therefore,

triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

Construct the triangle with the angles measures 90, 45, 45.

ANSWER:

Neither; m G = or 55. ANSWER: 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. SOLUTION:

It is not possible because a triangle cannot have more than one obtuse angle. 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? ANSWER: It is not possible because a triangle cannot have more SOLUTION: than one obtuse angle. If a triangle is already classified, you can use the previously proven properties of that type of triangle 50. REASONING In isosceles m B = 90. in the proof. For example, if you know that a Draw the triangle. Indicate the congruent sides and label each angle with its measure. triangle is an equilateral triangle, you can use Corollary 4.3 and 4.4 in the proof. Doing this can SOLUTION: save you steps when writing the proof. The sum of the angle measures in a triangle must be 180, ANSWER: Sample answer: If a triangle is already classified, Since is isosceles, you can use the previously proven properties of that And given that m∠B = 90. type of triangle in the proof. Doing this can save you Therefore, steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

Construct the triangle with the angles measures 90, 45, 45.

a. Which statement is not necessarily true from the information given?

ANSWER: A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE?

51. WRITING IN MATH How can triangle c. If m∠ADE = 50, what is the measure of ∠BEC? classifications help you prove triangle congruence? d. What measure would ∠AEB need to be so that SOLUTION: ? If a triangle is already classified, you can use the previously proven properties of that type of triangle SOLUTION: in the proof. For example, if you know that a a. Of the statements given, only choice B is not triangle is an equilateral triangle, you can use necessarily true from the given information. Corollary 4.3 and 4.4 in the proof. Doing this can b. The angles are congruent, so ∠DCE is 40°. save you steps when writing the proof. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. ANSWER: d. 60; All three angles in △AEB are 60° because Sample answer: If a triangle is already classified, base angles of an isosceles triangle are congruent. So you can use the previously proven properties of that △AEB is equiangular and equilateral. △AEB ≅ type of triangle in the proof. Doing this can save you △DEC by SAS, so AB ≅ EC. steps when writing the proof. ANSWER: 52. MULTI-STEP △AED and △BEC are congruent a. B and isosceles. b. 40 c. 80 d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m FEG? ∠

a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC

D A 34

B 56 b. If m∠EAB = 40, what is the measure of ∠DCE? C 60

D 62 c. If m∠ADE = 50, what is the measure of ∠BEC? E 124

d. What measure would ∠AEB need to be so that ? SOLUTION: SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. Consider triangle DEH. Since one angle is 56°, and d. 60; All three angles in △AEB are 60° because the triangle is isosceles, then the remaining two base angles of an isosceles triangle are congruent. So angles would be 180 – 56 or 124°. △AEB is equiangular and equilateral. △AEB ≅ So, m∠FEG is , or 62. The correct answer is △DEC by SAS, so AB ≅ EC. choice D.

ANSWER: ANSWER: a. B D b. 40 c. 80 54. A jeweler bends wire to make charms in the shape d. 60; Sample answer: All three angles in △AEB of isosceles triangles. The base of the triangle are 60° because base angles of an isosceles triangle measures 2 centimeters and another side measures are congruent. So △AEB is equiangular and 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. SOLUTION: 53. What is m∠FEG? Because we know that the shape of each charm is an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm.

To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms.

A 34 ANSWER: B 56 7 C 60 55. In the figure below, NDG LGD. D 62

E 124

SOLUTION:

a. What is m∠L? b. What is the value of x? Consider triangle DEH. Since one angle is 56°, and c. What is m∠DGN? the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. SOLUTION: So, m∠FEG is , or 62. The correct answer is a. In this figure, angles N and L are congruent so choice D. m∠L = 36°. b. Because we know two angle measures, and that ANSWER: the sum of the measures of the angles of a triangle is D 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle ANSWER: measures 2 centimeters and another side measures a. 36 3.5 centimeters. How many charms can be made b. 54 from a piece of wire that is 63 centimeters long? c. 54 SOLUTION: 56. An equilateral triangle has a perimeter of 216 inches. Because we know that the shape of each charm is What is the length of one side of the triangle? an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, SOLUTION: the perimeter of each triangle is 9 cm. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 To find how many charms can be made, divide 63 to find the length of one side. cm by 9 cm. So, the jeweler can make 7 charms. 216 ÷ 3 = 72 ANSWER: 7 ANSWER: 72 55. In the figure below, NDG LGD. 57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

a. What is m∠L? Because the x-terms cancel each other out: 2x + 2x b. What is the value of x? – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or c. What is m∠DGN? 16. SOLUTION: So, the perimeter of △JKL is 16. a. In this figure, angles N and L are congruent so m∠L = 36°. ANSWER: b. Because we know two angle measures, and that 16 the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. 58. △ABC is an isosceles triangle with base angles c. ∠DGN is congruent to x°, so m∠DGN is 54°. measuring (5x + 5)° and (7x – 19)°. Find the value of x. ANSWER: SOLUTION: a. 36 b. 54 In an isosceles triangle, the base angles are congruent. c. 54

56. An equilateral triangle has a perimeter of 216 inches. So, 5x + 5 = 7x – 19. What is the length of one side of the triangle? Simplify. SOLUTION: Because the triangle is equilateral that means each –2x = –24 side has the same length, so divide the perimeter by 3 x = 12 to find the length of one side. ANSWER: 216 ÷ 3 = 72 12

ANSWER: 59. QRV is an equilateral triangle. What additional 72 information would be enough to prove VRT is an equilateral triangle? 57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or A 16. B m∠VRT = 60

C So, the perimeter of △JKL is 16. QRV VTU D VRT RST ANSWER: SOLUTION: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION: In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19. Consider each choice. Simplify. A does not provide enough information to –2x = –24 determine how sides VT and TR are related. x = 12 B If we know that m∠VRT = 60, we still would not ANSWER: have enough information to prove that the other two 12 measures are 60 degrees.

59. QRV is an equilateral triangle. What additional C information would be enough to prove VRT is an equilateral triangle?

If , then m∠RVT = 60 because

∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. A B m∠VRT = 60 So, . C QRV VTU Therefore, is equilateral by Theorem 4.3.

D VRT RST D Since we do not have any measures to indicate SOLUTION: that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

Refer to the figure.

SOLUTION: By the Triangle Angle-Sum Theorem,

1. If name two congruent angles. Since the measures of all the three angles are 60°; SOLUTION: the triangle must be equiangular. All the equiangular Isosceles Triangle Theorem states that if two sides triangles are equilateral. Therefore, FH = GH = 12. of the triangle are congruent, then the angles opposite those sides are congruent. ANSWER: Therefore, in triangle ABC, 12

ANSWER: 4. m MRP BAC and BCA

2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, SOLUTION: in triangle EAC, Since all the sides are congruent, is an ANSWER: equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: Find each measure. 3. FH 60

SENSE-MAKING Find the value of each variable.

SOLUTION: 5. By the Triangle Angle-Sum Theorem, SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem, Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12. That is, .

ANSWER: 12

4. m MRP ANSWER: 12

SOLUTION: 6. Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral SOLUTION:

triangle measures 60°. Therefore, m MRP = 60°. In the figure, Therefore, triangle WXY is ANSWER: an isosceles triangle.

60 By the Isosceles Triangle Theorem, .

SENSE-MAKING Find the value of each variable.

ANSWER: 16

5. 7. PROOF Write a two-column proof. SOLUTION: In the figure, . Therefore, triangle RST is Given: is isosceles; bisects ABC. an isosceles triangle. Prove: By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: ANSWER: 12

6. ANSWER: SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

8. ROLLER COASTERS The roller coaster track ANSWER: appears to be composed of congruent triangles. A portion of the track is shown. 16

7. PROOF Write a two-column proof.

Given: is isosceles; bisects ABC. Prove:

a. If and are perpendicular to is isosceles with base , and prove that

SOLUTION: b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

ANSWER:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. a. If and are perpendicular to is ANSWER: isosceles with base , and prove that a.

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. 9. If name two congruent angles. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, ANSWER: a. ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. SOLUTION:

b. The Pythagorean Theorem tells us that By the Isosceles Triangle Theorem, in triangle ACD,

By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + ANSWER: VT = QT. By substitution, we have 1.5 + 1.5 = QT. ACD and ADC So QT = 3 m. 12. If DAE DEA, name two congruent Refer to the figure. segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

9. If name two congruent angles. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ABE and AEB ANSWER:

10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: 14. If name two congruent angles. By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: ANSWER: AFH and AHF 11. If name two congruent angles. Find each measure. SOLUTION: 15. m BAC By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC 12. If DAE DEA, name two congruent SOLUTION: segments. Here SOLUTION: By Isosceles Triangle Theorem, .

By the Converse of Isosceles Triangle Theorem, in Apply the Triangle Sum Theorem. triangle ADE, ANSWER:

ANSWER: 13. If BCF BFC, name two congruent segments. 60 SOLUTION: By the Converse of Isosceles Triangle Theorem, in 16. m SRT triangle BCF,

ANSWER:

14. If name two congruent angles. SOLUTION: SOLUTION: Given: By the Isosceles Triangle Theorem, in triangle AFH, By Isosceles Triangle Theorem, .

ANSWER: Apply Triangle Angle-Sum Theorem.

AFH and AHF

Find each measure. 15. m BAC ANSWER: 65 17. TR

SOLUTION: Here By Isosceles Triangle Theorem, .

SOLUTION: Apply the Triangle Sum Theorem. Since the triangle is isosceles, Therefore,

ANSWER: 60 All the angles are congruent. Therefore, it is an equiangular triangle.

16. m SRT Since the equiangular triangle is an equilateral,

ANSWER: 4

18. CB SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, ANSWER: That is, CB = 3. 65 ANSWER: 17. TR 3 REGULARITY Find the value of each variable.

SOLUTION: 19. Since the triangle is isosceles, SOLUTION: Therefore, Since all the angles are congruent, the sides are also

congruent to each other. Therefore, Solve for x.

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, ANSWER: x = 5 ANSWER: 4 18. CB

20. SOLUTION: Given:

By the Isosceles Triangle Theorem, SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, And That is, CB = 3. We know that, ANSWER: 3 REGULARITY Find the value of each variable. ANSWER: x = 13

19. SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. 21.

Therefore, SOLUTION: Solve for x. Given: By the Isosceles Triangle Theorem,

That is, ANSWER: x = 5 Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 2:

20. Add the equations 1 and 2. SOLUTION: Given: By the Isosceles Triangle Theorem,

And We know Substitute the value of x in one of the two equations that, to find the value of y.

ANSWER: x = 13

ANSWER: x = 11, y = 11 21. SOLUTION: Given: By the Isosceles Triangle Theorem, 22. That is, SOLUTION:

Then, let Equation 1 be: . Given: By the Triangle Angle-Sum

By the Triangle Sum Theorem, we can find Equation Theorem, 2: So, the triangle is an equilateral triangle. Therefore,

Add the equations 1 and 2. Set up two equations to solve for x and y.

ANSWER:

x = 1.4, y = –1 Substitute the value of x in one of the two equations to find the value of y. PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

ANSWER:

x = 11, y = 11 SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and 22. MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. SOLUTION: Because is an equilateral triangle, we know Given: HLK LKH KHL and By the Triangle Angle-Sum JKH, HKL and HLK,

Theorem, MLH are supplementary, and HKL HLK,

we know JKH MLH by the Congruent

So, the triangle is an equilateral triangle. Supplements Theorem. By AAS,

Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y. ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles ANSWER: Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know x = 1.4, y = –1 HLK LKH KHL and PROOF Write a paragraph proof. JKH, HKL and HLK, 23. Given: is isosceles, and is MLH are supplementary, and HKL HLK, equilateral. JKH and HKL are supplementary we know JKH ∠MLH by the Congruent and HLK and MLH are supplementary. Supplements Theorem. By AAS, Prove: JHK MHL By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove: SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and SOLUTION: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY ANSWER: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = 24. Given: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ.

W is the midpoint of If we divide each side by 2, we have WY = QZ. The Q is the midpoint of Isosceles Triangle Theorem says XYZ XZY. Prove: by the Reflexive Property. By SAS, So, by CPCTC.

by the Reflexive Property. By SAS, estimate, what is m∠ABC? Explain. b. So, by CPCTC. If show that is isosceles. c. If and show that is ANSWER: equilateral. Proof: We are given W is the midpoint of d. If is isosceles, what is the minimum information needed to prove that and Q is the midpoint of Because W is the Explain your reasoning. midpoint of we know that Similarly, because Q is the midpoint of The SOLUTION: Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = a. Given: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, . If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS,

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

d. One pair of congruent corresponding sides and b. one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS. c. ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: d. One pair of congruent corresponding sides and a. 65°; Because is isosceles, ABC one pair of congruent corresponding angles; since ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. you know that the triangle is isosceles, if one leg is b. congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

c. 26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION: 27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 27. CONSTRUCTION Construct three different 2.3 cm, the triangles are isosceles. I used a isosceles right triangles. Explain your method. Then protractor to confirm that A, D, and G are all verify your constructions using measurement and right angles. mathematics. ANSWER: SOLUTION:

Sample answer: I constructed a pair of perpendicular Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to segments and then used the same compass setting to mark points equidistant from their intersection. I mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all protractor to confirm that A, D, and G are all right angles. right angles. 28. PROOF Based on your construction in Exercise 27, ANSWER: make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, Solve for x

REGULARITY Find each measure.

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship By the Triangle Sum Theorem, between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an ANSWER: isosceles right triangle are 45. 44 Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are 30. m ACD complementary, Solve for x SOLUTION:

We know that ANSWER: Therefore, . Conjecture: The measures of the base angles of an ANSWER: isosceles right triangle are 45. 44 Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute 31. m ACB angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. SOLUTION: The angles in a straight line add to 180°. REGULARITY Find each measure. Therefore,

We know that

29. m∠CAD ANSWER: 136 SOLUTION: From the figure, 32. m ABC Therefore, is an isosceles triangle. SOLUTION:

From the figure, So, is an isosceles

triangle. Therefore, By the Triangle Sum Theorem,

By the Triangle Angle Sum Theorem,

ANSWER: 44 ANSWER: 22 30. m ACD 33. FITNESS In the diagram, the rider will use his bike SOLUTION: to hop across the tops of each of the concrete solids We know shown. If each triangle is isosceles with vertex that angles G, H, and J, and G Therefore, . H, and H J, show that the distance ANSWER: from B to F is three times the distance from D to F.

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that SOLUTION:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, ANSWER:

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F. 34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION: SOLUTION:

ANSWER: ANSWER:

4-6 Isosceles and Equilateral Triangles

34. Given: is isosceles; PROOF Write a two-column proof of each Prove: X and YZV are complementary. corollary or theorem. 35. Corollary 4.3 SOLUTION:

SOLUTION:

ANSWER:

36. Corollary 4.4 SOLUTION:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

ANSWER:

eSolutions Manual - Powered by Cognero Page 12

ANSWER:

37. Theorem 4.11 SOLUTION:

ANSWER: 36. Corollary 4.4 SOLUTION:

Find the value of each variable.

ANSWER: 38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because 37. Theorem 4.11 . SOLUTION: ANSWER: 3

39. SOLUTION: ANSWER: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Find the value of each variable. The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 38. SOLUTION: GAMES Use the diagram of a game timer shown to find each measure. By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because

ANSWER: 40. m LPM 3 SOLUTION: Angles at a point in a straight line add up to 180 .

39.

SOLUTION: Substitute x = 45 in (3x – 55) to find By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

ANSWER: 80

41. m LMP

The measure of an angle cannot be negative, and 2(– SOLUTION: 18) – 5 = –41, so y = 14. Since the triangle LMP is isosceles, ANSWER: Angles at a point in a straight line

14 add up to 180 .

GAMES Use the diagram of a game timer shown to find each measure.

Substitute x = 45 in (3x – 55) to find

Therefore,

ANSWER: 40. m LPM 80 SOLUTION: 42. m JLK Angles at a point in a straight line add up to 180 . SOLUTION: Vertical angles are congruent. Therefore,

First we need to find Substitute x = 45 in (3x – 55) to find We know that,

ANSWER: So, 80 ANSWER: 20 41. m LMP SOLUTION: 43. m JKL Since the triangle LMP is isosceles, SOLUTION: Angles at a point in a straight line Vertical angles are congruent. Therefore, add up to 180 . First we need to find

We know that,

Substitute x = 45 in (3x – 55) to find

So,

In Therefore, Since the triangle JKL is isosceles, ANSWER: 80

42. m JLK SOLUTION: ANSWER: Vertical angles are congruent. Therefore, 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the First we need to find interior angles of an isosceles triangle given the measure of one exterior angle. We know that,

So,

ANSWER:

20 a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of 43. m JKL the sides adjacent to the vertex angle and to one of SOLUTION: the base angles, and labeling as shown. Vertical angles are congruent. Therefore, b. Tabular Use a protractor to measure and record First we need to find m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and We know that, record m 2 and use it to calculate these same measures. Organize your results in two tables.

c. Verbal Explain how you used m 1 to find the So, measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. In d. Algebraic If m 1 = x, write an expression for Since the triangle JKL is isosceles, the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

ANSWER: SOLUTION: 80 a.

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. b.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so b. m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

b. SOLUTION:

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the ANSWER: angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that PRECISION Determine whether the following statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65;

SOLUTION: vertex angle = 55, base angles = 62.5. ANSWER: Sometimes; only if the measure of the vertex angle is even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that PRECISION Determine whether the following m G = 35, while Miguela says that m G = 60. Is statements are sometimes, always, or never true. either of them correct? Explain your reasoning. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; SOLUTION: vertex angle = 55, base angles = 62.5. Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. Sometimes; only if the measure of the vertex angle is even.

48. ERROR ANALYSIS Alexis and Miguela are ANSWER: finding m G in the figure shown. Alexis says that It is not possible because a triangle cannot have more m G = 35, while Miguela says that m G = 60. Is than one obtuse angle. either of them correct? Explain your reasoning. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

SOLUTION: Since is isosceles, Neither of them is correct. This is an isosceles And given that m∠B = 90. triangle with a vertex angle of 70. Since this is an Therefore,

isosceles triangle, the base angles are congruent.

Construct the triangle with the angles measures 90, 45, 45.

ANSWER:

Neither; m G = or 55.

49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not possible, explain why not. SOLUTION: It is not possible because a triangle cannot have more

than one obtuse angle. 51. WRITING IN MATH How can triangle ANSWER: classifications help you prove triangle congruence? It is not possible because a triangle cannot have more than one obtuse angle. SOLUTION: If a triangle is already classified, you can use the REASONING 50. In isosceles m B = 90. previously proven properties of that type of triangle Draw the triangle. Indicate the congruent sides and in the proof. For example, if you know that a label each angle with its measure. triangle is an equilateral triangle, you can use SOLUTION: Corollary 4.3 and 4.4 in the proof. Doing this can The sum of the angle measures in a triangle must be save you steps when writing the proof. 180, ANSWER: Since is isosceles, Sample answer: If a triangle is already classified, And given that m∠B = 90. you can use the previously proven properties of that Therefore, type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

Construct the triangle with the angles measures 90, 45, 45.

a. Which statement is not necessarily true from the information given? ANSWER: A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? c. If m∠ADE = 50, what is the measure of ∠BEC?

SOLUTION: d. What measure would ∠AEB need to be so that If a triangle is already classified, you can use the ? previously proven properties of that type of triangle in the proof. For example, if you know that a SOLUTION: triangle is an equilateral triangle, you can use a. Of the statements given, only choice B is not Corollary 4.3 and 4.4 in the proof. Doing this can necessarily true from the given information. save you steps when writing the proof. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a ANSWER: triangle is 180, so ∠BEC must be 80°. Sample answer: If a triangle is already classified, d. 60; All three angles in △AEB are 60° because you can use the previously proven properties of that base angles of an isosceles triangle are congruent. So type of triangle in the proof. Doing this can save you △AEB is equiangular and equilateral. △AEB ≅ steps when writing the proof. △DEC by SAS, so AB ≅ EC.

52. MULTI-STEP △AED and △BEC are congruent ANSWER: and isosceles. a. B b. 40 c. 80 d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m FEG? a. Which statement is not necessarily true from the ∠

information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

A 34 b. If m∠EAB = 40, what is the measure of ∠DCE? B 56

C 60 c. If m∠ADE = 50, what is the measure of ∠BEC? D 62

E 124 d. What measure would ∠AEB need to be so that ? SOLUTION: SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because Consider triangle DEH. Since one angle is 56°, and base angles of an isosceles triangle are congruent. So the triangle is isosceles, then the remaining two △AEB is equiangular and equilateral. △AEB ≅ angles would be 180 – 56 or 124°. △DEC by SAS, so AB ≅ EC. So, m∠FEG is , or 62. The correct answer is choice D. ANSWER: a. B ANSWER: b. 40 D c. 80 d. 60; Sample answer: All three angles in △AEB 54. A jeweler bends wire to make charms in the shape are 60° because base angles of an isosceles triangle of isosceles triangles. The base of the triangle are congruent. So △AEB is equiangular and measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. from a piece of wire that is 63 centimeters long? 53. What is m∠FEG? SOLUTION: Because we know that the shape of each charm is an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm.

To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. A 34 B 56 ANSWER: C 60 7 D 62 55. In the figure below, NDG LGD. E 124

SOLUTION:

a. What is m∠L? Consider triangle DEH. Since one angle is 56°, and b. What is the value of x? the triangle is isosceles, then the remaining two c. What is m∠DGN? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is SOLUTION: choice D. a. In this figure, angles N and L are congruent so m∠L = 36°. ANSWER: b. Because we know two angle measures, and that D the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. 54. A jeweler bends wire to make charms in the shape c. ∠DGN is congruent to x°, so m∠DGN is 54°. of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures ANSWER: 3.5 centimeters. How many charms can be made a. 36 from a piece of wire that is 63 centimeters long? b. 54 SOLUTION: c. 54 Because we know that the shape of each charm is 56. An equilateral triangle has a perimeter of 216 inches. an isosceles triangle, then we can determine the side What is the length of one side of the triangle? lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. SOLUTION: Because the triangle is equilateral that means each To find how many charms can be made, divide 63 side has the same length, so divide the perimeter by 3 cm by 9 cm. So, the jeweler can make 7 charms. to find the length of one side.

ANSWER: 216 ÷ 3 = 72 7 ANSWER: 55. In the figure below, NDG LGD. 72

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side

lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. a. What is m L? ∠ b. What is the value of x? Because the x-terms cancel each other out: 2x + 2x c. What is m∠DGN? – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or SOLUTION: 16.

a. In this figure, angles N and L are congruent so So, the perimeter of △JKL is 16. m∠L = 36°. b. Because we know two angle measures, and that ANSWER: the sum of the measures of the angles of a triangle is 16 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of ANSWER: x. a. 36 SOLUTION: b. 54 c. 54 In an isosceles triangle, the base angles are congruent. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? So, 5x + 5 = 7x – 19.

SOLUTION: Simplify. Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 –2x = –24 to find the length of one side. x = 12

216 ÷ 3 = 72 ANSWER: 12 ANSWER: 72 59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an 57. △JKL is an isosceles triangle, and . JK = equilateral triangle? 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x

– 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. A

B m∠VRT = 60 So, the perimeter of △JKL is 16. C QRV VTU ANSWER: D VRT RST 16 SOLUTION: 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION: In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify. Consider each choice.

–2x = –24 A does not provide enough information to x = 12 determine how sides VT and TR are related.

ANSWER: B If we know that m∠VRT = 60, we still would not 12 have enough information to prove that the other two measures are 60 degrees. 59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an C equilateral triangle?

If , then m∠RVT = 60 because

QVR + RVT + TVU = 180. Since , A ∠ ∠ ∠ by the Isosceles Triangle Theorem. B m∠VRT = 60 C QRV VTU So, . Therefore, is equilateral by Theorem 4.3. D VRT RST

SOLUTION: D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; Refer to the figure. the triangle must be equiangular. All the equiangular

triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

4. m MRP

ANSWER: SOLUTION: BAC and BCA Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral 2. If EAC ECA, name two congruent segments. triangle measures 60°. Therefore, m MRP = 60°. SOLUTION: ANSWER: Converse of Isosceles Triangle Theorem states that 60 if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, SENSE-MAKING Find the value of each in triangle EAC, variable.

ANSWER:

Find each measure. 3. FH 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . SOLUTION: By the Triangle Angle-Sum Theorem,

ANSWER: Since the measures of all the three angles are 60°; 12 the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

4. m MRP 6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: 16 ANSWER: 60 7. PROOF Write a two-column proof.

SENSE-MAKING Find the value of each variable. Given: is isosceles; bisects ABC. Prove:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. SOLUTION: By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER: ANSWER: 12

6. SOLUTION: 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A In the figure, Therefore, triangle WXY is portion of the track is shown. an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: 16

7. PROOF Write a two-column proof. a. If and are perpendicular to is

isosceles with base , and prove that Given: is isosceles; bisects ABC.

Prove: b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a.

SOLUTION:

ANSWER:

b. The Pythagorean Theorem tells us that:

8. ROLLER COASTERS The roller coaster track By CPCTC we know that VT = 1.5 m. The appears to be composed of congruent triangles. A Segment Addition Postulate says QV + VT = QT. portion of the track is shown. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: b. The Pythagorean Theorem tells us that a. By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, b. The Pythagorean Theorem tells us that:

ANSWER: ABE and AEB By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 10. If ∠ABF ∠AFB, name two congruent segments. 3 m. SOLUTION: By the Converse of Isosceles Triangle Theorem, in ANSWER: triangle ABF, a. ANSWER:

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: b. The Pythagorean Theorem tells us that By the Converse of Isosceles Triangle Theorem, in By CPCTC we know triangle ADE, that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. ANSWER: So QT = 3 m.

Refer to the figure. 13. If BCF BFC, name two congruent segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER:

9. If name two congruent angles. SOLUTION: 14. If name two congruent angles. By the Isosceles Triangle Theorem, in triangle ABE, SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, ANSWER: ABE and AEB ANSWER: AFH and AHF 10. If ∠ABF ∠AFB, name two congruent segments. Find each measure. SOLUTION: 15. m BAC By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. SOLUTION: Here SOLUTION: By Isosceles Triangle Theorem, . By the Isosceles Triangle Theorem, in triangle ACD, Apply the Triangle Sum Theorem. ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. ANSWER: 60 SOLUTION: By the Converse of Isosceles Triangle Theorem, in 16. m SRT triangle ADE,

ANSWER:

13. If BCF BFC, name two congruent segments. SOLUTION: SOLUTION: By the Converse of Isosceles Triangle Theorem, in Given: triangle BCF, By Isosceles Triangle Theorem, .

ANSWER: Apply Triangle Angle-Sum Theorem.

14. If name two congruent angles.

SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle AFH, 65

17. TR ANSWER: AFH and AHF

Find each measure. 15. m BAC

SOLUTION: Since the triangle is isosceles, Therefore,

SOLUTION: Here By Isosceles Triangle Theorem, . All the angles are congruent. Therefore, it is an Apply the Triangle Sum Theorem. equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: 4 ANSWER: 60 18. CB

16. m SRT

SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABC, Given: That is, CB = 3. By Isosceles Triangle Theorem, . ANSWER: Apply Triangle Angle-Sum Theorem. 3

REGULARITY Find the value of each variable.

ANSWER: 65 19. 17. TR SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, Solve for x.

SOLUTION: Since the triangle is isosceles, Therefore, ANSWER: x = 5

All the angles are congruent. Therefore, it is an equiangular triangle.

Since the equiangular triangle is an equilateral, 20. SOLUTION:

ANSWER: Given:

4 By the Isosceles Triangle Theorem,

18. CB And We know that,

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in x = 13 triangle ABC, That is, CB = 3.

ANSWER: 3 REGULARITY Find the value of each variable. 21. SOLUTION: Given: By the Isosceles Triangle Theorem, 19. That is, SOLUTION:

Since all the angles are congruent, the sides are also Then, let Equation 1 be: . congruent to each other.

Therefore, By the Triangle Sum Theorem, we can find Equation Solve for x. 2:

Add the equations 1 and 2. ANSWER: x = 5

Substitute the value of x in one of the two equations 20. to find the value of y. SOLUTION: Given: By the Isosceles Triangle Theorem,

And We know that, ANSWER: x = 11, y = 11

ANSWER: x = 13

22. SOLUTION: Given: 21. By the Triangle Angle-Sum SOLUTION: Theorem,

Given: So, the triangle is an equilateral triangle. By the Isosceles Triangle Theorem, Therefore,

That is, Set up two equations to solve for x and y.

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 2: ANSWER: x = 1.4, y = –1

Add the equations 1 and 2. PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

Substitute the value of x in one of the two equations to find the value of y.

SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH

and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles ANSWER: Triangle Theorem, we know that HJK HML. x = 11, y = 11 Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent Supplements Theorem. By AAS, 22. By CPCTC, JHK MHL. SOLUTION: ANSWER: Given: Proof: We are given that is an isosceles By the Triangle Angle-Sum triangle and is an equilateral triangle, JKH Theorem, and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles So, the triangle is an equilateral triangle. Triangle Theorem, we know that HJK HML. Therefore, Because is an equilateral triangle, we know HLK LKH KHL and Set up two equations to solve for x and y. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL. ANSWER: x = 1.4, y = –1 24. Given: W is the midpoint of PROOF Write a paragraph proof.

23. Given: is isosceles, and is Q is the midpoint of equilateral. JKH and HKL are supplementary Prove: and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: SOLUTION: Proof: We are given W is the midpoint of Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and Q is the midpoint of Because W is the and HKL are supplementary and HLK and midpoint of we know that Similarly, MLH are supplementary. From the Isosceles because Q is the midpoint of The Triangle Theorem, we know that HJK HML. Segment Addition Postulate gives us XW + WY = XY Because is an equilateral triangle, we know and XQ + QZ = XZ. Substitution gives XW + WY = HLK LKH KHL and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. JKH, HKL and HLK, If we divide each side by 2, we have WY = QZ. The MLH are supplementary, and HKL HLK, Isosceles Triangle Theorem says XYZ XZY. we know JKH MLH by the Congruent by the Reflexive Property. By SAS, Supplements Theorem. By AAS, So, by CPCTC. By CPCTC, JHK MHL. ANSWER: ANSWER: Proof: We are given W is the midpoint of Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and Q is the midpoint of Because W is the and HKL are supplementary and HLK and midpoint of we know that Similarly, MLH are supplementary. From the Isosceles because Q is the midpoint of The Triangle Theorem, we know that HJK HML. Segment Addition Postulate gives us XW + WY = XY Because is an equilateral triangle, we know and XQ + QZ = XZ. Substitution gives XW + WY = HLK LKH KHL and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. JKH, HKL and HLK, If we divide each side by 2, we have WY = QZ. The MLH are supplementary, and HKL HLK, Isosceles Triangle Theorem says XYZ XZY. we know JKH ∠MLH by the Congruent by the Reflexive Property. By SAS, Supplements Theorem. By AAS, So, by CPCTC. By CPCTC, JHK MHL. 25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either 24. Given: side of a park swing set form two sets of triangles. W is the midpoint of Using a jump rope to measure, Elisa is able to Q is the midpoint of determine that but Prove:

SOLUTION:

Proof: We are given W is the midpoint of a. Elisa estimates m BAC to be 50. Based on this and Q is the midpoint of Because W is the estimate, what is m∠ABC? Explain. midpoint of we know that Similarly, b. If show that is isosceles. because Q is the midpoint of The c. If and show that is Segment Addition Postulate gives us XW + WY = XY equilateral. and XQ + QZ = XZ. Substitution gives XW + WY = d. If is isosceles, what is the minimum XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. information needed to prove that If we divide each side by 2, we have WY = QZ. The Explain your reasoning. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, SOLUTION: So, by CPCTC. a. Given: ANSWER: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem. Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly,

because Q is the midpoint of The b. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC.

BABYSITTING 25. While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. c. Using a jump rope to measure, Elisa is able to determine that but

d. One pair of congruent corresponding sides and SOLUTION: one pair of congruent corresponding angles; since a. Given: you know that the triangle is isosceles, if one leg is By Isosceles Triangle Theorem, . congruent to a leg of then you know that Apply Triangle Sum Theorem. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and b. one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base d. One pair of congruent corresponding sides and angles of an isosceles triangle are congruent, if you one pair of congruent corresponding angles; since know that you know that you know that the triangle is isosceles, if one leg is Therefore, congruent to a leg of then you know that with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the both pairs of legs are congruent. Because the base triangles can be proved congruent using either ASA angles of an isosceles triangle are congruent, if you or SAS. know that , you know that Therefore, ANSWER: with one pair of congruent corresponding sides and a. 65°; Because is isosceles, ABC one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. or SAS. b. 26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, Sample answer: I constructed a pair of perpendicular make and prove a conjecture about the relationship segments and then used the same compass setting to between the base angles of an isosceles right mark points equidistant from their intersection. I triangle. measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = SOLUTION: 2.3 cm, the triangles are isosceles. I used a Conjecture: The measures of the base angles of an protractor to confirm that A, D, and G are all isosceles right triangle are 45. right angles. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute ANSWER: angle be x. The acute angles of a right triangle are complementary, Solve for x

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to By the Triangle Sum Theorem, mark points equidistant from their intersection. I

measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all ANSWER: right angles. 44 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship 30. m ACD between the base angles of an isosceles right SOLUTION: triangle. We know SOLUTION: that Conjecture: The measures of the base angles of an Therefore, . isosceles right triangle are 45. Proof: The base angles are congruent because it is ANSWER: an isosceles triangle. Let the measure of each acute 44 angle be x. The acute angles of a right triangle are complementary, Solve for x 31. m ACB SOLUTION: The angles in a straight line add to 180°. ANSWER: Therefore,

Conjecture: The measures of the base angles of an isosceles right triangle are 45. We know that Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. ANSWER: REGULARITY Find each measure. 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

29. m∠CAD By the Triangle Angle Sum Theorem, SOLUTION: From the figure, Therefore, is an isosceles triangle. ANSWER: 22 By the Triangle Sum Theorem, FITNESS 33. In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G ANSWER: H, and H J, show that the distance 44 from B to F is three times the distance from D to F.

30. m ACD SOLUTION: We know that Therefore, . ANSWER: SOLUTION: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

ANSWER: ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER: Given: 22 34. is isosceles; Prove: X and YZV are complementary. 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION: ANSWER:

ANSWER: 36. Corollary 4.4 SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER:

4-6 Isosceles and Equilateral Triangles

36. Corollary 4.4 Find the value of each variable. SOLUTION:

38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

ANSWER: Note that x can equal –8 here because .

ANSWER: 3

37. Theorem 4.11 39. SOLUTION: SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

ANSWER: The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

38. SOLUTION: eSolutions Manual - Powered by Cognero Page 13 By the converse of Isosceles Triangle theorem, 40. m LPM SOLUTION: Solve the equation for x. Angles at a point in a straight line add up to 180 .

Note that x can equal –8 here because

. ANSWER: Substitute x = 45 in (3x – 55) to find 3

ANSWER: 39. 80 SOLUTION: 41. m LMP By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . SOLUTION: Since the triangle LMP is isosceles, The interior angles of a triangle add up to 180 . Angles at a point in a straight line add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Substitute x = 45 in (3x – 55) to find

ANSWER: 14 GAMES Use the diagram of a game timer Therefore, shown to find each measure. ANSWER: 80

42. m JLK SOLUTION: Vertical angles are congruent. Therefore,

First we need to find 40. m LPM

SOLUTION: We know that, Angles at a point in a straight line add up to 180 .

So,

ANSWER: Substitute x = 45 in (3x – 55) to find 20

43. m JKL SOLUTION: Vertical angles are congruent. Therefore, ANSWER: 80 First we need to find

We know that, 41. m LMP

SOLUTION: Since the triangle LMP is isosceles, Angles at a point in a straight line So, add up to 180 . In

Since the triangle JKL is isosceles,

Substitute x = 45 in (3x – 55) to find

ANSWER: 80

Therefore, 44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the ANSWER: interior angles of an isosceles triangle given the 80 measure of one exterior angle.

42. m JLK SOLUTION: Vertical angles are congruent. Therefore,

First we need to find a. Geometric Use a ruler and a protractor to draw We know that, three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

So, b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the ANSWER: measures of 3, 4, and 5. Then find and 20 record m 2 and use it to calculate these same measures. Organize your results in two tables. 43. m JKL SOLUTION: c. Verbal Explain how you used m 1 to find the Vertical angles are congruent. Therefore, measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. First we need to find d. Algebraic If m 1 = x, write an expression for We know that, the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

So, SOLUTION:

a. In

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

b. Tabular Use a protractor to measure and record b. m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle c. 5 is supplementary to 1, so m 5 = 180 – measures. m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = SOLUTION: 180 – m 4 – 5. 2 is supplementary to 3, so a. m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – c. 5 is supplementary to 1, so m 5 = 180 – 180; m 3 = 180 – x, m 4 = m 5 = m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = ANSWER: 180 – m 4 – 5. 2 is supplementary to 3, so a. m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER:

4 and m 5, so m 4 = m 5 = statements are sometimes, always, or never true. Explain. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base 180; m 3 = 180 – x, m 4 = m 5 = angle is an integer. SOLUTION: 45. CHALLENGE In the figure, if is equilateral The statement is sometimes true. It is true only if the and ZWP WJM JZL, prove that measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. ANSWER: Sometimes; only if the measure of the vertex angle is even.

47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex angle is odd. SOLUTION: SOLUTION: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even.

ANSWER: Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will ANSWER: be even. 48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

PRECISION Determine whether the following statements are sometimes, always, or never true. SOLUTION: Explain. Neither of them is correct. This is an isosceles 46. If the measure of the vertex angle of an isosceles triangle with a vertex angle of 70. Since this is an triangle is an integer, then the measure of each base isosceles triangle, the base angles are congruent. angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is ANSWER: even. Neither; m G = or 55. 47. If the measures of the base angles of an isosceles triangle are integers, then the measure of its vertex 49. OPEN-ENDED If possible, draw an isosceles angle is odd. triangle with base angles that are obtuse. If it is not possible, explain why not. SOLUTION: The statement is never true. The measure of the SOLUTION: vertex angle will be 180 – 2(measure of the base It is not possible because a triangle cannot have more angle) so if the base angles are integers, then 2 than one obtuse angle. (measure of the base angle) will be even and 180 – 2 (measure of the base angle) will be even. ANSWER: It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles 50. REASONING In isosceles m B = 90. are integers, then 2(measure of the base angle) will Draw the triangle. Indicate the congruent sides and be even and 180 – 2(measure of the base angle) will label each angle with its measure. be even. SOLUTION: 48. ERROR ANALYSIS Alexis and Miguela are The sum of the angle measures in a triangle must be finding m G in the figure shown. Alexis says that 180, m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. Since is isosceles, And given that m∠B = 90. Therefore,

Construct the triangle with the angles measures 90,

SOLUTION: 45, 45.

Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

ANSWER:

Neither; m G = or 55. 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not SOLUTION: possible, explain why not. If a triangle is already classified, you can use the SOLUTION: previously proven properties of that type of triangle in the proof. For example, if you know that a It is not possible because a triangle cannot have more than one obtuse angle. triangle is an equilateral triangle, you can use Corollary 4.3 and 4.4 in the proof. Doing this can ANSWER: save you steps when writing the proof. It is not possible because a triangle cannot have more than one obtuse angle. ANSWER: Sample answer: If a triangle is already classified, 50. REASONING In isosceles m B = 90. you can use the previously proven properties of that Draw the triangle. Indicate the congruent sides and type of triangle in the proof. Doing this can save you label each angle with its measure. steps when writing the proof. SOLUTION: 52. MULTI-STEP △AED and △BEC are congruent The sum of the angle measures in a triangle must be and isosceles.

180,

Since is isosceles, And given that m∠B = 90. Therefore,

a. Which statement is not necessarily true from the information given? Construct the triangle with the angles measures 90, 45, 45. A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE?

c. ANSWER: If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. WRITING IN MATH How can triangle 51. b. The angles are congruent, so ∠DCE is 40°. classifications help you prove triangle congruence? c. The sum of the measures of the angles of a SOLUTION: triangle is 180, so ∠BEC must be 80°. If a triangle is already classified, you can use the d. 60; All three angles in △AEB are 60° because previously proven properties of that type of triangle base angles of an isosceles triangle are congruent. So △ △ in the proof. For example, if you know that a AEB is equiangular and equilateral. AEB ≅ △ triangle is an equilateral triangle, you can use DEC by SAS, so AB ≅ EC. Corollary 4.3 and 4.4 in the proof. Doing this can ANSWER: save you steps when writing the proof. a. B b. 40 ANSWER: c. 80 Sample answer: If a triangle is already classified, d. 60; Sample answer: All three angles in △AEB you can use the previously proven properties of that are 60° because base angles of an isosceles triangle type of triangle in the proof. Doing this can save you are congruent. So △AEB is equiangular and steps when writing the proof. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

52. MULTI-STEP △AED and △BEC are congruent 53. What is m∠FEG? and isosceles.

A 34 a. Which statement is not necessarily true from the B 56 information given? C 60 D 62 A △AEB ≅ △DEC E 124 B C ∠CAD ≅ ∠DBC SOLUTION: D

b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that Consider triangle DEH. Since one angle is 56°, and ? the triangle is isosceles, then the remaining two SOLUTION: angles would be 180 – 56 or 124°. So, m FEG is , or 62. The correct answer is a. Of the statements given, only choice B is not ∠ necessarily true from the given information. choice D. b. The angles are congruent, so ∠DCE is 40°. ANSWER: c. The sum of the measures of the angles of a D triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because 54. A jeweler bends wire to make charms in the shape base angles of an isosceles triangle are congruent. So of isosceles triangles. The base of the triangle △AEB is equiangular and equilateral. △AEB ≅ measures 2 centimeters and another side measures △DEC by SAS, so AB ≅ EC. 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? ANSWER: SOLUTION: a. B b. 40 Because we know that the shape of each charm is c. 80 an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, d. 60; Sample answer: All three angles in △AEB the perimeter of each triangle is 9 cm. are 60° because base angles of an isosceles triangle

are congruent. So △AEB is equiangular and To find how many charms can be made, divide 63 △ △ equilateral. AEB ≅ DEC by SAS, so AB ≅ EC. cm by 9 cm. So, the jeweler can make 7 charms.

53. What is m∠FEG? ANSWER: 7

55. In the figure below, NDG LGD.

A 34 B 56 C 60

D 62 E 124 a. What is m∠L? b. What is the value of x? SOLUTION: c. What is m∠DGN? SOLUTION: a. In this figure, angles N and L are congruent so m∠L = 36°. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is

180, then we can solve 36 + 90 + x = 180, or x = 54. Consider triangle DEH. Since one angle is 56°, and c. the triangle is isosceles, then the remaining two ∠DGN is congruent to x°, so m∠DGN is 54°. angles would be 180 – 56 or 124°. ANSWER: So, m∠FEG is , or 62. The correct answer is a. 36 choice D. b. 54 c. 54 ANSWER: D 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle SOLUTION: measures 2 centimeters and another side measures Because the triangle is equilateral that means each 3.5 centimeters. How many charms can be made side has the same length, so divide the perimeter by 3 from a piece of wire that is 63 centimeters long? to find the length of one side.

SOLUTION: 216 ÷ 3 = 72 Because we know that the shape of each charm is an isosceles triangle, then we can determine the side ANSWER: lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, 72 the perimeter of each triangle is 9 cm. 57. △JKL is an isosceles triangle, and . JK = To find how many charms can be made, divide 63 2x + 3, and KL = 10 – 4x. What is the perimeter of cm by 9 cm. So, the jeweler can make 7 charms. △JKL? ANSWER: SOLUTION: 7 The perimeter can be found by adding all three side lengths. In this situation, since the triangle is 55. In the figure below, NDG LGD. isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16.

ANSWER:

16 a. What is m∠L? b. What is the value of x? 58. △ABC is an isosceles triangle with base angles c. What is m∠DGN? measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION: a. In this figure, angles N and L are congruent so SOLUTION: m∠L = 36°. In an isosceles triangle, the base angles are b. Because we know two angle measures, and that congruent. the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. So, 5x + 5 = 7x – 19. c. ∠DGN is congruent to x°, so m∠DGN is 54°. Simplify. ANSWER: a. 36 –2x = –24 b. 54 x = 12 c. 54 ANSWER: 56. An equilateral triangle has a perimeter of 216 inches. 12 What is the length of one side of the triangle? 59. QRV is an equilateral triangle. What additional SOLUTION: information would be enough to prove VRT is an Because the triangle is equilateral that means each equilateral triangle? side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

ANSWER: 72

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? A SOLUTION: B m∠VRT = 60 The perimeter can be found by adding all three side C QRV VTU lengths. In this situation, since the triangle is D VRT RST isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. SOLUTION:

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of Consider each choice. x. A does not provide enough information to SOLUTION: determine how sides VT and TR are related. In an isosceles triangle, the base angles are congruent. B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two So, 5x + 5 = 7x – 19. measures are 60 degrees.

Simplify. C

–2x = –24 x = 12

ANSWER: 12

59. QRV is an equilateral triangle. What additional information would be enough to prove VRT is an

equilateral triangle? If , then m∠RVT = 60 because

∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral. A B m∠VRT = 60 The correct answer is choice C. C QRV VTU ANSWER: D VRT RST C SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: Refer to the figure. 12

4. m MRP

1. If name two congruent angles.

2. If EAC ECA, name two congruent segments. SENSE-MAKING Find the value of each variable. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC, 5. ANSWER: SOLUTION: In the figure, . Therefore, triangle RST is Find each measure. an isosceles triangle. 3. FH By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: ANSWER: By the Triangle Angle-Sum Theorem, 12

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 6. 12 SOLUTION: In the figure, Therefore, triangle WXY is 4. m MRP an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: SOLUTION: 16 Since all the sides are congruent, is an PROOF Write a two-column proof. equilateral triangle. Each angle of an equilateral 7.

triangle measures 60°. Therefore, m MRP = 60°. Given: is isosceles; bisects ABC. ANSWER: Prove: 60 SENSE-MAKING Find the value of each variable.

SOLUTION: 5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER:

ANSWER: 12

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

8. ROLLER COASTERS The roller coaster track ANSWER: appears to be composed of congruent triangles. A portion of the track is shown. a.

a. If and are perpendicular to is isosceles with base , and prove that

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER:

ABE and AEB

b. The Pythagorean Theorem tells us that: 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By CPCTC we know that VT = 1.5 m. The By the Converse of Isosceles Triangle Theorem, in Segment Addition Postulate says QV + VT = QT. triangle ABF, By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ANSWER:

ANSWER: a. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + 13. If BCF BFC, name two congruent segments. VT = QT. By substitution, we have 1.5 + 1.5 = QT. SOLUTION: So QT = 3 m. By the Converse of Isosceles Triangle Theorem, in

Refer to the figure. triangle BCF,

ANSWER:

14. If name two congruent angles. SOLUTION: 9. If name two congruent angles. By the Isosceles Triangle Theorem, in triangle AFH,

SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, ANSWER: AFH and AHF

ANSWER: Find each measure. ABE and AEB 15. m BAC

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ABF, SOLUTION: ANSWER: Here By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem. 11. If name two congruent angles.

SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ANSWER: 60 ACD and ADC

12. If DAE DEA, name two congruent 16. m SRT segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: SOLUTION: Given: By Isosceles Triangle Theorem, . 13. If BCF BFC, name two congruent segments.

SOLUTION: Apply Triangle Angle-Sum Theorem.

By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: 65 14. If name two congruent angles. 17. TR SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: AFH and AHF SOLUTION: Find each measure. Since the triangle is isosceles, 15. m BAC Therefore,

All the angles are congruent. Therefore, it is an

SOLUTION: equiangular triangle. Since the equiangular triangle is an equilateral, Here By Isosceles Triangle Theorem, . ANSWER: Apply the Triangle Sum Theorem. 4

18. CB

ANSWER: 60

16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3 SOLUTION: Given: REGULARITY Find the value of each variable. By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

19. SOLUTION: Since all the angles are congruent, the sides are also ANSWER: congruent to each other. 65 Therefore, Solve for x. 17. TR

ANSWER: x = 5 SOLUTION: Since the triangle is isosceles, Therefore,

20. SOLUTION:

All the angles are congruent. Therefore, it is an Given: equiangular triangle. By the Isosceles Triangle Theorem, Since the equiangular triangle is an equilateral,

And ANSWER: We know 4 that, 18. CB

ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3. 21.

ANSWER: SOLUTION: 3 Given: By the Isosceles Triangle Theorem, REGULARITY Find the value of each variable. That is,

ANSWER: x = 5 Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11 And We know that,

ANSWER: 22. x = 13 SOLUTION: Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. 21. Therefore,

SOLUTION: Set up two equations to solve for x and y. Given: By the Isosceles Triangle Theorem,

That is,

ANSWER: Then, let Equation 1 be: . x = 1.4, y = –1 By the Triangle Sum Theorem, we can find Equation PROOF Write a paragraph proof. 2: Given: 23. is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Add the equations 1 and 2. Prove: JHK MHL

JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK, x = 11, y = 11 we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, SOLUTION: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent Proof: We are given W is the midpoint of Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The MLH are supplementary. From the Isosceles Isosceles Triangle Theorem says XYZ XZY. Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, 25. BABYSITTING While babysitting her neighbor’s MLH are supplementary, and HKL HLK, children, Elisa observes that the supports on either

we know JKH ∠MLH by the Congruent side of a park swing set form two sets of triangles. Supplements Theorem. By AAS, Using a jump rope to measure, Elisa is able to By CPCTC, JHK MHL. determine that but

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is equilateral. d. One pair of congruent corresponding sides and d. If is isosceles, what is the minimum one pair of congruent corresponding angles; since information needed to prove that you know that the triangle is isosceles, if one leg is

Explain your reasoning. congruent to a leg of then you know that both pairs of legs are congruent. Because the base SOLUTION: angles of an isosceles triangle are congruent, if you know that you know that a. Given: Therefore, By Isosceles Triangle Theorem, . with one pair of congruent corresponding sides and Apply Triangle Sum Theorem. one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS. ANSWER: b. a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

c. c.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

measured both legs for each triangle. Because AB = an isosceles triangle. Let the measure of each acute AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = angle be x. The acute angles of a right triangle are 2.3 cm, the triangles are isosceles. I used a complementary, Solve for x protractor to confirm that A, D, and G are all right angles.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to ANSWER: mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = 44 AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 30. m ACD protractor to confirm that A, D, and G are all right angles. SOLUTION: We know 28. PROOF Based on your construction in Exercise 27, that make and prove a conjecture about the relationship Therefore, . between the base angles of an isosceles right triangle. ANSWER: 44 SOLUTION: Conjecture: The measures of the base angles of an 31. m ACB isosceles right triangle are 45. Proof: The base angles are congruent because it is SOLUTION: an isosceles triangle. Let the measure of each acute The angles in a straight line add to 180°. angle be x. The acute angles of a right triangle are Therefore, complementary, Solve for x We know that

complementary, so x + x = 90 and x = 45. SOLUTION: REGULARITY Find each measure. From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

29. m∠CAD ANSWER: SOLUTION: 22 From the figure, 33. FITNESS In the diagram, the rider will use his bike Therefore, is an isosceles triangle. to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G By the Triangle Sum Theorem, H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD

SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that ANSWER:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, 34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER: 22

SOLUTION: ANSWER:

ANSWER: PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

36. Corollary 4.4 38. SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . 37. Theorem 4.11 SOLUTION: The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM 38. SOLUTION: SOLUTION: Angles at a point in a straight line add up to 180 . By the converse of Isosceles Triangle theorem,

Solve the equation for x. Substitute x = 45 in (3x – 55) to find

Note that x can equal –8 here because .

ANSWER: ANSWER: 3 80

41. m LMP SOLUTION: 39. Since the triangle LMP is isosceles, SOLUTION: Angles at a point in a straight line By the Isosceles Triangle Theorem, the third angle is add up to 180 . equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Therefore,

4-6 IsoscelesANSWER: and Equilateral Triangles ANSWER: 14 80

GAMES Use the diagram of a game timer 42. m JLK shown to find each measure. SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM SOLUTION: So, Angles at a point in a straight line add up to 180 . ANSWER: 20

43. m JKL SOLUTION: Substitute x = 45 in (3x – 55) to find Vertical angles are congruent. Therefore,

First we need to find

We know that,

ANSWER:

80 So, 41. m LMP SOLUTION: In Since the triangle LMP is isosceles, Angles at a point in a straight line Since the triangle JKL is isosceles, add up to 180 .

ANSWER: 80 Substitute x = 45 in (3x – 55) to find 44. MULTIPLE REPRESENTATIONS In this

problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

Therefore,

ANSWER: 80

42. m JLK eSolutionsSOLUTION: Manual - Powered by Cognero Page 14 Vertical angles are congruent. Therefore, a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of

the base angles, and labeling as shown. First we need to find b. Tabular Use a protractor to measure and record We know that, m∠1 for each triangle. Use m 1 to calculate the

measures of 3, 4, and 5. Then find and

measures. First we need to find

We know that, SOLUTION: a.

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw b. three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – SOLUTION:

a. 180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

180; m 3 = 180 – x, m 4 = m 5 = 4 and m 5, so m 4 = m 5 =

ANSWER: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

a. 180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

b. ANSWER:

angle is an integer. 4 and m 5, so m 4 = m 5 = SOLUTION: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, 180; m 3 = 180 – x, m 4 = m 5 = vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 45. CHALLENGE In the figure, if is equilateral ANSWER: and ZWP WJM JZL, prove that Sometimes; only if the measure of the vertex angle is

even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning.

statements are sometimes, always, or never true.

be even. Since is isosceles,

48. ERROR ANALYSIS Alexis and Miguela are And given that m∠B = 90. Therefore, finding m G in the figure shown. Alexis says that

m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. ANSWER:

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? ANSWER: SOLUTION:

Neither; m G = or 55. If a triangle is already classified, you can use the previously proven properties of that type of triangle in the proof. For example, if you know that a 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not triangle is an equilateral triangle, you can use possible, explain why not. Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. SOLUTION: It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Sample answer: If a triangle is already classified, you can use the previously proven properties of that ANSWER: type of triangle in the proof. Doing this can save you It is not possible because a triangle cannot have more than one obtuse angle. steps when writing the proof. MULTI-STEP △ △ 50. REASONING In isosceles m B = 90. 52. AED and BEC are congruent and isosceles. Draw the triangle. Indicate the congruent sides and

label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, And given that m∠B = 90. a. Which statement is not necessarily true from the Therefore, information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC Construct the triangle with the angles measures 90, D 45, 45. b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

A 34 B 56 C 60 D 62 E 124

a. Which statement is not necessarily true from the information given? SOLUTION:

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two c. If m∠ADE = 50, what is the measure of ∠BEC? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is d. What measure would ∠AEB need to be so that choice D. ? ANSWER: SOLUTION: D a. Of the statements given, only choice B is not necessarily true from the given information. 54. A jeweler bends wire to make charms in the shape b. The angles are congruent, so ∠DCE is 40°. of isosceles triangles. The base of the triangle c. The sum of the measures of the angles of a measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made triangle is 180, so BEC must be 80°. ∠ from a piece of wire that is 63 centimeters long? d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So SOLUTION: △AEB is equiangular and equilateral. △AEB ≅ Because we know that the shape of each charm is △DEC by SAS, so AB ≅ EC. an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, ANSWER: the perimeter of each triangle is 9 cm. a. B b. 40 To find how many charms can be made, divide 63 c. 80 cm by 9 cm. So, the jeweler can make 7 charms. d. 60; Sample answer: All three angles in △AEB ANSWER: are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 7 △ △ equilateral. AEB ≅ DEC by SAS, so AB ≅ EC. 55. In the figure below, NDG LGD.

53. What is m∠FEG?

A 34 a. What is m∠L? B 56 b. What is the value of x? C 60 c. What is m∠DGN? D 62 SOLUTION: E 124 a. In this figure, angles N and L are congruent so

a. 36 Consider triangle DEH. Since one angle is 56°, and b. 54 the triangle is isosceles, then the remaining two c. 54 angles would be 180 – 56 or 124°. 56. An equilateral triangle has a perimeter of 216 inches. So, m∠FEG is , or 62. The correct answer is What is the length of one side of the triangle? choice D. SOLUTION: ANSWER: Because the triangle is equilateral that means each D side has the same length, so divide the perimeter by 3 to find the length of one side. 54. A jeweler bends wire to make charms in the shape of isosceles triangles. The base of the triangle 216 ÷ 3 = 72 measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made ANSWER: from a piece of wire that is 63 centimeters long? 72 SOLUTION: 57. △JKL is an isosceles triangle, and . JK = Because we know that the shape of each charm is 2x + 3, and KL = 10 – 4x. What is the perimeter of an isosceles triangle, then we can determine the side △JKL? lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. SOLUTION: The perimeter can be found by adding all three side To find how many charms can be made, divide 63 lengths. In this situation, since the triangle is cm by 9 cm. So, the jeweler can make 7 charms. isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

ANSWER: Because the x-terms cancel each other out: 2x + 2x 7 – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 55. In the figure below, NDG LGD.

So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of

x. a. What is m∠L? SOLUTION: b. What is the value of x? In an isosceles triangle, the base angles are c. What is m∠DGN? congruent.

SOLUTION: So, 5x + 5 = 7x – 19. a. In this figure, angles N and L are congruent so m∠L = 36°. Simplify. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is –2x = –24 180, then we can solve 36 + 90 + x = 180, or x = 54. x = 12 c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: 12 a. 36 b. 54 59. QRV is an equilateral triangle. What additional c. 54 information would be enough to prove VRT is an equilateral triangle? 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

Refer to the figure. 4. m MRP

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that

ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the 16 distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

So QT = 3 m. SOLUTION: a. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

ANSWER: 11. If name two congruent angles. a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER:

60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR 14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION: AFH and AHF Since the triangle is isosceles, Therefore, Find each measure. 15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION: Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER:

60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3

REGULARITY Find the value of each variable. SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, ANSWER: Solve for x. 65 17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11

And We know that,

22. SOLUTION: ANSWER: Given: x = 13 By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y. SOLUTION: Given: By the Isosceles Triangle Theorem,

Add the equations 1 and 2.

SOLUTION:

Substitute the value of x in one of the two equations Proof: We are given that is an isosceles to find the value of y. triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent ANSWER: Supplements Theorem. By AAS, x = 11, y = 11 By CPCTC, JHK MHL. ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles 22. Triangle Theorem, we know that HJK HML. SOLUTION: Because is an equilateral triangle, we know Given: HLK LKH KHL and By the Triangle Angle-Sum JKH, HKL and HLK, Theorem, MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent So, the triangle is an equilateral triangle. Supplements Theorem. By AAS, Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y.

24. Given: W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary

and HLK and MLH are supplementary. Prove: JHK MHL SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly,

because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY SOLUTION: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, Proof: We are given W is the midpoint of MLH are supplementary, and HKL HLK, and Q is the midpoint of Because W is the we know JKH MLH by the Congruent Supplements Theorem. By AAS, midpoint of we know that Similarly, By CPCTC, JHK MHL. because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY ANSWER: and XQ + QZ = XZ. Substitution gives XW + WY = Proof: We are given that is an isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. triangle and is an equilateral triangle, JKH If we divide each side by 2, we have WY = QZ. The and HKL are supplementary and HLK and Isosceles Triangle Theorem says XYZ XZY. MLH are supplementary. From the Isosceles by the Reflexive Property. By SAS, Triangle Theorem, we know that HJK HML. So, by CPCTC. Because is an equilateral triangle, we know HLK LKH KHL and 25. BABYSITTING While babysitting her neighbor’s JKH, HKL and HLK, children, Elisa observes that the supports on either MLH are supplementary, and HKL HLK, side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

Isosceles Triangle Theorem says XYZ XZY. c. by the Reflexive Property. By SAS, So, by CPCTC.

ANSWER: a. 65°; Because is isosceles, ABC

∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. b.

c. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that , you know that angles of an isosceles triangle are congruent, if you Therefore, know that you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. bisects the angle formed by the sloped sides of b. the roof, ABC.

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

ANSWER: ANSWER: Conjecture: The measures of the base angles of an 136 isosceles right triangle are 45. Proof: The base angles are congruent because it is 32. m ABC an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles triangle. Therefore, REGULARITY Find each measure.

By the Triangle Angle Sum Theorem,

H, and H J, show that the distance from B to F is three times the distance from D to F. By the Triangle Sum Theorem,

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. SOLUTION: 36. Corollary 4.4 By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

37. Theorem 4.11 The interior angles of a triangle add up to 180 .

SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 .

SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find

Solve the equation for x.

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

So, 40. m LPM SOLUTION: ANSWER: 20 Angles at a point in a straight line add up to 180 . 43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP In

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

Therefore,

ANSWER: 80

First we need to find SOLUTION: a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

4-6 Isosceles and Equilateral Triangles

b. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the c. 5 is supplementary to 1, so m 5 = 180 – measures of 3, 4, and 5. Then explain how m 1. 4 5, so m 4 = m 5. The sum of the you used m 2 to find these same measures. angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. Algebraic If m 1 = x, write an expression for m 3 = 180 – m 2. m 2 is twice as much as m the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle 4 and m 5, so m 4 = m 5 =

measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

b. eSolutions Manual - Powered by Cognero Page 15 b.

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4 and m 5, so m 4 = m 5 =

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

ANSWER: 180; m 3 = 180 – x, m 4 = m 5 = a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

measure of the vertex angle is even. For example, Neither; m G = or 55. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not Sometimes; only if the measure of the vertex angle is possible, explain why not. even. SOLUTION: It is not possible because a triangle cannot have more 47. If the measures of the base angles of an isosceles than one obtuse angle. triangle are integers, then the measure of its vertex angle is odd. ANSWER: SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 50. REASONING In isosceles m B = 90. angle) so if the base angles are integers, then 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even and 180 – 2 label each angle with its measure. (measure of the base angle) will be even. SOLUTION: ANSWER: The sum of the angle measures in a triangle must be Never; the measure of the vertex angle will be 180 – 180, 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will

be even and 180 – 2(measure of the base angle) will Since is isosceles, be even. And given that m∠B = 90. Therefore,

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

180, a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC

B C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION:

52. MULTI-STEP △AED and △BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B

ANSWER: To find how many charms can be made, divide 63 a. B cm by 9 cm. So, the jeweler can make 7 charms. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB 7 are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 55. In the figure below, NDG LGD.

equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

△ 55. In the figure below, NDG LGD. So, the perimeter of JKL is 16. ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. Consider each choice. So, the perimeter of △JKL is 16. A does not provide enough information to ANSWER: determine how sides VT and TR are related. 16 B If we know that m VRT = 60, we still would not 58. △ABC is an isosceles triangle with base angles ∠ have enough information to prove that the other two measuring (5x + 5)° and (7x – 19)°. Find the value of measures are 60 degrees. x.

SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

Refer to the figure. 4. m MRP

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular

triangles are equilateral. Therefore, FH = GH = 12. 6. SOLUTION: ANSWER: 12 In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, . 4. m MRP

ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that

ANSWER: b. 16 If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

ANSWER: b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

ANSWER: 11. If name two congruent angles. a. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

Refer to the figure. ANSWER:

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER:

60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, SOLUTION: ANSWER: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR 14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION: AFH and AHF Since the triangle is isosceles, Therefore, Find each measure. 15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION:

Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER: 60 SOLUTION: 16. m SRT By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 3

REGULARITY Find the value of each variable. SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other.

ANSWER: Therefore, Solve for x. 65 17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

SOLUTION: Given: By the Isosceles Triangle Theorem, All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, And

We know ANSWER: that, 4 18. CB ANSWER: x = 13

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 11, y = 11

And We know that,

22. SOLUTION: ANSWER: Given: x = 13 By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y. SOLUTION: Given: By the Isosceles Triangle Theorem,

Add the equations 1 and 2.

SOLUTION:

we know JKH ∠MLH by the Congruent So, the triangle is an equilateral triangle. Supplements Theorem. By AAS, Therefore, By CPCTC, JHK MHL.

Set up two equations to solve for x and y. 24. Given: W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary.

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles.

c. If and show that is SOLUTION: equilateral. Proof: We are given W is the midpoint of d. If is isosceles, what is the minimum information needed to prove that and Q is the midpoint of Because W is the Explain your reasoning. midpoint of we know that Similarly, because Q is the midpoint of The SOLUTION: Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = a. Given: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, . If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem. Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. b. ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The

Isosceles Triangle Theorem says XYZ XZY. c. by the Reflexive Property. By SAS, So, by CPCTC.

a. Elisa estimates m BAC to be 50. Based on this

estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is d. One pair of congruent corresponding sides and equilateral. one pair of congruent corresponding angles; since d. If is isosceles, what is the minimum you know that the triangle is isosceles, if one leg is information needed to prove that congruent to a leg of then you know that Explain your reasoning. both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that SOLUTION: Therefore, a. Given: with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the By Isosceles Triangle Theorem, . triangles can be proved congruent using either ASA Apply Triangle Sum Theorem. or SAS.

ANSWER: a. 65°; Because is isosceles, ABC

∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. b.

c. c.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that , you know that angles of an isosceles triangle are congruent, if you Therefore, know that you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA or SAS. 26. CHIMNEYS In the picture, and ANSWER: is an isosceles triangle with base Show that the chimney of the house, represented by a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. bisects the angle formed by the sloped sides of b. the roof, ABC.

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. SOLUTION: Conjecture: The measures of the base angles of an Sample answer: I constructed a pair of perpendicular isosceles right triangle are 45. segments and then used the same compass setting to Proof: The base angles are congruent because it is mark points equidistant from their intersection. I an isosceles triangle. Let the measure of each acute measured both legs for each triangle. Because AB = angle be x. The acute angles of a right triangle are AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = complementary, Solve for x 2.3 cm, the triangles are isosceles. I used a protractor to confirm that A, D, and G are all right angles. ANSWER: ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

ANSWER: ANSWER: Conjecture: The measures of the base angles of an 136 isosceles right triangle are 45. Proof: The base angles are congruent because it is 32. m ABC an isosceles triangle. Let the measure of each acute angle be x. The acute angles of a right triangle are SOLUTION: complementary, so x + x = 90 and x = 45. From the figure, So, is an isosceles triangle. Therefore, REGULARITY Find each measure.

By the Triangle Angle Sum Theorem,

ANSWER:

22 29. m∠CAD SOLUTION: 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids From the figure, shown. If each triangle is isosceles with vertex Therefore, is an isosceles triangle. angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F. By the Triangle Sum Theorem,

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38.

36. Corollary 4.4 SOLUTION: By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

37. Theorem 4.11 The interior angles of a triangle add up to 180 .

SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 . SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find Solve the equation for x.

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

So, 40. m LPM SOLUTION: ANSWER: Angles at a point in a straight line add up to 180 . 20

43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

41. m LMP In

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

Therefore,

ANSWER: 80

So, c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how ANSWER: you used m 2 to find these same measures. 20 d. Algebraic If m 1 = x, write an expression for

43. m JKL the measures of 3, 4, and 5. Likewise, if m SOLUTION: 2 = x, write an expression for these same angle Vertical angles are congruent. Therefore, measures.

First we need to find SOLUTION:

a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

b. a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the c. 5 is supplementary to 1, so m 5 = 180 – measures of 3, 4, and 5. Then explain how m 1. 4 5, so m 4 = m 5. The sum of the you used m 2 to find these same measures. angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so d. Algebraic If m 1 = x, write an expression for m 3 = 180 – m 2. m 2 is twice as much as m the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle 4 and m 5, so m 4 = m 5 = measures. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4-6 Isosceles and Equilateral Triangles ANSWER: 180; m 3 = 180 – x, m 4 = m 5 = a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

4 and m 5, so m 4 = m 5 = The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 180; m 3 = 180 – x, m 4 = m 5 = ANSWER: eSolutions45. CHALLENGEManual - Powered In bytheCognero figure, if is equilateral Sometimes; only if the measure of the vertex anglePage is16 and ZWP WJM JZL, prove that even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

measure of the vertex angle is even. For example, Neither; m G = or 55. vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 49. OPEN-ENDED If possible, draw an isosceles ANSWER: triangle with base angles that are obtuse. If it is not Sometimes; only if the measure of the vertex angle is possible, explain why not. even. SOLUTION: It is not possible because a triangle cannot have more 47. If the measures of the base angles of an isosceles than one obtuse angle. triangle are integers, then the measure of its vertex angle is odd. ANSWER: SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle. The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base 50. REASONING In isosceles m B = 90. angle) so if the base angles are integers, then 2 Draw the triangle. Indicate the congruent sides and (measure of the base angle) will be even and 180 – 2 label each angle with its measure. (measure of the base angle) will be even. SOLUTION: ANSWER: The sum of the angle measures in a triangle must be Never; the measure of the vertex angle will be 180 – 180, 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will be even and 180 – 2(measure of the base angle) will Since is isosceles, be even. And given that m∠B = 90. Therefore, ERROR ANALYSIS 48. Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning. Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

180, a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC

B C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ?

SOLUTION: a. Of the statements given, only choice B is not ANSWER: necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ 51. WRITING IN MATH How can triangle △DEC by SAS, so AB ≅ EC. classifications help you prove triangle congruence? ANSWER: SOLUTION: a. B If a triangle is already classified, you can use the b. 40 previously proven properties of that type of triangle c. 80 in the proof. For example, if you know that a d. 60; Sample answer: All three angles in △AEB triangle is an equilateral triangle, you can use are 60° because base angles of an isosceles triangle Corollary 4.3 and 4.4 in the proof. Doing this can are congruent. So △AEB is equiangular and save you steps when writing the proof. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

52. MULTI-STEP △AED and △BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B

d. What measure would ∠AEB need to be so that ANSWER: ? D

SOLUTION: 54. A jeweler bends wire to make charms in the shape a. Of the statements given, only choice B is not of isosceles triangles. The base of the triangle necessarily true from the given information. measures 2 centimeters and another side measures b. The angles are congruent, so ∠DCE is 40°. 3.5 centimeters. How many charms can be made c. The sum of the measures of the angles of a from a piece of wire that is 63 centimeters long? triangle is 180, so ∠BEC must be 80°. SOLUTION: d. △ 60; All three angles in AEB are 60° because Because we know that the shape of each charm is base angles of an isosceles triangle are congruent. So an isosceles triangle, then we can determine the side △AEB is equiangular and equilateral. △AEB ≅ lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, △DEC by SAS, so AB ≅ EC. the perimeter of each triangle is 9 cm.

ANSWER: To find how many charms can be made, divide 63 a. B cm by 9 cm. So, the jeweler can make 7 charms. b. 40 c. 80 ANSWER: d. 60; Sample answer: All three angles in △AEB 7 are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 55. In the figure below, NDG LGD. equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

a. What is m∠L? b. What is the value of x? A 34 c. What is m∠DGN? B 56 C 60 SOLUTION: D 62 a. In this figure, angles N and L are congruent so E 124 m∠L = 36°. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is SOLUTION: 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: a. 36 b. 54 c. 54 Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two 56. An equilateral triangle has a perimeter of 216 inches. angles would be 180 – 56 or 124°. What is the length of one side of the triangle? So, m∠FEG is , or 62. The correct answer is SOLUTION: choice D. Because the triangle is equilateral that means each ANSWER: side has the same length, so divide the perimeter by 3 to find the length of one side. D 54. A jeweler bends wire to make charms in the shape 216 ÷ 3 = 72 of isosceles triangles. The base of the triangle ANSWER: measures 2 centimeters and another side measures 72 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of Because we know that the shape of each charm is △JKL? an isosceles triangle, then we can determine the side SOLUTION: lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, The perimeter can be found by adding all three side the perimeter of each triangle is 9 cm. lengths. In this situation, since the triangle is

isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. To find how many charms can be made, divide 63

cm by 9 cm. So, the jeweler can make 7 charms. Because the x-terms cancel each other out: 2x + 2x ANSWER: – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. 7

55. In the figure below, NDG LGD. So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. SOLUTION:

In an isosceles triangle, the base angles are a. What is m∠L? congruent. b. What is the value of x?

c. What is m∠DGN? So, 5x + 5 = 7x – 19. SOLUTION: Simplify. a. In this figure, angles N and L are congruent so

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16. Consider each choice. So, the perimeter of △JKL is 16. A does not provide enough information to ANSWER: determine how sides VT and TR are related. 16 B 58. △ABC is an isosceles triangle with base angles If we know that m∠VRT = 60, we still would not measuring (5x + 5)° and (7x – 19)°. Find the value of have enough information to prove that the other two x. measures are 60 degrees.

SOLUTION: C In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

Refer to the figure.

SOLUTION: By the Triangle Angle-Sum Theorem,

1. If name two congruent angles.

SOLUTION: Isosceles Triangle Theorem states that if two sides Since the measures of all the three angles are 60°; of the triangle are congruent, then the angles opposite the triangle must be equiangular. All the equiangular those sides are congruent. triangles are equilateral. Therefore, FH = GH = 12. Therefore, in triangle ABC, ANSWER: ANSWER: 12 BAC and BCA 4. m MRP 2. If EAC ECA, name two congruent segments. SOLUTION: Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore,

in triangle EAC, ANSWER: SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral Find each measure. triangle measures 60°. Therefore, m MRP = 60°. 3. FH ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION: By the Triangle Angle-Sum Theorem,

5. SOLUTION:

Since the measures of all the three angles are 60°; In the figure, . Therefore, triangle RST is the triangle must be equiangular. All the equiangular an isosceles triangle. triangles are equilateral. Therefore, FH = GH = 12. By the Converse of Isosceles Triangle Theorem,

ANSWER: That is, . 12

4. m MRP

ANSWER: 12

SOLUTION: Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. 6.

ANSWER: SOLUTION: 60 In the figure, Therefore, triangle WXY is an isosceles triangle. SENSE-MAKING Find the value of each By the Isosceles Triangle Theorem, . variable.

5. ANSWER: 16 SOLUTION: In the figure, . Therefore, triangle RST is 7. PROOF Write a two-column proof. an isosceles triangle. By the Converse of Isosceles Triangle Theorem, Given: is isosceles; bisects ABC. Prove: That is, .

ANSWER: 12 SOLUTION:

6. SOLUTION: In the figure, Therefore, triangle WXY is ANSWER: an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: 16 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A 7. PROOF Write a two-column proof. portion of the track is shown.

Given: is isosceles; bisects ABC. Prove:

a. If and are perpendicular to is SOLUTION: isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning.

SOLUTION: a. ANSWER:

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. a. If and are perpendicular to is By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. isosceles with base , and prove that ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the a. distance between and Explain your reasoning.

SOLUTION: a.

b. The Pythagorean Theorem tells us that By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure. b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. 9. If name two congruent angles. ANSWER: SOLUTION: a. By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABF,

ANSWER:

11. If name two congruent angles. b. The Pythagorean Theorem tells us that By CPCTC we know SOLUTION: that VT = 1.5 m. The Seg. Add. Post. says QV + By the Isosceles Triangle Theorem, in triangle ACD,

VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ANSWER:

Refer to the figure. ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

9. If name two congruent angles. ANSWER: SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, 13. If BCF BFC, name two congruent segments. SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in ABE and AEB triangle BCF,

10. If ∠ABF ∠AFB, name two congruent segments. ANSWER: SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ABF, 14. If name two congruent angles. SOLUTION: ANSWER: By the Isosceles Triangle Theorem, in triangle AFH,

11. If name two congruent angles. ANSWER: AFH and AHF SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, Find each measure. 15. m BAC ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: SOLUTION: By the Converse of Isosceles Triangle Theorem, in Here triangle ADE, By Isosceles Triangle Theorem, .

ANSWER: Apply the Triangle Sum Theorem.

13. If BCF BFC, name two congruent segments.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in 60 triangle BCF,

ANSWER: 16. m SRT

14. If name two congruent angles.

SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH, SOLUTION:

Given: ANSWER: By Isosceles Triangle Theorem, . AFH and AHF Apply Triangle Angle-Sum Theorem. Find each measure. 15. m BAC

ANSWER: 65

17. TR SOLUTION: Here By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem.

SOLUTION: Since the triangle is isosceles, Therefore, ANSWER:

16. m SRT All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: SOLUTION: 4 Given: 18. CB By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

SOLUTION: ANSWER: By the Converse of Isosceles Triangle Theorem, in 65 triangle ABC, 17. TR That is, CB = 3. ANSWER: 3 REGULARITY Find the value of each variable.

SOLUTION: Since the triangle is isosceles, Therefore, 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, All the angles are congruent. Therefore, it is an Solve for x. equiangular triangle. Since the equiangular triangle is an equilateral,

ANSWER: ANSWER: 4 x = 5 18. CB

20.

SOLUTION: SOLUTION: Given: By the Converse of Isosceles Triangle Theorem, in By the Isosceles Triangle Theorem, triangle ABC, That is, CB = 3.

ANSWER: And We know 3 that, REGULARITY Find the value of each variable.

ANSWER: x = 13 19. SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, Solve for x. 21.

SOLUTION: Given: By the Isosceles Triangle Theorem, ANSWER: x = 5 That is,

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 2: 20. SOLUTION:

Given: Add the equations 1 and 2. By the Isosceles Triangle Theorem,

And We know

that, Substitute the value of x in one of the two equations to find the value of y.

ANSWER: x = 13

ANSWER: 21. x = 11, y = 11 SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, 22. Then, let Equation 1 be: . SOLUTION: By the Triangle Sum Theorem, we can find Equation Given: 2: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Add the equations 1 and 2. Therefore,

Set up two equations to solve for x and y.

Substitute the value of x in one of the two equations to find the value of y. ANSWER: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

ANSWER: x = 11, y = 11

SOLUTION: Proof: We are given that is an isosceles 22. triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and SOLUTION: MLH are supplementary. From the Isosceles Given: Triangle Theorem, we know that HJK HML. By the Triangle Angle-Sum Because is an equilateral triangle, we know Theorem, HLK LKH KHL and

So, the triangle is an equilateral triangle. JKH, HKL and HLK, Therefore, MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent Supplements Theorem. By AAS, Set up two equations to solve for x and y. By CPCTC, JHK MHL. ANSWER: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH ANSWER: and HKL are supplementary and HLK and x = 1.4, y = –1 MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. PROOF Write a paragraph proof. Because is an equilateral triangle, we know 23. Given: is isosceles, and is HLK LKH KHL and equilateral. JKH and HKL are supplementary JKH, HKL and HLK, and HLK and MLH are supplementary. MLH are supplementary, and HKL HLK, Prove: JHK MHL we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

24. Given: W is the midpoint of

SOLUTION: Q is the midpoint of

Proof: We are given that is an isosceles Prove: triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML. Because is an equilateral triangle, we know HLK LKH KHL and JKH, HKL and HLK, MLH are supplementary, and HKL HLK, SOLUTION: we know JKH MLH by the Congruent Proof: Supplements Theorem. By AAS, We are given W is the midpoint of By CPCTC, JHK MHL. and Q is the midpoint of Because W is the midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, Proof: We are given W is the midpoint of By CPCTC, JHK MHL. and Q is the midpoint of Because W is the midpoint of we know that Similarly, 24. Given: because Q is the midpoint of The W is the midpoint of Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = Q is the midpoint of XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Prove: If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC.

25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to SOLUTION: determine that but Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. ANSWER: b. If show that is isosceles. Proof: We are given W is the midpoint of c. If and show that is and Q is the midpoint of Because W is the equilateral. d. midpoint of we know that Similarly, If is isosceles, what is the minimum information needed to prove that because Q is the midpoint of The Explain your reasoning. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. SOLUTION: If we divide each side by 2, we have WY = QZ. The a. Given: Isosceles Triangle Theorem says XYZ XZY. By Isosceles Triangle Theorem, . by the Reflexive Property. By SAS, Apply Triangle Sum Theorem. So, by CPCTC.

25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. b. Using a jump rope to measure, Elisa is able to determine that but

SOLUTION: a. Given: By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and

one pair of congruent corresponding angles, the c. triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. d. b. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA c. or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of ANSWER: the roof, ABC.

SOLUTION:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

ANSWER:

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I Sample answer: I constructed a pair of perpendicular measured both legs for each triangle. Because AB = segments and then used the same compass setting to AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = mark points equidistant from their intersection. I 2.3 cm, the triangles are isosceles. I used a measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = protractor to confirm that A, D, and G are all 2.3 cm, the triangles are isosceles. I used a right angles. protractor to confirm that A, D, and G are all ANSWER: right angles.

REGULARITY Find each measure.

Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to mark points equidistant from their intersection. I measured both legs for each triangle. Because AB = AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = 2.3 cm, the triangles are isosceles. I used a 29. m∠CAD protractor to confirm that A, D, and G are all right angles. SOLUTION: From the figure, 28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship Therefore, is an isosceles triangle. between the base angles of an isosceles right triangle. By the Triangle Sum Theorem, SOLUTION: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute ANSWER: angle be x. The acute angles of a right triangle are 44 complementary, Solve for x 30. m ACD SOLUTION: ANSWER: We know Conjecture: The measures of the base angles of an that isosceles right triangle are 45. Therefore, . Proof: The base angles are congruent because it is an isosceles triangle. Let the measure of each acute ANSWER: angle be x. The acute angles of a right triangle are 44 complementary, so x + x = 90 and x = 45. 31. m ACB REGULARITY Find each measure. SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

29. m∠CAD SOLUTION: ANSWER:

From the figure, 136 Therefore, is an isosceles triangle. 32. m ABC

SOLUTION: By the Triangle Sum Theorem, From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, ANSWER: 44

30. m ACD ANSWER: SOLUTION: 22 We know FITNESS that 33. In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids Therefore, . shown. If each triangle is isosceles with vertex ANSWER: angles G, H, and J, and G 44 H, and H J, show that the distance from B to F is three times the distance from D to F. 31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

SOLUTION:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

ANSWER:

ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G H, and H J, show that the distance from B to F is three times the distance from D to F.

34. Given: is isosceles; Prove: X and YZV are complementary.

SOLUTION:

ANSWER:

34. Given: is isosceles; Prove: X and YZV are complementary. PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

SOLUTION:

ANSWER:

36. Corollary 4.4 PROOF Write a two-column proof of each SOLUTION: corollary or theorem. 35. Corollary 4.3 SOLUTION:

ANSWER:

37. Theorem 4.11 SOLUTION:

36. Corollary 4.4 SOLUTION: ANSWER:

Find the value of each variable. ANSWER:

38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x. 37. Theorem 4.11 SOLUTION: Note that x can equal –8 here because .

ANSWER: 3

ANSWER: 39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Find the value of each variable.

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. 38. ANSWER: SOLUTION: 14 By the converse of Isosceles Triangle theorem, GAMES Use the diagram of a game timer shown to find each measure.

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 40. m LPM SOLUTION: Angles at a point in a straight line add up to 180 .

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is

equal to (2y – 5) . Substitute x = 45 in (3x – 55) to find

The interior angles of a triangle add up to 180 .

ANSWER: 80

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. 41. m LMP

ANSWER: SOLUTION: 14 Since the triangle LMP is isosceles, Angles at a point in a straight line GAMES Use the diagram of a game timer add up to 180 . shown to find each measure.

Substitute x = 45 in (3x – 55) to find

40. m LPM Therefore, SOLUTION: ANSWER: Angles at a point in a straight line add up to 180 . 80

42. m JLK SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

ANSWER: 80 So, 41. m LMP ANSWER: SOLUTION: 20 Since the triangle LMP is isosceles, Angles at a point in a straight line 43. m JKL add up to 180 . SOLUTION:

Vertical angles are congruent. Therefore,

First we need to find

Substitute x = 45 in (3x – 55) to find We know that,

So, Therefore, ANSWER: In

80 Since the triangle JKL is isosceles, 42. m JLK SOLUTION: Vertical angles are congruent. Therefore, ANSWER: 80 First we need to find 44. MULTIPLE REPRESENTATIONS In this We know that, problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

So,

ANSWER: 20

43. m JKL a. Geometric Use a ruler and a protractor to draw SOLUTION: three different isosceles triangles, extending one of Vertical angles are congruent. Therefore, the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown. First we need to find b. Tabular Use a protractor to measure and record We know that, m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same measures. Organize your results in two tables. So, c. Verbal Explain how you used m 1 to find the In measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. Since the triangle JKL is isosceles, d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle

ANSWER: measures.

80 SOLUTION: MULTIPLE REPRESENTATIONS 44. In this a. problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures.

d. Algebraic If m 1 = x, write an expression for b. the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle measures.

SOLUTION:

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = b. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so ANSWER: m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

The statement is sometimes true. It is true only if the SOLUTION: measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5.

ANSWER: Sometimes; only if the measure of the vertex angle is even.

ANSWER: Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2(measure of the base angle) will 4-6 Isosceles and Equilateral Triangles be even and 180 – 2(measure of the base angle) will be even.

PRECISION Determine whether the following 48. ERROR ANALYSIS Alexis and Miguela are statements are sometimes, always, or never true. finding m G in the figure shown. Alexis says that Explain. m G = 35, while Miguela says that m G = 60. Is 46. If the measure of the vertex angle of an isosceles either of them correct? Explain your reasoning. triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. ANSWER: SOLUTION: Sometimes; only if the measure of the vertex angle is Neither of them is correct. This is an isosceles even. triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

Neither; m G = or 55. ANSWER: Never; the measure of the vertex angle will be 180 – 2(measure of the base angle) so if the base angles 49. OPEN-ENDED If possible, draw an isosceles are integers, then 2(measure of the base angle) will triangle with base angles that are obtuse. If it is not be even and 180 – 2(measure of the base angle) will possible, explain why not. be even. SOLUTION: 48. ERROR ANALYSIS Alexis and Miguela are It is not possible because a triangle cannot have more than one obtuse angle. finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning. It is not possible because a triangle cannot have more than one obtuse angle.

50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be SOLUTION: 180, Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an Since is isosceles, isosceles triangle, the base angles are congruent. And given that m∠B = 90. Therefore, eSolutions Manual - Powered by Cognero Page 17

Construct the triangle with the angles measures 90, 45, 45.

ANSWER:

Neither; m G = or 55.

OPEN-ENDED 49. If possible, draw an isosceles triangle with base angles that are obtuse. If it is not possible, explain why not. ANSWER: SOLUTION: It is not possible because a triangle cannot have more than one obtuse angle.

ANSWER: It is not possible because a triangle cannot have more than one obtuse angle. 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? 50. REASONING In isosceles m B = 90. SOLUTION: Draw the triangle. Indicate the congruent sides and label each angle with its measure. If a triangle is already classified, you can use the previously proven properties of that type of triangle SOLUTION: in the proof. For example, if you know that a The sum of the angle measures in a triangle must be triangle is an equilateral triangle, you can use 180, Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. Since is isosceles, And given that m∠B = 90. ANSWER: Therefore, Sample answer: If a triangle is already classified,

you can use the previously proven properties of that type of triangle in the proof. Doing this can save you steps when writing the proof.

Construct the triangle with the angles measures 90, 52. MULTI-STEP △AED and △BEC are congruent 45, 45. and isosceles.

ANSWER: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D 51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? b. If m∠EAB = 40, what is the measure of ∠DCE? SOLUTION: If a triangle is already classified, you can use the c. If m∠ADE = 50, what is the measure of ∠BEC?

previously proven properties of that type of triangle d. What measure would AEB need to be so that in the proof. For example, if you know that a ∠ triangle is an equilateral triangle, you can use ? Corollary 4.3 and 4.4 in the proof. Doing this can SOLUTION: save you steps when writing the proof. a. Of the statements given, only choice B is not necessarily true from the given information. ANSWER: b. The angles are congruent, so ∠DCE is 40°. Sample answer: If a triangle is already classified, c. The sum of the measures of the angles of a you can use the previously proven properties of that triangle is 180, so ∠BEC must be 80°. type of triangle in the proof. Doing this can save you d. 60; All three angles in △AEB are 60° because steps when writing the proof. base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ MULTI-STEP △ △ 52. AED and BEC are congruent △DEC by SAS, so AB ≅ EC. and isosceles. ANSWER: a. B b. 40 c. 80 d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and a. Which statement is not necessarily true from the equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. information given? 53. What is m∠FEG? A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC? A 34 B 56 d. What measure would ∠AEB need to be so that C 60 D ? 62 E 124 SOLUTION: a. Of the statements given, only choice B is not SOLUTION: necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because

base angles of an isosceles triangle are congruent. So △AEB is equiangular and equilateral. △AEB ≅ Consider triangle DEH. Since one angle is 56°, and △DEC by SAS, so AB ≅ EC. the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. ANSWER: So, m∠FEG is , or 62. The correct answer is a. B choice D. b. 40 c. 80 ANSWER: △ d. 60; Sample answer: All three angles in AEB D are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and 54. A jeweler bends wire to make charms in the shape equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures 53. What is m∠FEG? 3.5 centimeters. How many charms can be made from a piece of wire that is 63 centimeters long? SOLUTION: Because we know that the shape of each charm is an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm.

A 34 To find how many charms can be made, divide 63 B 56 cm by 9 cm. So, the jeweler can make 7 charms. C 60 D 62 ANSWER: E 124 7

55. In the figure below, NDG LGD. SOLUTION:

Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. a. What is m∠L? b. What is the value of x? So, m∠FEG is , or 62. The correct answer is c. What is m∠DGN? choice D. SOLUTION: ANSWER: a. In this figure, angles N and L are congruent so D m∠L = 36°. 54. A jeweler bends wire to make charms in the shape b. Because we know two angle measures, and that of isosceles triangles. The base of the triangle the sum of the measures of the angles of a triangle is measures 2 centimeters and another side measures 180, then we can solve 36 + 90 + x = 180, or x = 54. 3.5 centimeters. How many charms can be made c. ∠DGN is congruent to x°, so m∠DGN is 54°. from a piece of wire that is 63 centimeters long? ANSWER: SOLUTION: a. 36 Because we know that the shape of each charm is b. 54 an isosceles triangle, then we can determine the side c. 54 lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, the perimeter of each triangle is 9 cm. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. SOLUTION: Because the triangle is equilateral that means each ANSWER: side has the same length, so divide the perimeter by 3 7 to find the length of one side.

55. In the figure below, NDG LGD. 216 ÷ 3 = 72

ANSWER: 72

57. △JKL is an isosceles triangle, and . JK = 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL?

SOLUTION: a. What is m∠L? The perimeter can be found by adding all three side b. What is the value of x? lengths. In this situation, since the triangle is c. What is m∠DGN? isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

SOLUTION: Because the x-terms cancel each other out: 2x + 2x a. In this figure, angles N and L are congruent so – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or m∠L = 36°. 16. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is So, the perimeter of △JKL is 16. 180, then we can solve 36 + 90 + x = 180, or x = 54. ANSWER: c. ∠DGN is congruent to x°, so m∠DGN is 54°. 16 ANSWER: 58. △ABC is an isosceles triangle with base angles a. 36 measuring (5x + 5)° and (7x – 19)°. Find the value of b. 54 x. c. 54 SOLUTION: 56. An equilateral triangle has a perimeter of 216 inches. In an isosceles triangle, the base angles are What is the length of one side of the triangle? congruent. SOLUTION: So, 5x + 5 = 7x – 19. Because the triangle is equilateral that means each

side has the same length, so divide the perimeter by 3 Simplify. to find the length of one side.

–2x = –24 216 ÷ 3 = 72 x = 12 ANSWER: ANSWER: 72 12 57. △JKL is an isosceles triangle, and . JK = 59. QRV is an equilateral triangle. What additional 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? information would be enough to prove VRT is an equilateral triangle? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. A ANSWER: B m∠VRT = 60 16 C QRV VTU D VRT RST 58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of SOLUTION: x. SOLUTION: In an isosceles triangle, the base angles are congruent.

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 Consider each choice. x = 12

ANSWER: A does not provide enough information to 12 determine how sides VT and TR are related.

59. QRV is an equilateral triangle. What additional B If we know that m∠VRT = 60, we still would not information would be enough to prove VRT is an have enough information to prove that the other two equilateral triangle? measures are 60 degrees.

A B m∠VRT = 60 If , then m∠RVT = 60 because C QRV VTU ∠QVR + ∠RVT + ∠TVU = 180. Since , D VRT RST by the Isosceles Triangle Theorem. So, . SOLUTION: Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: Refer to the figure. 12

4. m MRP

1. If name two congruent angles. SOLUTION:

Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite SOLUTION: those sides are congruent. Since all the sides are congruent, is an Therefore, in triangle ABC, equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. ANSWER: BAC and BCA ANSWER: 60 2. If EAC ECA, name two congruent segments. SENSE-MAKING Find the value of each SOLUTION: variable. Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC, 5. ANSWER: SOLUTION: In the figure, . Therefore, triangle RST is Find each measure. an isosceles triangle. 3. FH By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: By the Triangle Angle-Sum Theorem, ANSWER: 12

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 6. 12 SOLUTION: In the figure, Therefore, triangle WXY is 4. m MRP an isosceles triangle. By the Isosceles Triangle Theorem, .

5. SOLUTION: SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER:

ANSWER: 12

a. If and are perpendicular to is ANSWER: isosceles with base , and prove that 16 7. PROOF Write a two-column proof. b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your Given: is isosceles; bisects ABC. reasoning. Prove: SOLUTION: a.

SOLUTION:

ANSWER:

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. 8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A ANSWER: portion of the track is shown. a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your b. The Pythagorean Theorem tells us that reasoning. By CPCTC we know

that VT = 1.5 m. The Seg. Add. Post. says QV + SOLUTION: VT = QT. By substitution, we have 1.5 + 1.5 = QT. a. So QT = 3 m. Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB b. The Pythagorean Theorem tells us that: 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By CPCTC we know that VT = 1.5 m. The By the Converse of Isosceles Triangle Theorem, in Segment Addition Postulate says QV + VT = QT. triangle ABF, By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. ANSWER:

ANSWER: a. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: b. The Pythagorean Theorem tells us that

By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + 13. If BCF BFC, name two congruent segments. VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m. SOLUTION: By the Converse of Isosceles Triangle Theorem, in Refer to the figure. triangle BCF,

ANSWER:

14. If name two congruent angles. SOLUTION: 9. If name two congruent angles. By the Isosceles Triangle Theorem, in triangle AFH, SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE, ANSWER:

AFH and AHF ANSWER: Find each measure. ABE and AEB 15. m BAC

10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ABF, SOLUTION: ANSWER: Here By Isosceles Triangle Theorem, .

11. If name two congruent angles. Apply the Triangle Sum Theorem.

SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ANSWER: ACD and ADC 60

12. If DAE DEA, name two congruent 16. m SRT segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: SOLUTION:

Given: By Isosceles Triangle Theorem, . 13. If BCF BFC, name two congruent segments. SOLUTION: Apply Triangle Angle-Sum Theorem. By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER: ANSWER: 65 14. If name two congruent angles. 17. TR SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER:

AFH and AHF SOLUTION: Find each measure. Since the triangle is isosceles, 15. m BAC Therefore,

All the angles are congruent. Therefore, it is an SOLUTION: equiangular triangle. Since the equiangular triangle is an equilateral, Here

By Isosceles Triangle Theorem, . ANSWER:

Apply the Triangle Sum Theorem. 4

18. CB

ANSWER: 60

16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: SOLUTION: 3 Given: REGULARITY Find the value of each variable. By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem.

19. SOLUTION: ANSWER: Since all the angles are congruent, the sides are also congruent to each other. 65 Therefore, 17. TR Solve for x.

ANSWER:

x = 5 SOLUTION: Since the triangle is isosceles, Therefore,

20. SOLUTION: All the angles are congruent. Therefore, it is an Given: equiangular triangle. By the Isosceles Triangle Theorem, Since the equiangular triangle is an equilateral,

ANSWER: And 4 We know that, 18. CB

ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3. 21. ANSWER: SOLUTION: 3 Given:

REGULARITY Find the value of each variable. By the Isosceles Triangle Theorem,

That is,

ANSWER:

x = 5 Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION: Given: By the Isosceles Triangle Theorem,

ANSWER: And x = 11, y = 11 We know that,

ANSWER: 22. x = 13 SOLUTION: Given: By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. 21. Therefore, SOLUTION:

Set up two equations to solve for x and y. Given: By the Isosceles Triangle Theorem,

That is,

Then, let Equation 1 be: . ANSWER: x = 1.4, y = –1 By the Triangle Sum Theorem, we can find Equation 2: PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Add the equations 1 and 2. Prove: JHK MHL

HLK LKH KHL and JKH, HKL and HLK, ANSWER: MLH are supplementary, and HKL HLK, x = 11, y = 11 we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

Therefore, we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, Set up two equations to solve for x and y. By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of ANSWER: Prove: x = 1.4, y = –1

PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the midpoint of we know that Similarly, SOLUTION: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, ANSWER: we know JKH MLH by the Congruent Proof: We are given W is the midpoint of Supplements Theorem. By AAS, and Q is the midpoint of Because W is the By CPCTC, JHK MHL. midpoint of we know that Similarly, ANSWER: because Q is the midpoint of The Proof: We are given that is an isosceles Segment Addition Postulate gives us XW + WY = XY triangle and is an equilateral triangle, JKH and XQ + QZ = XZ. Substitution gives XW + WY = and HKL are supplementary and HLK and XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. MLH are supplementary. From the Isosceles If we divide each side by 2, we have WY = QZ. The Triangle Theorem, we know that HJK HML. Isosceles Triangle Theorem says XYZ XZY. Because is an equilateral triangle, we know by the Reflexive Property. By SAS, HLK LKH KHL and So, by CPCTC. JKH, HKL and HLK, MLH are supplementary, and HKL HLK, 25. BABYSITTING While babysitting her neighbor’s we know JKH ∠MLH by the Congruent children, Elisa observes that the supports on either Supplements Theorem. By AAS, side of a park swing set form two sets of triangles. By CPCTC, JHK MHL. Using a jump rope to measure, Elisa is able to determine that but

24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. SOLUTION: b. If show that is isosceles. Proof: We are given W is the midpoint of c. If and show that is equilateral. and Q is the midpoint of Because W is the d. If is isosceles, what is the minimum midpoint of we know that Similarly, information needed to prove that because Q is the midpoint of The Explain your reasoning. Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = SOLUTION: XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The a. Given: Isosceles Triangle Theorem says XYZ XZY. By Isosceles Triangle Theorem, . Apply Triangle Sum Theorem. by the Reflexive Property. By SAS,

So, by CPCTC.

by the Reflexive Property. By SAS, So, by CPCTC. c.

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is equilateral. d. If is isosceles, what is the minimum d. One pair of congruent corresponding sides and information needed to prove that one pair of congruent corresponding angles; since Explain your reasoning. you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that both pairs of legs are congruent. Because the base SOLUTION: angles of an isosceles triangle are congruent, if you a. Given: know that you know that By Isosceles Triangle Theorem, . Therefore, Apply Triangle Sum Theorem. with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

ANSWER: b. a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

SOLUTION:

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle.

Sample answer: I constructed a pair of perpendicular SOLUTION: segments and then used the same compass setting to Conjecture: The measures of the base angles of an mark points equidistant from their intersection. I isosceles right triangle are 45. measured both legs for each triangle. Because AB = Proof: The base angles are congruent because it is AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = an isosceles triangle. Let the measure of each acute 2.3 cm, the triangles are isosceles. I used a angle be x. The acute angles of a right triangle are protractor to confirm that A, D, and G are all complementary, Solve for x right angles.

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

isosceles right triangle are 45. 31. m ACB Proof: The base angles are congruent because it is SOLUTION: an isosceles triangle. Let the measure of each acute The angles in a straight line add to 180°. angle be x. The acute angles of a right triangle are Therefore, complementary, Solve for x

We know that

triangle. Therefore,

By the Triangle Angle Sum Theorem,

angles G, H, and J, and G By the Triangle Sum Theorem, H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD

SOLUTION: We know SOLUTION: that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that ANSWER:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem, 34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER: 22

SOLUTION: ANSWER:

ANSWER: PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION: ANSWER:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

36. Corollary 4.4 38. SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: ANSWER: 3

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is 37. Theorem 4.11 equal to (2y – 5) .

SOLUTION: The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM 38. SOLUTION: SOLUTION: Angles at a point in a straight line add up to 180 . By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Substitute x = 45 in (3x – 55) to find

Note that x can equal –8 here because .

ANSWER: 3 ANSWER: 80

41. m LMP SOLUTION: 39. Since the triangle LMP is isosceles, SOLUTION: Angles at a point in a straight line By the Isosceles Triangle Theorem, the third angle is add up to 180 . equal to (2y – 5) .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. Therefore,

ANSWER: ANSWER: 14 80 GAMES Use the diagram of a game timer 42. m JLK shown to find each measure. SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM SOLUTION: So, Angles at a point in a straight line add up to 180 . ANSWER:

43. m JKL SOLUTION: Substitute x = 45 in (3x – 55) to find Vertical angles are congruent. Therefore,

First we need to find

We know that, ANSWER: 80

41. m LMP So,

SOLUTION: In Since the triangle LMP is isosceles, Angles at a point in a straight line Since the triangle JKL is isosceles, add up to 180 .

ANSWER: 80 Substitute x = 45 in (3x – 55) to find 44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle. Therefore,

ANSWER: 80

42. m JLK

SOLUTION: Vertical angles are congruent. Therefore, a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of First we need to find the base angles, and labeling as shown.

We know that, b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and record m 2 and use it to calculate these same

So, measures. Organize your results in two tables.

ANSWER: c. Verbal Explain how you used m 1 to find the 20 measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. 43. m JKL SOLUTION: d. Algebraic If m 1 = x, write an expression for Vertical angles are congruent. Therefore, the measures of 3, 4, and 5. Likewise, if m 2 = x, write an expression for these same angle First we need to find measures.

We know that, SOLUTION:

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of b. the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the d. Algebraic If m 1 = x, write an expression for angle measures in a triangle must be 180, so m 3 = the measures of 3, 4, and 5. Likewise, if m 180 – m 4 – 5. 2 is supplementary to 3, so 2 = x, write an expression for these same angle m 3 = 180 – m 2. m 2 is twice as much as m measures. 4 and m 5, so m 4 = m 5 = SOLUTION: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

a. 180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the 4 and m 5, so m 4 = m 5 = angle measures in a triangle must be 180, so m 3 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m 180; m 3 = 180 – x, m 4 = m 5 = 4 and m 5, so m 4 = m 5 = ANSWER: d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – a. 180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

b. ANSWER:

180; m 3 = 180 – x, m 4 = m 5 = measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that ANSWER: Sometimes; only if the measure of the vertex angle is

even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is ANSWER: either of them correct? Explain your reasoning.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an PRECISION Determine whether the following isosceles triangle, the base angles are congruent. statements are sometimes, always, or never true. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; ANSWER: vertex angle = 55, base angles = 62.5. Neither; m G = or 55. ANSWER: Sometimes; only if the measure of the vertex angle is 49. OPEN-ENDED If possible, draw an isosceles even. triangle with base angles that are obtuse. If it is not possible, explain why not. 47. If the measures of the base angles of an isosceles SOLUTION: triangle are integers, then the measure of its vertex It is not possible because a triangle cannot have more

angle is odd. than one obtuse angle. SOLUTION: ANSWER: The statement is never true. The measure of the vertex angle will be 180 – 2(measure of the base It is not possible because a triangle cannot have more

angle) so if the base angles are integers, then 2 than one obtuse angle. (measure of the base angle) will be even and 180 – 2 50. REASONING In isosceles m B = 90. (measure of the base angle) will be even. Draw the triangle. Indicate the congruent sides and ANSWER: label each angle with its measure. Never; the measure of the vertex angle will be 180 – SOLUTION: 2(measure of the base angle) so if the base angles The sum of the angle measures in a triangle must be are integers, then 2(measure of the base angle) will 180, be even and 180 – 2(measure of the base angle) will be even. Since is isosceles, 48. ERROR ANALYSIS Alexis and Miguela are And given that m∠B = 90. finding m G in the figure shown. Alexis says that Therefore,

m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. ANSWER:

51. WRITING IN MATH How can triangle classifications help you prove triangle congruence? ANSWER: SOLUTION: Neither; m G = or 55. If a triangle is already classified, you can use the previously proven properties of that type of triangle 49. OPEN-ENDED If possible, draw an isosceles in the proof. For example, if you know that a triangle with base angles that are obtuse. If it is not triangle is an equilateral triangle, you can use possible, explain why not. Corollary 4.3 and 4.4 in the proof. Doing this can SOLUTION: save you steps when writing the proof. It is not possible because a triangle cannot have more ANSWER: than one obtuse angle. Sample answer: If a triangle is already classified, ANSWER: you can use the previously proven properties of that 4-6 IsoscelesIt is not possible and Equilateral because a triangle Triangles cannot have more type of triangle in the proof. Doing this can save you than one obtuse angle. steps when writing the proof.

50. REASONING In isosceles m B = 90. 52. MULTI-STEP △AED and △BEC are congruent Draw the triangle. Indicate the congruent sides and and isosceles. label each angle with its measure. SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles, And given that m∠B = 90. a. Which statement is not necessarily true from the Therefore, information given?

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC Construct the triangle with the angles measures 90, D 45, 45. b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? ANSWER: SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. d. 60; All three angles in △AEB are 60° because 51. WRITING IN MATH How can triangle base angles of an isosceles triangle are congruent. So classifications help you prove triangle congruence? △AEB is equiangular and equilateral. △AEB ≅ SOLUTION: △DEC by SAS, so AB ≅ EC. If a triangle is already classified, you can use the ANSWER: previously proven properties of that type of triangle a. B in the proof. For example, if you know that a b. 40 triangle is an equilateral triangle, you can use c. 80 Corollary 4.3 and 4.4 in the proof. Doing this can d. 60; Sample answer: All three angles in △AEB save you steps when writing the proof. are 60° because base angles of an isosceles triangle are congruent. So △AEB is equiangular and ANSWER: equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. Sample answer: If a triangle is already classified, you can use the previously proven properties of that 53. What is m∠FEG? type of triangle in the proof. Doing this can save you steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles. eSolutions Manual - Powered by Cognero Page 18

A 34 B 56 C 60 D 62 E 124 a. Which statement is not necessarily true from the information given? SOLUTION:

A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? Consider triangle DEH. Since one angle is 56°, and the triangle is isosceles, then the remaining two c. If m∠ADE = 50, what is the measure of ∠BEC? angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is d. What measure would ∠AEB need to be so that choice D. ? ANSWER: SOLUTION: D a. Of the statements given, only choice B is not necessarily true from the given information. 54. A jeweler bends wire to make charms in the shape b. The angles are congruent, so ∠DCE is 40°. of isosceles triangles. The base of the triangle c. The sum of the measures of the angles of a measures 2 centimeters and another side measures triangle is 180, so ∠BEC must be 80°. 3.5 centimeters. How many charms can be made d. 60; All three angles in △AEB are 60° because from a piece of wire that is 63 centimeters long? base angles of an isosceles triangle are congruent. So SOLUTION: △ △ AEB is equiangular and equilateral. AEB ≅ Because we know that the shape of each charm is △DEC by SAS, so AB ≅ EC. an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, ANSWER: the perimeter of each triangle is 9 cm. a. B b. 40 To find how many charms can be made, divide 63 c. 80 cm by 9 cm. So, the jeweler can make 7 charms. d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle ANSWER: are congruent. So △AEB is equiangular and 7 equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. 55. In the figure below, NDG LGD. 53. What is m∠FEG?

A 34 a. What is m∠L? B 56 b. What is the value of x? C 60 c. What is m∠DGN? D 62 E 124 SOLUTION: a. In this figure, angles N and L are congruent so m∠L = 36°. SOLUTION: b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°.

ANSWER: a. 36 Consider triangle DEH. Since one angle is 56°, and b. 54 the triangle is isosceles, then the remaining two c. 54 angles would be 180 – 56 or 124°. So, m∠FEG is , or 62. The correct answer is 56. An equilateral triangle has a perimeter of 216 inches. choice D. What is the length of one side of the triangle? SOLUTION: ANSWER: Because the triangle is equilateral that means each D side has the same length, so divide the perimeter by 3 54. A jeweler bends wire to make charms in the shape to find the length of one side. of isosceles triangles. The base of the triangle measures 2 centimeters and another side measures 216 ÷ 3 = 72 3.5 centimeters. How many charms can be made ANSWER: from a piece of wire that is 63 centimeters long? 72 SOLUTION: Because we know that the shape of each charm is 57. △JKL is an isosceles triangle, and . JK = an isosceles triangle, then we can determine the side 2x + 3, and KL = 10 – 4x. What is the perimeter of lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, △JKL? the perimeter of each triangle is 9 cm. SOLUTION:

To find how many charms can be made, divide 63 The perimeter can be found by adding all three side cm by 9 cm. So, the jeweler can make 7 charms. lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. ANSWER: 7 Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 55. In the figure below, NDG LGD. 16.

So, the perimeter of △JKL is 16.

ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of x. a. What is m L? ∠ SOLUTION: b. What is the value of x? In an isosceles triangle, the base angles are c. What is m DGN? ∠ congruent. SOLUTION: a. In this figure, angles N and L are congruent so So, 5x + 5 = 7x – 19.

m∠L = 36°. Simplify. b. Because we know two angle measures, and that

the sum of the measures of the angles of a triangle is –2x = –24 180, then we can solve 36 + 90 + x = 180, or x = 54. x = 12 c. ∠DGN is congruent to x°, so m∠DGN is 54°. ANSWER: ANSWER: 12 a. 36 b. 54 59. QRV is an equilateral triangle. What additional c. 54 information would be enough to prove VRT is an equilateral triangle? 56. An equilateral triangle has a perimeter of 216 inches.

What is the length of one side of the triangle? SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

ANSWER: 72 A B m∠VRT = 60 57. △JKL is an isosceles triangle, and . JK = C QRV VTU 2x + 3, and KL = 10 – 4x. What is the perimeter of D VRT RST △JKL? SOLUTION: SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. Consider each choice. ANSWER: 16 A does not provide enough information to 58. △ABC is an isosceles triangle with base angles determine how sides VT and TR are related. measuring (5x + 5)° and (7x – 19)°. Find the value of x. B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two SOLUTION: measures are 60 degrees. In an isosceles triangle, the base angles are congruent. C

So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

ANSWER:

12 If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , 59. QRV is an equilateral triangle. What additional by the Isosceles Triangle Theorem. information would be enough to prove VRT is an So, . equilateral triangle? Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: A C B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

Refer to the figure. ANSWER: 12

4. m MRP

1. If name two congruent angles. SOLUTION: Isosceles Triangle Theorem states that if two sides of the triangle are congruent, then the angles opposite those sides are congruent. SOLUTION: Therefore, in triangle ABC, Since all the sides are congruent, is an equilateral triangle. Each angle of an equilateral ANSWER: triangle measures 60°. Therefore, m MRP = 60°. BAC and BCA ANSWER: 60 2. If EAC ECA, name two congruent segments. SOLUTION: SENSE-MAKING Find the value of each Converse of Isosceles Triangle Theorem states that variable. if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Therefore, in triangle EAC,

ANSWER: 5.

SOLUTION: Find each measure. In the figure, . Therefore, triangle RST is 3. FH an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

SOLUTION: By the Triangle Angle-Sum Theorem, ANSWER: 12

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12 6. SOLUTION: 4. m MRP In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

SOLUTION: ANSWER: Since all the sides are congruent, is an 16 equilateral triangle. Each angle of an equilateral triangle measures 60°. Therefore, m MRP = 60°. 7. PROOF Write a two-column proof.

ANSWER: Given: is isosceles; bisects ABC. 60 Prove: SENSE-MAKING Find the value of each variable.

5. SOLUTION: SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, .

ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A 6. portion of the track is shown.

SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

ANSWER: a. If and are perpendicular to is 16 isosceles with base , and prove that 7. PROOF Write a two-column proof. b. If VR = 2.5 meters and QR = 2 meters, find the Given: is isosceles; bisects ABC. distance between and Explain your Prove: reasoning.

SOLUTION: a.

SOLUTION:

ANSWER:

b. The Pythagorean Theorem tells us that:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 8. ROLLER COASTERS The roller coaster track 3 m. appears to be composed of congruent triangles. A portion of the track is shown. ANSWER: a.

a. If and are perpendicular to is isosceles with base , and prove that

b. If VR = 2.5 meters and QR = 2 meters, find the distance between and Explain your reasoning. b. The Pythagorean Theorem tells us that By CPCTC we know SOLUTION: that VT = 1.5 m. The Seg. Add. Post. says QV + a. VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

Refer to the figure.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: b. The Pythagorean Theorem tells us that: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. By CPCTC we know that VT = 1.5 m. The SOLUTION: Segment Addition Postulate says QV + VT = QT. By the Converse of Isosceles Triangle Theorem, in By substitution, we have 1.5 + 1.5 = QT. So QT = triangle ABF, 3 m. ANSWER: ANSWER: a. 11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ADE, b. The Pythagorean Theorem tells us that ANSWER: By CPCTC we know that VT = 1.5 m. The Seg. Add. Post. says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. 13. If BCF BFC, name two congruent segments. So QT = 3 m. SOLUTION: Refer to the figure. By the Converse of Isosceles Triangle Theorem, in triangle BCF,

ANSWER:

14. If name two congruent angles. 9. If name two congruent angles. SOLUTION: SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

By the Isosceles Triangle Theorem, in triangle ABE, ANSWER: ANSWER: AFH and AHF ABE and AEB Find each measure. 15. m BAC 10. If ∠ABF ∠AFB, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in

triangle ABF, ANSWER: SOLUTION: Here By Isosceles Triangle Theorem, . 11. If name two congruent angles. Apply the Triangle Sum Theorem. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD,

ANSWER: ANSWER: ACD and ADC 60 12. If DAE DEA, name two congruent segments. 16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER: SOLUTION: Given: 13. If BCF BFC, name two congruent segments. By Isosceles Triangle Theorem, . SOLUTION: By the Converse of Isosceles Triangle Theorem, in Apply Triangle Angle-Sum Theorem. triangle BCF,

ANSWER:

ANSWER: 14. If name two congruent angles. 65 SOLUTION: 17. TR By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: AFH and AHF Find each measure. SOLUTION: 15. m BAC Since the triangle is isosceles, Therefore,

SOLUTION: All the angles are congruent. Therefore, it is an equiangular triangle. Here Since the equiangular triangle is an equilateral, By Isosceles Triangle Theorem, .

Apply the Triangle Sum Theorem. ANSWER: 4 18. CB

ANSWER: 60

16. m SRT SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

SOLUTION: ANSWER: Given: 3 By Isosceles Triangle Theorem, . REGULARITY Find the value of each variable.

Apply Triangle Angle-Sum Theorem.

19. SOLUTION: ANSWER: Since all the angles are congruent, the sides are also 65 congruent to each other. 17. TR Therefore, Solve for x.

ANSWER: SOLUTION: x = 5 Since the triangle is isosceles, Therefore,

20.

All the angles are congruent. Therefore, it is an SOLUTION: equiangular triangle. Given: Since the equiangular triangle is an equilateral, By the Isosceles Triangle Theorem,

ANSWER: 4 And We know 18. CB that,

ANSWER: x = 13

SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ABC, That is, CB = 3.

ANSWER: 21. 3 SOLUTION:

REGULARITY Find the value of each variable. Given: By the Isosceles Triangle Theorem,

That is,

19. Then, let Equation 1 be: .

SOLUTION: By the Triangle Sum Theorem, we can find Equation Since all the angles are congruent, the sides are also 2: congruent to each other. Therefore, Solve for x. Add the equations 1 and 2.

ANSWER: x = 5

Substitute the value of x in one of the two equations to find the value of y.

20. SOLUTION: Given: By the Isosceles Triangle Theorem,

ANSWER: And We know x = 11, y = 11 that,

ANSWER: x = 13 22. SOLUTION: Given: By the Triangle Angle-Sum Theorem,

21. So, the triangle is an equilateral triangle.

SOLUTION: Therefore,

Given: Set up two equations to solve for x and y. By the Isosceles Triangle Theorem,

That is,

Then, let Equation 1 be: . ANSWER: By the Triangle Sum Theorem, we can find Equation x = 1.4, y = –1 2: PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary Add the equations 1 and 2. and HLK and MLH are supplementary. Prove: JHK MHL

Substitute the value of x in one of the two equations to find the value of y. SOLUTION: Proof: We are given that is an isosceles triangle and is an equilateral triangle, JKH and HKL are supplementary and HLK and MLH are supplementary. From the Isosceles Triangle Theorem, we know that HJK HML.

Because is an equilateral triangle, we know HLK LKH KHL and ANSWER: JKH, HKL and HLK, x = 11, y = 11 MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent Supplements Theorem. By AAS, By CPCTC, JHK MHL.

ANSWER: Proof: We are given that is an isosceles 22. triangle and is an equilateral triangle, JKH SOLUTION: and HKL are supplementary and HLK and Given: MLH are supplementary. From the Isosceles By the Triangle Angle-Sum Triangle Theorem, we know that HJK HML. Theorem, Because is an equilateral triangle, we know HLK LKH KHL and So, the triangle is an equilateral triangle. JKH, HKL and HLK, Therefore, MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent Set up two equations to solve for x and y. Supplements Theorem. By AAS, By CPCTC, JHK MHL.

24. Given: W is the midpoint of ANSWER: Q is the midpoint of x = 1.4, y = –1 Prove: PROOF Write a paragraph proof. 23. Given: is isosceles, and is equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

SOLUTION: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the SOLUTION: midpoint of we know that Similarly, Proof: We are given that is an isosceles because Q is the midpoint of The triangle and is an equilateral triangle, JKH Segment Addition Postulate gives us XW + WY = XY and HKL are supplementary and HLK and and XQ + QZ = XZ. Substitution gives XW + WY = MLH are supplementary. From the Isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Triangle Theorem, we know that HJK HML. If we divide each side by 2, we have WY = QZ. The Because is an equilateral triangle, we know Isosceles Triangle Theorem says XYZ XZY. HLK LKH KHL and by the Reflexive Property. By SAS, JKH, HKL and HLK, So, by CPCTC. MLH are supplementary, and HKL HLK, we know JKH MLH by the Congruent ANSWER: Supplements Theorem. By AAS, Proof: We are given W is the midpoint of By CPCTC, JHK MHL. and Q is the midpoint of Because W is the ANSWER: midpoint of we know that Similarly, Proof: We are given that is an isosceles because Q is the midpoint of The triangle and is an equilateral triangle, JKH Segment Addition Postulate gives us XW + WY = XY and HKL are supplementary and HLK and and XQ + QZ = XZ. Substitution gives XW + WY = MLH are supplementary. From the Isosceles XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. Triangle Theorem, we know that HJK HML. If we divide each side by 2, we have WY = QZ. The Because is an equilateral triangle, we know Isosceles Triangle Theorem says XYZ XZY. HLK LKH KHL and by the Reflexive Property. By SAS, JKH, HKL and HLK, So, by CPCTC. MLH are supplementary, and HKL HLK, we know JKH ∠MLH by the Congruent 25. BABYSITTING While babysitting her neighbor’s Supplements Theorem. By AAS, children, Elisa observes that the supports on either By CPCTC, JHK MHL. side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to

determine that but 24. Given: W is the midpoint of Q is the midpoint of Prove:

a. Elisa estimates m BAC to be 50. Based on this

SOLUTION: estimate, what is m∠ABC? Explain. b. If show that is isosceles. Proof: We are given W is the midpoint of c. If and show that is and Q is the midpoint of Because W is the equilateral. midpoint of we know that Similarly, d. If is isosceles, what is the minimum because Q is the midpoint of The information needed to prove that Segment Addition Postulate gives us XW + WY = XY Explain your reasoning. and XQ + QZ = XZ. Substitution gives XW + WY =

XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. SOLUTION: If we divide each side by 2, we have WY = QZ. The

Isosceles Triangle Theorem says XYZ XZY. a. Given:

by the Reflexive Property. By SAS, By Isosceles Triangle Theorem, .

Apply Triangle Sum Theorem. So, by CPCTC. ANSWER: Proof: We are given W is the midpoint of and Q is the midpoint of Because W is the b. midpoint of we know that Similarly, because Q is the midpoint of The Segment Addition Postulate gives us XW + WY = XY and XQ + QZ = XZ. Substitution gives XW + WY = XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. If we divide each side by 2, we have WY = QZ. The Isosceles Triangle Theorem says XYZ XZY. by the Reflexive Property. By SAS, So, by CPCTC. c. 25. BABYSITTING While babysitting her neighbor’s children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that but

a. Elisa estimates m BAC to be 50. Based on this estimate, what is m∠ABC? Explain. b. If show that is isosceles. c. If and show that is equilateral. d. If is isosceles, what is the minimum information needed to prove that d. One pair of congruent corresponding sides and Explain your reasoning. one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is congruent to a leg of then you know that SOLUTION: both pairs of legs are congruent. Because the base a. Given: angles of an isosceles triangle are congruent, if you By Isosceles Triangle Theorem, . know that you know that Apply Triangle Sum Theorem. Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS. b. ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b.

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since d. One pair of congruent corresponding sides and you know that the triangle is isosceles, if one leg is one pair of congruent corresponding angles; since congruent to a leg of then you know that you know that the triangle is isosceles, if one leg is both pairs of legs are congruent. Because the base congruent to a leg of then you know that angles of an isosceles triangle are congruent, if you both pairs of legs are congruent. Because the base know that you know that angles of an isosceles triangle are congruent, if you Therefore, know that , you know that with one pair of congruent corresponding sides and Therefore, one pair of congruent corresponding angles, the with one pair of congruent corresponding sides and triangles can be proved congruent using either ASA one pair of congruent corresponding angles, the or SAS. triangles can be proved congruent using either ASA

ANSWER: or SAS. a. 65°; Because is isosceles, ABC 26. CHIMNEYS In the picture, and ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. is an isosceles triangle with base Show b. that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

SOLUTION:

d. One pair of congruent corresponding sides and one pair of congruent corresponding angles; since you know that the triangle is isosceles, if one leg is ANSWER: congruent to a leg of then you know that both pairs of legs are congruent. Because the base angles of an isosceles triangle are congruent, if you know that , you know that Therefore, with one pair of congruent corresponding sides and one pair of congruent corresponding angles, the triangles can be proved congruent using either ASA or SAS.

26. CHIMNEYS In the picture, and is an isosceles triangle with base Show that the chimney of the house, represented by bisects the angle formed by the sloped sides of the roof, ABC.

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

28. PROOF Based on your construction in Exercise 27, make and prove a conjecture about the relationship between the base angles of an isosceles right triangle. Sample answer: I constructed a pair of perpendicular segments and then used the same compass setting to SOLUTION: mark points equidistant from their intersection. I Conjecture: The measures of the base angles of an measured both legs for each triangle. Because AB = isosceles right triangle are 45. AC = 1.3 cm, DE = DF = 1.9 cm, and GH = GJ = Proof: The base angles are congruent because it is 2.3 cm, the triangles are isosceles. I used a an isosceles triangle. Let the measure of each acute protractor to confirm that A, D, and G are all angle be x. The acute angles of a right triangle are right angles. complementary, Solve for x

ANSWER:

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

triangle. Therefore, . SOLUTION: ANSWER: Conjecture: The measures of the base angles of an 44 isosceles right triangle are 45.

Proof: The base angles are congruent because it is 31. m ACB an isosceles triangle. Let the measure of each acute SOLUTION: angle be x. The acute angles of a right triangle are The angles in a straight line add to 180°. complementary, Solve for x Therefore,

We know that ANSWER: Conjecture: The measures of the base angles of an isosceles right triangle are 45. Proof: The base angles are congruent because it is ANSWER: an isosceles triangle. Let the measure of each acute 136 angle be x. The acute angles of a right triangle are complementary, so x + x = 90 and x = 45. 32. m ABC REGULARITY Find each measure. SOLUTION:

From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

29. m∠CAD SOLUTION: ANSWER: From the figure, 22 Therefore, is an isosceles triangle. 33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex By the Triangle Sum Theorem, angles G, H, and J, and G

H, and H J, show that the distance from B to F is three times the distance from D to F.

ANSWER: 44

30. m ACD

SOLUTION: We know that SOLUTION: Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore,

We know that

ANSWER:

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore,

By the Triangle Angle Sum Theorem,

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER: 22

33. FITNESS In the diagram, the rider will use his bike to hop across the tops of each of the concrete solids shown. If each triangle is isosceles with vertex angles G, H, and J, and G SOLUTION: H, and H J, show that the distance from B to F is three times the distance from D to F.

SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary.

ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION: ANSWER:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION:

37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

36. Corollary 4.4

SOLUTION: 38. SOLUTION: By the converse of Isosceles Triangle theorem,

Solve the equation for x.

Note that x can equal –8 here because

ANSWER: . ANSWER: 3

39. SOLUTION: 37. Theorem 4.11 By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) . SOLUTION: The interior angles of a triangle add up to 180 .

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14. ANSWER: ANSWER: 14 GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

38. 40. m LPM SOLUTION: SOLUTION: By the converse of Isosceles Triangle theorem, Angles at a point in a straight line add up to 180 .

Solve the equation for x.

Substitute x = 45 in (3x – 55) to find Note that x can equal –8 here because .

ANSWER: 3 ANSWER: 80

41. m LMP 39. SOLUTION: SOLUTION: Since the triangle LMP is isosceles, By the Isosceles Triangle Theorem, the third angle is Angles at a point in a straight line equal to (2y – 5) . add up to 180 .

The interior angles of a triangle add up to 180 .

Substitute x = 45 in (3x – 55) to find

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: Therefore, 14 ANSWER: GAMES Use the diagram of a game timer 80 shown to find each measure. 42. m JLK

SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

We know that,

40. m LPM SOLUTION: Angles at a point in a straight line add up to 180 . So,

ANSWER: 20

43. m JKL Substitute x = 45 in (3x – 55) to find SOLUTION: Vertical angles are congruent. Therefore,

First we need to find

ANSWER: We know that, 80

41. m LMP So, SOLUTION: Since the triangle LMP is isosceles, In Angles at a point in a straight line add up to 180 . Since the triangle JKL is isosceles,

ANSWER: Substitute x = 45 in (3x – 55) to find 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the Therefore, measure of one exterior angle.

ANSWER: 80

42. m JLK SOLUTION: Vertical angles are congruent. Therefore, a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of First we need to find the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown. We know that, b. Tabular Use a protractor to measure and record m∠1 for each triangle. Use m 1 to calculate the measures of 3, 4, and 5. Then find and So, record m 2 and use it to calculate these same measures. Organize your results in two tables. ANSWER: 20 c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how 43. m JKL you used m 2 to find these same measures. SOLUTION: Vertical angles are congruent. Therefore, d. Algebraic If m 1 = x, write an expression for the measures of 3, 4, and 5. Likewise, if m First we need to find 2 = x, write an expression for these same angle measures. We know that, SOLUTION: a.

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of b. the base angles, and labeling as shown.

c. Verbal Explain how you used m 1 to find the measures of 3, 4, and 5. Then explain how you used m 2 to find these same measures. c. 5 is supplementary to 1, so m 5 = 180 – d. Algebraic If m 1 = x, write an expression for m 1. 4 5, so m 4 = m 5. The sum of the the measures of 3, 4, and 5. Likewise, if m angle measures in a triangle must be 180, so m 3 = 2 = x, write an expression for these same angle 180 – m 4 – 5. 2 is supplementary to 3, so measures. m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = SOLUTION: a. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

ANSWER: a.

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so m 3 = 180 – m 2. m 2 is twice as much as m c. 5 is supplementary to 1, so m 5 = 180 – 4 and m 5, so m 4 = m 5 = m 1. 4 5, so m 4 = m 5. The sum of the d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – angle measures in a triangle must be 180, so m 3 = 180 – m 4 – 5. 2 is supplementary to 3, so 180; m 3 = 180 – x, m 4 = m 5 = m 3 = 180 – m 2. m 2 is twice as much as m

4 and m 5, so m 4 = m 5 = ANSWER: a. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 =

45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER:

c. 5 is supplementary to 1, so m 5 = 180 – m 1. 4 5, so m 4 = m 5. The sum of the PRECISION Determine whether the following angle measures in a triangle must be 180, so m 3 = statements are sometimes, always, or never true. 180 – m 4 – 5. 2 is supplementary to 3, so Explain. m 3 = 180 – m 2. m 2 is twice as much as m 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base 4 and m 5, so m 4 = m 5 = angle is an integer. d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – SOLUTION:

180; m 3 = 180 – x, m 4 = m 5 = The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; 45. CHALLENGE In the figure, if is equilateral vertex angle = 55, base angles = 62.5. and ZWP WJM JZL, prove that ANSWER: Sometimes; only if the measure of the vertex angle is even.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that ANSWER: m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

SOLUTION: Neither of them is correct. This is an isosceles PRECISION Determine whether the following triangle with a vertex angle of 70. Since this is an statements are sometimes, always, or never true. isosceles triangle, the base angles are congruent. Explain. 46. If the measure of the vertex angle of an isosceles triangle is an integer, then the measure of each base angle is an integer. SOLUTION: The statement is sometimes true. It is true only if the measure of the vertex angle is even. For example, vertex angle = 50, base angles = 65; vertex angle = 55, base angles = 62.5. ANSWER:

ANSWER: Neither; m G = or 55. Sometimes; only if the measure of the vertex angle is even. 49. OPEN-ENDED If possible, draw an isosceles triangle with base angles that are obtuse. If it is not 47. If the measures of the base angles of an isosceles possible, explain why not. triangle are integers, then the measure of its vertex SOLUTION: angle is odd. It is not possible because a triangle cannot have more SOLUTION: than one obtuse angle. The statement is never true. The measure of the ANSWER: vertex angle will be 180 – 2(measure of the base angle) so if the base angles are integers, then 2 It is not possible because a triangle cannot have more (measure of the base angle) will be even and 180 – 2 than one obtuse angle. (measure of the base angle) will be even. 50. REASONING In isosceles m B = 90. ANSWER: Draw the triangle. Indicate the congruent sides and Never; the measure of the vertex angle will be 180 – label each angle with its measure. 2(measure of the base angle) so if the base angles SOLUTION: are integers, then 2(measure of the base angle) will The sum of the angle measures in a triangle must be be even and 180 – 2(measure of the base angle) will

be even. 180,

48. ERROR ANALYSIS Alexis and Miguela are Since is isosceles, finding m G in the figure shown. Alexis says that And given that m∠B = 90. m G = 35, while Miguela says that m G = 60. Is Therefore,

either of them correct? Explain your reasoning.

Construct the triangle with the angles measures 90, 45, 45.

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent. ANSWER:

51. WRITING IN MATH How can triangle ANSWER: classifications help you prove triangle congruence?

Neither; m G = or 55. SOLUTION: If a triangle is already classified, you can use the 49. OPEN-ENDED If possible, draw an isosceles previously proven properties of that type of triangle triangle with base angles that are obtuse. If it is not in the proof. For example, if you know that a possible, explain why not. triangle is an equilateral triangle, you can use SOLUTION: Corollary 4.3 and 4.4 in the proof. Doing this can save you steps when writing the proof. It is not possible because a triangle cannot have more than one obtuse angle. ANSWER: ANSWER: Sample answer: If a triangle is already classified, It is not possible because a triangle cannot have more you can use the previously proven properties of that than one obtuse angle. type of triangle in the proof. Doing this can save you steps when writing the proof. 50. REASONING In isosceles m B = 90. Draw the triangle. Indicate the congruent sides and 52. MULTI-STEP △AED and △BEC are congruent label each angle with its measure. and isosceles.

SOLUTION: The sum of the angle measures in a triangle must be 180,

Since is isosceles,

And given that m∠B = 90. Therefore, a. Which statement is not necessarily true from the

information given?

A △AEB ≅ △DEC B Construct the triangle with the angles measures 90, C ∠CAD ≅ ∠DBC 45, 45. D

b. If m∠EAB = 40, what is the measure of ∠DCE?

c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? ANSWER: SOLUTION: a. Of the statements given, only choice B is not necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. c. The sum of the measures of the angles of a triangle is 180, so ∠BEC must be 80°. 51. WRITING IN MATH How can triangle d. 60; All three angles in △AEB are 60° because classifications help you prove triangle congruence? base angles of an isosceles triangle are congruent. So △ △ SOLUTION: AEB is equiangular and equilateral. AEB ≅ △ If a triangle is already classified, you can use the DEC by SAS, so AB ≅ EC. previously proven properties of that type of triangle ANSWER: in the proof. For example, if you know that a a. B triangle is an equilateral triangle, you can use b. 40 Corollary 4.3 and 4.4 in the proof. Doing this can c. 80 save you steps when writing the proof. d. 60; Sample answer: All three angles in △AEB are 60° because base angles of an isosceles triangle ANSWER: are congruent. So △AEB is equiangular and Sample answer: If a triangle is already classified, equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. you can use the previously proven properties of that type of triangle in the proof. Doing this can save you 53. What is m∠FEG? steps when writing the proof.

52. MULTI-STEP △AED and △BEC are congruent and isosceles.

A 34 B 56 C 60 D 62 a. Which statement is not necessarily true from the E 124 information given?

SOLUTION: A △AEB ≅ △DEC B C ∠CAD ≅ ∠DBC D

b. If m∠EAB = 40, what is the measure of ∠DCE? Consider triangle DEH. Since one angle is 56°, and c. If m∠ADE = 50, what is the measure of ∠BEC? the triangle is isosceles, then the remaining two angles would be 180 – 56 or 124°. d. What measure would ∠AEB need to be so that So, m∠FEG is , or 62. The correct answer is ? choice D.

SOLUTION: ANSWER: a. Of the statements given, only choice B is not D necessarily true from the given information. b. The angles are congruent, so ∠DCE is 40°. 54. A jeweler bends wire to make charms in the shape c. The sum of the measures of the angles of a of isosceles triangles. The base of the triangle triangle is 180, so ∠BEC must be 80°. measures 2 centimeters and another side measures d. 60; All three angles in △AEB are 60° because 3.5 centimeters. How many charms can be made base angles of an isosceles triangle are congruent. So from a piece of wire that is 63 centimeters long? △AEB is equiangular and equilateral. △AEB ≅ SOLUTION: △ DEC by SAS, so AB ≅ EC. Because we know that the shape of each charm is ANSWER: an isosceles triangle, then we can determine the side lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, a. B the perimeter of each triangle is 9 cm. b. 40 c. 80 To find how many charms can be made, divide 63 d. 60; Sample answer: All three angles in △AEB cm by 9 cm. So, the jeweler can make 7 charms. are 60° because base angles of an isosceles triangle 4-6 Isoscelesare congruent. and EquilateralSo △AEB is Trianglesequiangular and ANSWER: equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC. 7

53. What is m∠FEG? 55. In the figure below, NDG LGD.

A 34 B 56 a. What is m∠L? C 60 b. What is the value of x? D 62 c. What is m∠DGN? E 124 SOLUTION: a. In this figure, angles N and L are congruent so SOLUTION: m∠L = 36°. b. Because we know two angle measures, and that the sum of the measures of the angles of a triangle is 180, then we can solve 36 + 90 + x = 180, or x = 54. c. ∠DGN is congruent to x°, so m∠DGN is 54°.

ANSWER: Consider triangle DEH. Since one angle is 56°, and a. 36 the triangle is isosceles, then the remaining two b. 54 angles would be 180 – 56 or 124°. c. 54 So, m∠FEG is , or 62. The correct answer is choice D. 56. An equilateral triangle has a perimeter of 216 inches. What is the length of one side of the triangle? ANSWER: SOLUTION: D Because the triangle is equilateral that means each 54. A jeweler bends wire to make charms in the shape side has the same length, so divide the perimeter by 3 of isosceles triangles. The base of the triangle to find the length of one side. measures 2 centimeters and another side measures 3.5 centimeters. How many charms can be made 216 ÷ 3 = 72 from a piece of wire that is 63 centimeters long? ANSWER: SOLUTION: 72 Because we know that the shape of each charm is an isosceles triangle, then we can determine the side 57. △JKL is an isosceles triangle, and . JK = lengths to be 2 cm, 3.5 cm, and 3.5 cm. Therefore, 2x + 3, and KL = 10 – 4x. What is the perimeter of the perimeter of each triangle is 9 cm. △JKL?

SOLUTION: To find how many charms can be made, divide 63 cm by 9 cm. So, the jeweler can make 7 charms. The perimeter can be found by adding all three side lengths. In this situation, since the triangle is ANSWER: isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x. 7 Because the x-terms cancel each other out: 2x + 2x 55. In the figure below, NDG LGD. – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16. eSolutions Manual - Powered by Cognero Page 19 ANSWER: 16

58. △ABC is an isosceles triangle with base angles measuring (5x + 5)° and (7x – 19)°. Find the value of a. What is m∠L? x. b. What is the value of x? SOLUTION: c. What is m∠DGN? In an isosceles triangle, the base angles are SOLUTION: congruent. a. In this figure, angles N and L are congruent so So, 5x + 5 = 7x – 19. m∠L = 36°.

b. Because we know two angle measures, and that Simplify. the sum of the measures of the angles of a triangle is

180, then we can solve 36 + 90 + x = 180, or x = 54. –2x = –24 c. ∠DGN is congruent to x°, so m∠DGN is 54°. x = 12

ANSWER: ANSWER: a. 36 12 b. 54 c. 54 59. QRV is an equilateral triangle. What additional 56. An equilateral triangle has a perimeter of 216 inches. information would be enough to prove VRT is an What is the length of one side of the triangle? equilateral triangle?

SOLUTION: Because the triangle is equilateral that means each side has the same length, so divide the perimeter by 3 to find the length of one side.

216 ÷ 3 = 72

ANSWER: 72 A 57. △JKL is an isosceles triangle, and . JK = B m∠VRT = 60 2x + 3, and KL = 10 – 4x. What is the perimeter of C QRV VTU △JKL? D VRT RST SOLUTION: SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

Because the x-terms cancel each other out: 2x + 2x – 4x = 4x – 4x = 0, we are left with 3 + 3 + 10, or 16.

So, the perimeter of △JKL is 16.

ANSWER: Consider each choice. 16 58. △ABC is an isosceles triangle with base angles A does not provide enough information to measuring (5x + 5)° and (7x – 19)°. Find the value of determine how sides VT and TR are related. x. B If we know that m∠VRT = 60, we still would not SOLUTION: have enough information to prove that the other two In an isosceles triangle, the base angles are measures are 60 degrees. congruent. C So, 5x + 5 = 7x – 19.

Simplify.

–2x = –24 x = 12

ANSWER: 12 If , then m∠RVT = 60 because 59. QRV is an equilateral triangle. What additional ∠QVR + ∠RVT + ∠TVU = 180. Since , information would be enough to prove VRT is an by the Isosceles Triangle Theorem. equilateral triangle? So, .

Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: A C B m∠VRT = 60 C QRV VTU D VRT RST SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C Refer to the figure.

ANSWER: BAC and BCA

ANSWER:

Find each measure. 3. FH

SOLUTION: By the Triangle Angle-Sum Theorem,

Since the measures of all the three angles are 60°; the triangle must be equiangular. All the equiangular triangles are equilateral. Therefore, FH = GH = 12.

ANSWER: 12

Refer to the figure. 4. m MRP

ANSWER: SOLUTION: 12 By the Triangle Angle-Sum Theorem,

ANSWER: 16

ANSWER: 60 SENSE-MAKING Find the value of each variable.

SOLUTION:

5. SOLUTION: In the figure, . Therefore, triangle RST is an isosceles triangle. By the Converse of Isosceles Triangle Theorem,

That is, . ANSWER:

ANSWER: 12

8. ROLLER COASTERS The roller coaster track appears to be composed of congruent triangles. A portion of the track is shown.

6. SOLUTION: In the figure, Therefore, triangle WXY is an isosceles triangle. By the Isosceles Triangle Theorem, .

a. If and are perpendicular to is isosceles with base , and prove that

ANSWER: b. If VR = 2.5 meters and QR = 2 meters, find the 16 distance between and Explain your 7. PROOF Write a two-column proof. reasoning.

Given: is isosceles; bisects ABC. SOLUTION: Prove: a.

SOLUTION:

b. The Pythagorean Theorem tells us that: ANSWER:

By CPCTC we know that VT = 1.5 m. The Segment Addition Postulate says QV + VT = QT. By substitution, we have 1.5 + 1.5 = QT. So QT = 3 m.

ANSWER: 8. ROLLER COASTERS The roller coaster track a. appears to be composed of congruent triangles. A portion of the track is shown.

a. If and are perpendicular to is isosceles with base , and prove that

b. The Pythagorean Theorem tells us that b. If VR = 2.5 meters and QR = 2 meters, find the By CPCTC we know distance between and Explain your that VT = 1.5 m. The Seg. Add. Post. says QV +

reasoning. VT = QT. By substitution, we have 1.5 + 1.5 = QT.

So QT = 3 m. SOLUTION: Refer to the figure. a.

9. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ABE,

ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments. b. The Pythagorean Theorem tells us that: SOLUTION:

ANSWER: ACD and ADC

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE,

ANSWER:

So QT = 3 m. triangle BCF, ANSWER: Refer to the figure.

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

9. If name two congruent angles. ANSWER: SOLUTION: AFH and AHF By the Isosceles Triangle Theorem, in triangle ABE, Find each measure. 15. m BAC ANSWER: ABE and AEB

10. If ∠ABF ∠AFB, name two congruent segments.

SOLUTION: By the Converse of Isosceles Triangle Theorem, in SOLUTION: triangle ABF, Here By Isosceles Triangle Theorem, . ANSWER: Apply the Triangle Sum Theorem.

11. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle ACD, ANSWER: 60 ANSWER: ACD and ADC 16. m SRT

12. If DAE DEA, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle ADE, SOLUTION: Given: ANSWER: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 13. If BCF BFC, name two congruent segments. SOLUTION: By the Converse of Isosceles Triangle Theorem, in triangle BCF, ANSWER: ANSWER: 65

17. TR

14. If name two congruent angles. SOLUTION: By the Isosceles Triangle Theorem, in triangle AFH,

ANSWER: SOLUTION:

AFH and AHF Since the triangle is isosceles, Therefore, Find each measure.

15. m BAC

All the angles are congruent. Therefore, it is an equiangular triangle. Since the equiangular triangle is an equilateral, SOLUTION: Here ANSWER: By Isosceles Triangle Theorem, . 4

Apply the Triangle Sum Theorem. 18. CB

ANSWER:

60 SOLUTION: By the Converse of Isosceles Triangle Theorem, in 16. m SRT triangle ABC, That is, CB = 3.

ANSWER: 3

REGULARITY Find the value of each variable. SOLUTION: Given: By Isosceles Triangle Theorem, .

Apply Triangle Angle-Sum Theorem. 19.

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Therefore, ANSWER: Solve for x. 65

17. TR

ANSWER: x = 5

SOLUTION: Since the triangle is isosceles, Therefore, 20.

Then, let Equation 1 be: .

By the Triangle Sum Theorem, we can find Equation 19. 2:

SOLUTION: Since all the angles are congruent, the sides are also congruent to each other. Add the equations 1 and 2. Therefore, Solve for x.

ANSWER: Substitute the value of x in one of the two equations x = 5 to find the value of y.

20.

SOLUTION:

Given: ANSWER: By the Isosceles Triangle Theorem, x = 11, y = 11

And We know that,

22. SOLUTION: ANSWER: Given: x = 13 By the Triangle Angle-Sum Theorem,

So, the triangle is an equilateral triangle. Therefore,

21. Set up two equations to solve for x and y.

SOLUTION: Given: By the Isosceles Triangle Theorem,

That is, ANSWER: x = 1.4, y = –1 Then, let Equation 1 be: . PROOF Write a paragraph proof.

Given: By the Triangle Sum Theorem, we can find Equation 23. is isosceles, and is 2: equilateral. JKH and HKL are supplementary and HLK and MLH are supplementary. Prove: JHK MHL

Add the equations 1 and 2.

SOLUTION:

Set up two equations to solve for x and y. Given: 24. W is the midpoint of Q is the midpoint of Prove: ANSWER: x = 1.4, y = –1

we know JKH ∠MLH by the Congruent Supplements Theorem. By AAS, determine that but By CPCTC, JHK MHL.

24. Given: W is the midpoint of Q is the midpoint of Prove:

XQ + QZ and WY + WY = QZ + QZ. So, 2WY = 2QZ. By Isosceles Triangle Theorem, .

If we divide each side by 2, we have WY = QZ. The Apply Triangle Sum Theorem.

ANSWER: a. 65°; Because is isosceles, ABC ∠ACB, so 180 – 50 = 130 and 130 ÷ 2 = 65. b. b.

c. c.

b. the roof, ABC.

SOLUTION:

ANSWER:

SOLUTION:

ANSWER:

27. CONSTRUCTION Construct three different isosceles right triangles. Explain your method. Then verify your constructions using measurement and mathematics. SOLUTION:

REGULARITY Find each measure.

29. m∠CAD SOLUTION: From the figure, Therefore, is an isosceles triangle.

By the Triangle Sum Theorem,

H, and H J, show that the distance from B to F is three times the distance from D to F. By the Triangle Sum Theorem,

ANSWER: 44

30. m ACD SOLUTION: SOLUTION: We know that Therefore, .

ANSWER: 44

31. m ACB SOLUTION: The angles in a straight line add to 180°. Therefore, ANSWER: We know that

ANSWER: 136

32. m ABC SOLUTION: From the figure, So, is an isosceles triangle. Therefore, 34. Given: is isosceles; By the Triangle Angle Sum Theorem, Prove: X and YZV are complementary.

ANSWER: 22

ANSWER: SOLUTION:

PROOF Write a two-column proof of each ANSWER: corollary or theorem. 35. Corollary 4.3 SOLUTION:

34. Given: is isosceles; Prove: X and YZV are complementary. ANSWER:

SOLUTION:

36. Corollary 4.4 SOLUTION:

ANSWER:

PROOF Write a two-column proof of each corollary or theorem. 35. Corollary 4.3 SOLUTION: 37. Theorem 4.11 SOLUTION:

ANSWER: ANSWER:

Find the value of each variable.

38. SOLUTION: 36. Corollary 4.4 By the converse of Isosceles Triangle theorem, SOLUTION:

Solve the equation for x.

Note that x can equal –8 here because .

ANSWER: 3 ANSWER:

39. SOLUTION: By the Isosceles Triangle Theorem, the third angle is equal to (2y – 5) .

The interior angles of a triangle add up to 180 . 37. Theorem 4.11 SOLUTION:

The measure of an angle cannot be negative, and 2(– 18) – 5 = –41, so y = 14.

ANSWER: 14 ANSWER: GAMES Use the diagram of a game timer shown to find each measure.

Find the value of each variable.

40. m LPM SOLUTION: 38. Angles at a point in a straight line add up to 180 .

SOLUTION: By the converse of Isosceles Triangle theorem,

Substitute x = 45 in (3x – 55) to find Solve the equation for x.

Note that x can equal –8 here because

The interior angles of a triangle add up to 180 . Substitute x = 45 in (3x – 55) to find

First we need to find

We know that,

So, 40. m LPM ANSWER: SOLUTION: 20 Angles at a point in a straight line add up to 180 . 43. m JKL SOLUTION: Vertical angles are congruent. Therefore,

Substitute x = 45 in (3x – 55) to find First we need to find

We know that,

ANSWER: 80 So,

In 41. m LMP

SOLUTION: Since the triangle JKL is isosceles, Since the triangle LMP is isosceles, Angles at a point in a straight line add up to 180 .

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the Substitute x = 45 in (3x – 55) to find interior angles of an isosceles triangle given the

measure of one exterior angle.

Therefore,

ANSWER: 80

measures. Organize your results in two tables.

First we need to find SOLUTION: a. We know that,

So,

Since the triangle JKL is isosceles,

ANSWER: 80

44. MULTIPLE REPRESENTATIONS In this problem, you will explore possible measures of the interior angles of an isosceles triangle given the measure of one exterior angle.

a. Geometric Use a ruler and a protractor to draw three different isosceles triangles, extending one of the sides adjacent to the vertex angle and to one of the base angles, and labeling as shown.

180; m 3 = 180 – x, m 4 = m 5 = SOLUTION: a. ANSWER: a.

d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x – 4 and m 5, so m 4 = m 5 =

180; m 3 = 180 – x, m 4 = m 5 = d. m 5 = 180 – x, m 4 = 180 – x, m 3 = 2x –

180; m 3 = 180 – x, m 4 = m 5 = ANSWER: a. 45. CHALLENGE In the figure, if is equilateral and ZWP WJM JZL, prove that

SOLUTION:

ANSWER: b.

48. ERROR ANALYSIS Alexis and Miguela are finding m G in the figure shown. Alexis says that m G = 35, while Miguela says that m G = 60. Is either of them correct? Explain your reasoning.

ANSWER:

SOLUTION: Neither of them is correct. This is an isosceles triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

SOLUTION: Neither of them is correct. This is an isosceles ANSWER: triangle with a vertex angle of 70. Since this is an isosceles triangle, the base angles are congruent.

180, a. Which statement is not necessarily true from the Since is isosceles, information given? And given that m∠B = 90. Therefore, A △AEB ≅ △DEC B

C ∠CAD ≅ ∠DBC D

Construct the triangle with the angles measures 90, b. If m∠EAB = 40, what is the measure of ∠DCE? 45, 45. c. If m∠ADE = 50, what is the measure of ∠BEC?

d. What measure would ∠AEB need to be so that ? SOLUTION:

52. MULTI-STEP △AED and △BEC are congruent and isosceles. A 34 B 56 C 60 D 62 E 124

SOLUTION: a. Which statement is not necessarily true from the information given?

A △AEB ≅ △DEC B

equilateral. △AEB ≅ △DEC by SAS, so AB ≅ EC.

53. What is m∠FEG?

ANSWER: 59. QRV is an equilateral triangle. What additional a. 36 information would be enough to prove VRT is an b. 54 equilateral triangle? c. 54

216 ÷ 3 = 72 A B m∠VRT = 60 4-6 IsoscelesANSWER: and Equilateral Triangles 72 C QRV VTU D VRT RST 57. △JKL is an isosceles triangle, and . JK = SOLUTION: 2x + 3, and KL = 10 – 4x. What is the perimeter of △JKL? SOLUTION: The perimeter can be found by adding all three side lengths. In this situation, since the triangle is isosceles, we add 2x + 3 + 2x + 3 + 10 – 4x.

So, 5x + 5 = 7x – 19.

Simplify.

The correct answer is choice C.

ANSWER: C

A B m∠VRT = 60 C QRV VTU D VRT RST eSolutions Manual - Powered by Cognero Page 20 SOLUTION:

Consider each choice.

A does not provide enough information to determine how sides VT and TR are related.

B If we know that m∠VRT = 60, we still would not have enough information to prove that the other two measures are 60 degrees.

If , then m∠RVT = 60 because ∠QVR + ∠RVT + ∠TVU = 180. Since , by the Isosceles Triangle Theorem. So, . Therefore, is equilateral by Theorem 4.3.

D Since we do not have any measures to indicate that either or is equilateral, knowing they are congruent would not help prove is equilateral.

The correct answer is choice C.

ANSWER: C