Activity Assess 8-2 CRITIQUE & EXPLAIN Manuel draws a diagram of kite PQRS Kites and ⟷ with QS​​ ​​ as the line of symmetry over Trapezoids a design of a kite-shaped key fob. He Q makes a list of conclusions based on the diagram. PearsonRealize.com P R • PR QS I CAN… use triangle • QP ~ QR congruence to understand = kites and trapezoids. • SP =~ SR • PR bisects QS. S VOCABULARY • ∆PQR is an equilateral • midsegment of a trapezoid triangle. • ∆PSR is an isosceles triangle.

A. Which of Manuel’s conclusions do you agree with? Which do you disagree with? Explain. B. Use Structure What other conclusions are supported by the diagram?

ESSENTIAL QUESTION How are diagonals and angle measures related in kites and trapezoids? CONCEPTUAL UNDERSTANDING EXAMPLE 1 Investigate the Diagonals of a Kite

How are the diagonals of a kite related? A A kite has two pairs of congruent adjacent sides.

X STUDY TIP D B Remember that you must show that both B and D are on the perpendicular bisector in order to show that one diagonal is C the perpendicular bisector of the other. It is not sufficient to Point B is equidistant from the endpoints of ¯AC​​ ​​, as is D, show that only one is on the so they lie on the perpendicular bisector of ¯AC​​ ​​. perpendicular bisector. The diagonals of a kite are perpendicular to each other. Exactly one diagonal bisects the other.

Try It! 1. a. What is the measure of ∠AXB? b. If ​AX = 3.8​, what is AC? c. If BD​ = 10​, does BX​ = 5​? Explain.

LESSON 8-2 Kites and Trapezoids 365 Activity Assess

THEOREM 8-3

The diagonals of a kite are If... X perpendicular.

E W Y

Z PROOF: SEE EXERCISE 12. Then... WY‾ ​ ⟂ XZ‾​ ​

EXAMPLE 2 Use the Diagonals of a Kite

Quadrilateral PQRS is a kite with diagonals QS‾​​ ​​ and PR‾​​ ​​. R 3 1 Q S 2 T 35

P

A. What is ​m∠1​? The diagonals of a kite are perpendicular, so ​m∠1 = 90​. COMMON ERROR B. What is ​m∠2​? You may incorrectly assume angles are congruent just from their The sum of the angles of △​ PQT​ is 180. appearance. Always check that you ​m 2 35 90 180​ can prove congruence first. ∠ + + = m ∠2 = 55​ C. What is m∠3? Since ​△PQR​ is an isosceles triangle, ​m∠3 ≅ m∠QPT.​ So, ​m∠3 = 35.​

Try It! 2. Quadrilateral WXYZ is a kite. X

a. What is ​m∠1​? 2 32 b. What is ​m∠2​?

1 W Y

Z

366 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess

APPLICATION EXAMPLE 3 Explore Parts of an Isosceles Trapezoid

Kiyo is designing a trapezoid-shaped roof. In order for the roof to be symmetric, the overlapping triangles △​ DAB​ B C and ​△ADC​ must be congruent. Will the roof be symmetric?

A D

Step 1 Show △​ ABE ≅ △DCF​. _ _ _ ​​BE ​ ≅_ CF​ ​​, because BE​​ ​​ and ​​CF ​​ are altitudes.

GENERALIZE B C How do the lengths of the The roof is an isosceles diagonals and the way they trapezoid, since​ DC‾​ ​ ≅ AB‾​ ​ . intersect relate to the sides of a quadrilateral? A E F D

​△ABE​ and △​ DCF​ are right triangles, so they are congruent by the HL Theorem.

Step 2 Show △​ DAB ≅ △ADC​. By CPCTC, ∠​ DAB ≅ ∠ADC​. By the Reflexive Property of Congruency, ​​AD‾ ​ ≅ DA‾​ ​ . So, △​ DAB ≅ △ADC​ by SAS. The overlapping triangles are congruent, so the roof is symmetric.

Try It! 3. a. Given isosceles trapezoid PQRS, Q R what are ​m∠P , m∠Q,​ 135 and ​m∠S​?

P S

b. Given ​​ST‾ ​ ∥ RU‾​ ​,​ what is the S T measure of ∠​ TUR​? 47

R U

LESSON 8-2 Kites and Trapezoids 367 Activity Assess

THEOREM 8-4

In an isosceles trapezoid, each pair If... B C of base angles is congruent.

A D PROOF: SEE EXERCISE 13. Then... ​∠BAD ≅ ∠CDA, ∠ABC ≅ ∠DCB​

THEOREM 8-5

The diagonals of an isosceles If... B C trapezoid are congruent.

A D PROOF: SEE EXERCISE 18. Then... AC‾ ​ ≅ DB‾​ ​

EXAMPLE 4 Solve Problems Involving Isosceles Trapezoids

All horizontal beams of the high-voltage transmission tower are parallel to the ground.

10 ft 1 10 ft 2 The top section is an The center section isosceles trapezoid. is an isosceles 14 ft 14 ft trapezoid.

A. If ​m∠1 = 138​, what is ​m∠2​? The sum of the interior angle measures of a quadrilateral is 360.

​m∠1 + m∠1 + m∠2 + m∠2 = 360​ ​138 + 138 + 2(m∠2) = 360​

​276 + 2(m∠2) = 360​ The base angles ​2(m∠2) = 84​ are congruent. MAKE SENSE AND PERSEVERE ​m∠2 = 42​ What other strategy might you use to solve this problem? The measure of ∠2 is ​42​. CONTINUED ON THE NEXT PAGE

368 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess EXAMPLE 4 CONTINUED B. One cross support in the center of the tower measures ​4c 3,​ and the GENERALIZE + Why might this strategy work for other measures ​6c − 5​. What is the length of each cross support? isosceles trapezoids but not for The cross supports are diagonals of trapezoids with noncongruent an isosceles trapezoid, so they are legs? congruent.

Step 1 Find the value of c. 14 ft 14 ft 4 c 3 6c 5​ + = − 4c + 3 6c – 5 ​8 = 2c ​ ​4 = c​

Step 2 Find the lengths of the diagonals.

4 c + 3 = 4(4) + 3​ ​ = 19 6 c − 5 = 6(4) − 5​ =​ 19​ Each cross support measures 19 ft in length.

Try It! 4. Given isosceles trapezoid MNOP where the given expressions represent the measures of the diagonals, what is the value of a? N

O 2a – 1 a + 13

P

M

THEOREM 8-6 Trapezoid Midsegment Theorem

In a trapezoid, the midsegment If... B C is parallel to the bases, and the length of the midsegment is half X Y the sum of the lengths of the bases. A D

PROOF: SEE LESSON 11-2. Then... XY‾ ​ ∥ AD‾​ ​, XY‾​ ​ ∥ BC‾​ ​,​ 1 and ​XY ​ __ ​ (AD BC)​ = 2 +

LESSON 8-2 Kites and Trapezoids 369 Activity Assess

APPLICATION EXAMPLE 5 Apply the Trapezoid Midsegment Theorem

Paxton makes trapezoidal handbags for her friends. She stiches decorative trim along the top, middle, and bottom on both sides of the handbags. How much trim does she need for three handbags? Explain.

6 in.

2 in. 2 in.

2 in. 2 in.

9 in.

Formulate The top and bottom sides of the handbag are the bases of a trapezoid. The left and right sides are the legs. Since the middle segment divides both legs in half, it is the midsegment of the trapezoid. The midsegment of a trapezoid is the segment that connects the midpoints of the legs. Let x represent the length of the midsegment in inches.

Compute Step 1 Find the value of x. 1 Apply the Trapezoid Midsegment Theorem ​x = ​ __ ​(6 + 9)​ 2 with the base lengths 6 and 9. ​x = 7.5​ The length of the midsegment is 7.5 in.

Step 2 Find the amount of trim that she needs. First, find the amount for one side.

6 + 9 + 7.5 = 22.5​ Then, multiply by 2 for the number of sides per handbag and by 3 for the number of handbags.

22.5 ∙ 2 ∙ 3 = 135​ Interpret Paxton needs 135 inches of trim.

Try It! 5. Given trapezoid JKLM, what is KL?

K L

25 m X Y

J 4 M 5 XY

370 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Concept Summary Assess CONCEPT SUMMARY Kites and Trapezoids

WORDS Kites Trapezoids A kite is a quadrilateral with two A trapezoid is a quadrilateral with exactly one pair pairs of adjacent sides congruent of parallel sides. The length of the midsegment and no pairs of opposite sides is the average of the lengths of the two bases. congruent. Exactly one diagonal A trapezoid with congruent legs is an isosceles is a perpendicular bisector of trapezoid that has congruent base angles and the other. congruent diagonals.

DIAGRAMS Quadrilateral ABCD is a kite. ​Quadrilateral RSTU is an isosceles trapezoid. B S T

E A B A C R U D ​SU‾ ​ ≅ TR‾​ ​​ AB‾​​ ​​ ∥ ST‾​​ ​​ ∥ ​​RU‾ ​​ ​AC‾ ​ ⟂ BD‾​ ​ 1 ​AB = ​ __ ​(ST + RU)​ m∠S = m∠T BE = ED 2 m∠R = m∠U

Do You UNDERSTAND? Do You KNOW HOW?

1. ESSENTIAL QUESTION How are For Exercises 5–7, use kite X diagonals and angle measures related in WXYZ to find the measures. kites and trapezoids? 5. m∠XQY Q W Y 56 12 mm 2. Error Analysis What is Reagan’s error? 6. m∠YZQ

7. WY Q R

Z

P S For Exercises 8–10, use trapezoid DEFG ✗ with ​EG = 21​ ft and ​m∠DGF = 77​ to find By Theorem 8-5, PR =~ QS each measure.

8. ED E F

9. DF 8 ft 3. Vocabulary If XY‾​​ ​​ is the midsegment of a trapezoid, what must be true about point X 10. m∠DEF and point Y? 77° 4. Construct Arguments Emaan says every D G kite is composed of 4 right triangles. Is he 11. What is the length 54 cm correct? Explain. B C of PQ‾​​ ​​? P Q

A D 33 cm

LESSON 8-2 Kites and Trapezoids 371 Scan for Practice Tutorial PRACTICE & PROBLEM SOLVING Multimedia Additional Exercises Available Online

UNDERSTAND PRACTICE

12. Construct Arguments Write a two-column 16. Given kite ABCD, in which B proof to show that the diagonals of a kite ​AN = 4.6 m​, what is ​AC​? are perpendicular. SEE EXAMPLE 1 N A C X 4.6 m

W Y

Z D

13. Mathematical Connections Write a paragraph 17. Given kite RSTU, what is ​m∠RUS​? SEE EXAMPLE 2 proof to show that each pair of base angles in an isosceles trapezoid is congruent. S Q R

27 R T

P S

14. Error Analysis What is Emery’s error? U 18. Write a two-column proof to show that the diagonals of an isosceles trapezoid are BD is the perpendicular B congruent. SEE EXAMPLES 3 AND 4 bisector of AC, so HC = 8 in. because B C AC = 16 in. H A C

A D D 19. Given trapezoid MNPQ, what is ​m∠MNP​? 16 in. SEE EXAMPLE 4 ✗ N P

78 M Q 15. Higher Order Thinking K Given kite JKLM with 20. Given trapezoid WXYZ, what is ​XY​? diagonal KM‾​​ ​​, ​JK < JM,​ and ​ SEE EXAMPLE 5 KL < LM​, prove that ∠JMK is J L congruent to ∠LMK. X Y

35 ft D E

W 4 Z 3 XY M

372 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Practice Tutorial PRACTICE & PROBLEM SOLVING Mixed Review Available Online

APPLY ASSESSMENT PRACTICE

21. Model With Mathematics Gregory plans to 24. The ? of a kite are always ? . make a kite like the one shown. He has 1,700 square inches of plastic sheeting. 25. SAT/ACT Given trapezoid ABCD, what is the Does Gregory have enough plastic to make length of ​​XY‾ ​​? the kite? Explain. B 6s 1 C

30 in. 4s  2 X Y

17 in. s 17 in. A D

3 2 3​​ __ ​ 4​​ __ ​ 5 11 18 Ⓐ 5 Ⓑ 3 Ⓒ Ⓓ Ⓔ 26. Performance Task Cindy is a member of a 39 in. volunteer group that built the play structure shown.

22. Reason Coach Murphy uses the map to plan a 2-mile run for the track team. How many times will the team run the route shown? 4 ft Brazos Ave.

48 yd

48 yd Elm St. 46 yd 12 ft Pecan St. 92 yd Mulberry St.

Start Part A Cindy wants to add three more 80 yd trapezoid boards evenly spaced between the bottom and top boards of the triangular frame. Based on the average lengths of the top and bottom boards shown, what will be the average lengths of each of the three additional Lamar Ave. 80 yd boards? Explain.

Part B The three boards will be trapezoids that have the same height as the top and bottom boards. How can Cindy use the lengths of the 23. Use Structure Abby builds a bench with the bases of the bottom board to determine the seat parallel to the ground. She bends pipe to lengths of the bases of the three new boards? make the leg and seat supports. At what angles should she bend the pipe? Explain.​ Part C What other measurements should Cindy find to be certain that the boards will fit exactly onto the triangular frame?

102°

LESSON 8-2 Kites and Trapezoids 373