Topic 6 Polygons and Quadrilaterals

TOPIC OVERVIEW VOCABULARY

6-1 The Polygon Angle-Sum Theorems English/Spanish Vocabulary Audio Online: 6-2 Properties of Parallelograms English Spanish 6-3 Proving That a Quadrilateral equiangular polygon, p. 249 polígono equiángulo Is a Parallelogram equilateral polygon, p. 249 polígono equilátero 6-4 Properties of Rhombuses, isosceles trapezoid, p. 281 trapecio isósceles Rectangles, and Squares kite, p. 282 cometa 6-5 Conditions for Rhombuses, midsegment of a trapezoid, p. 281 segmento medio de un trapecio Rectangles, and Squares parallelogram, p. 255 paralelogramo rectangle, p. 269 rectángulo 6-6 Trapezoids and Kites regular polygon, p. 249 polígono regular rhombus, p. 269 rombo square, p. 269 cuadrado trapezoid, p. 281 trapecio

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247 Technology Lab Exterior Angles of Polygons

Use With Lesson 6-1 teks (5)(A), (1)(E)

Use geometry software. Construct a polygon similar to the one at the right. Extend each side as shown. Mark a point on each ray so that you can measure the exterior angles. Use your figure to explore properties of a polygon. • Measure each exterior angle. • Calculate the sum of the measures of the exterior angles. • Manipulate the polygon. Observe the sum of the measures of the exterior angles of the new polygon.

hsm11gmse_0601a_t06178 Exercises 1. Write a conjecture about the sum of the measures of the exterior angles (one at each vertex) of a convex polygon. Test your conjecture with another polygon. 2. The figures below show a polygon that is decreasing in size until it finally becomes a point. Describe how you could use this to justify your conjecture in Exercise 1.

1 5 5 1 4 4 5 2 1 4 3 2 3 2 3

3. The figure at the right shows a square that has been copied several times. Notice that you can use the square to completely cover, or tile, a plane, without Ӈ gapshsm11gmse_0601a_t06179 or overlaps. hsm11gmse_0601a_t06180hsm11gmse_0601a_t06181 a. Using geometry software, make several copies of other regular polygons with 3, 5, 6, and 8 sides. Regular polygons have sides of equal length and angles of equal measure. Ӈ b. Which of the polygons you made can tile a plane? c. Measure one exterior angle of each polygon (including the square). d. Write a conjecture about the relationship between the measure of an exterior angle and your ability to tile a plane with the polygon. Test your conjecture with another regular polygon. hsm11gmse_0601a_t06182

248 Technology Lab Exterior Angles of Polygons 6-1 The Polygon Angle-Sum Theorems

TEKS FOCUS VOCABULARY TEKS (5)(A) Investigate patterns to make conjectures about • Equiangular polygon – An equiangular geometric relationships, including angles formed by parallel lines polygon is a polygon with all angles cut by a transversal, criteria required for triangle congruence, special congruent. segments of triangles, diagonals of quadrilaterals, interior and • Equilateral polygon – An equilateral polygon exterior angles of polygons, and special segments and angles of circles is a polygon with all sides congruent. choosing from a variety of tools. • Regular polygon – A regular polygon is TEKS (1)(C) Select tools, including real objects, manipulatives, paper and a polygon that is both equilateral and pencil, and technology as appropriate, and techniques, including mental equiangular. math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(E), (1)(F) • Number sense – the understanding of what numbers mean and how they are related

ESSENTIAL UNDERSTANDING The sum of the interior angle measures of a polygon depends on the number of sides the polygon has.

Key Concept Classifying Polygons Based on Sides and Angles

An equilateral polygon An equiangular polygon A regular polygon is is a polygon with all is a polygon with all a polygon that is both sides congruent. angles congruent. equilateral and equiangular.

Theorem 6-1 Polygonhsm11gmse_0601_t06300 Angle-Sum Theorem hsm11gmse_0601_t06299 hsm11gmse_0601_t06301 The sum of the measures of the interior angles of an n-gon is (n - 2)180. For a proof of Theorem 6-1, see the Reference section on page 683.

Corollary to the Polygon Angle-Sum Theorem

(n - 2)180 The measure of each interior angle of a regular n-gon is n .

You will prove the Corollary to the Polygon Angle-Sum Theorem in Exercise 16.

PearsonTEXAS.com 249 Theorem 6-2 Polygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a polygon, one at 3 2 each vertex, is 360.

For the pentagon, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360. 4 1 5

You will prove Theorem 6-2 in Exercise 9.

Problem 1 TEKShsm11gmse_0601_t06313.ai Process Standard (1)(C) Investigating Interior Angles of Polygons

A Choose from among a variety of tools (such as a ruler, a compass, or geometry software) to investigate the sums of the measures of the interior angles of different polygons. Explain your choice. Geometry software is a good way to identify the measures of the interior angles of polygons. You can quickly make many different polygons and use the software to find the measures of their angles.

B Use geometry software to make several triangles, quadrilaterals, pentagons, and hexagons. Then complete the table. Sum of Interior Angle Sum of Interior Angle Polygon Measures Polygon Measures Triangle 1 180 Pentagon 1 540 Triangle 2 180 Pentagon 2 540 Triangle 3 180 Pentagon 3 540 Quadrilateral 1 360 Hexagon 1 720 Quadrilateral 2 360 Hexagon 2 720 Quadrilateral 3 360 Hexagon 3 720

How can recording data in a table C Use the data in the table in part B to make a conjecture about the sum of the help you make a measures of the interior angles of a polygon. conjecture? Recording data in a table Notice that the numbers in the table are all multiples of 180. Look at the patterns: is an organized way to present and analyze Triangle 1 # 180 = 180 Pentagon 3 # 180 = 540 information. You can look Quadrilateral 2 180 360 Hexagon 4 180 720 for patterns in the data # = # = and make a conjecture. Conjecture: If you subtract 2 from the number of sides and multiply by 180, you will get the sum of the measures of the interior angles of any polygon.

250 Lesson 6-1 The Polygon Angle-Sum Theorems Problem 2

Finding a Polygon Angle Sum How many sides does What is the sum of the interior angle measures of a heptagon? a heptagon have? A heptagon has 7 sides. Sum = (n - 2)180 Polygon Angle-Sum Theorem = (7 - 2)180 Substitute 7 for n. = 5 # 180 Simplify. = 900 The sum of the interior angle measures of a heptagon is 900.

Problem 3

Using the Polygon Angle-Sum Theorem STEM Biology The common housefly, Musca domestica, has eyes How does the word that consist of approximately 4000 facets. Each facet is regular help you a regular hexagon. What is the measure of each answer the question? interior angle in one hexagonal facet? The word regular tells you that each angle has the same measure.

(n - 2)180 Measure of an angle = n Corollary to the Polygon Angle-Sum Theorem (6 - 2)180 = 6 Substitute 6 for n. 4 # 180 = 6 Simplify. = 120 The measure of each interior angle in one hexagonal facet is 120.

PearsonTEXAS.com 251 Problem 4 Using the Polygon Angle-Sum Theorem What is mjY in pentagon TODAY? How does the T diagram help you? Use the Polygon Angle-Sum Theorem for n = 5. 110Њ You know the number of O sides and four of the five m∠T + m∠O + m∠D + m∠A + m∠Y = (5 - 2)180 Y angle measures. 120Њ 150Њ 110 + 90 + 120 + 150 + m∠Y = 3 180 Substitute. # D A 470 + m∠Y = 540 Simplify. m∠Y = 70 Subtract 470 from each side.

Problem 5 hsm11gmse_0601_t06302 Investigating Exterior Angles of Polygons

A Choose from a variety of tools (such as a ruler, a protractor, or a graphing calculator) to investigate exterior angles of polygons. Explain your choice. A protractor is a useful tool for investigating exterior angles of polygons because you use protractors to measure angles.

B Draw an exterior angle at each vertex of three different polygons. Investigate patterns and write a conjecture about the exterior angles. What polygons can you draw to Step 1 Draw three different polygons. Then draw the exterior angles at each investigate patterns? vertex of the polygons as shown. If you draw a triangle, a quadrilateral, and 120° 63° 58° a pentagon, you can 135 90 investigate patterns for ° 79° ° 64° different numbers of exterior angles in each 88° 130° polygon. 58° 105° 90°

Step 2 Use the protractor to measure the exterior angles of each polygon. Observe any patterns. Write a conjecture about the exterior angles of polygons. Notice that for each polygon the sum of the measures of the exterior angles is 360.

Triangle: 135 + 120 + 105 = 360 Quadrilateral: 79 + 63 + 130 + 88 = 360 Pentagon: 90 + 90 + 58 + 58 + 64 = 360 Conjecture: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

252 Lesson 6-1 The Polygon Angle-Sum Theorems Problem 6 TEKS Process Standard (1)(F) Finding an Exterior Angle Measure What is mj1 in the regular octagon at the right?

What kind of angle By the Polygon Exterior Angle-Sum Theorem, the sum of the exterior 8 7 is j1? angle measures is 360. Since the octagon is regular, the interior angles are 1 Looking at the diagram, congruent. So their supplements, the exterior angles, are also congruent. 6 you know that ∠1 is an 360 2 exterior angle. m∠1 = 8 Divide 360 by 8, the number of sides in an octagon. 5 3 4 = 45 Simplify.

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H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O hsm11gmse_0601_t06303

Find the measure of one interior angle in each regular polygon. 1. 2. 3. For additional support when completing your homework, go to PearsonTEXAS.com.

4. Sketch an equilateral polygon that is not equiangular. 5. A triangle has two congruent interior angles and an exterior angle that measures 100. Find two possible sets of interior angle measures for the triangle. Analyze Mathematical Relationships (1)(F) Find the value of each variable. 6. 7. 8. y Њ 100 Њ z Њ 3x Њ 2x Њ 87 Њ x Њ 110Њ z Њ 4x Њ (z Ϫ 13)Њ w Њ y Њ x Њ (z ϩ 10)Њ

9. a. A polygon has n sides. An interior angle of the polygon and an adjacent hsm11gmse_0601_t06058.aiexterior angle form a straight angle. What is the sum of the measures of the hsm11gmse_0601_t06060.ai n straight angles? Of the nhsm11gmse_0601_t06059.ai interior angles? b. Using your answers in part (a), what is the sum of the measures of the n exterior angles? What theorem does this prove? 10. a. Use geometry software or other tool to explore the relationships among the interior angles of quadrilaterals. Draw several quadrilaterals with parallel opposite sides. Measure the interior angles. b. Make two conjectures about the interior angles of this type of quadrilateral.

PearsonTEXAS.com 253 11. Explain Mathematical Ideas (1)(G) Your friend says she has another way to find the sum of the interior angle measures of a polygon. She picks a point inside the polygon, draws a segment to each vertex, and counts the number of triangles. She multiplies the total by 180, and then subtracts 360 from the product. Does her method work? Explain. 12. The measure of an interior angle of a regular polygon is three times the measure of an exterior angle of the same polygon. What is the name of the polygon? hsm11gmse_0601_t06061.ai Apply Mathematics (1)(A) The gift package at the right contains fruit and cheese. The fruit is in a container that has the shape of a regular octagon. The fruit container fits in a square box. A triangular cheese wedge fills each corner of the box. 13. Find the measure of each interior angle of a cheese wedge. 14. Display Mathematical Ideas (1)(G) Show how to rearrange the four pieces of cheese to make a regular polygon. What is the measure of each interior angle of the polygon? 15. a. Select Tools to Solve Problems (1)(C) Choose from a variety of tools (such as a ruler, a compass, or geometry software) to investigate the exterior angles of regular polygons. Explain your choice. Draw three regular polygons, each with a different number of sides. Then draw the exterior angles at each vertex of the polygons. b. Make two conjectures about the exterior angles of regular polygons. 16. a. In the Corollary to the Polygon Angle-Sum Theorem, explain why the measure of an interior angle of a regular n-gon is given by the formulas 180(n - 2) 360 n and 180 - n . b. Use the second formula to explain what happens to the measures of the interior angles of regular n-gons as n becomes a large number. Explain also what happens to the polygons.

TEXAS Test Practice

17. The car at each vertex of a Ferris wheel holds a maximum of five people. The sum of the interior angle measures of the Ferris wheel is 7740. What is the maximum number of people the Ferris wheel can hold? A Maple Street C 18. The Public Garden is located between two parallel streets: 64Њ Maple Street and Oak Street. The garden faces Maple Street Public and is bordered by rows of shrubs that intersect Oak Street at Shrubs Garden point B. What is m∠ABC, the angle formed by the shrubs? Shrubs 19. △ABC ≅ △DEF. If m∠A = 3x + 4, m∠C = 2x, and 37Њ m∠E = 4x + 5, what is m∠B? Oak Street B

254 Lesson 6-1 The Polygon Angle-Sum Theorems hsm11gmse_0601_t12841.ai 6-2 Properties of Parallelograms

TEKS FOCUS VOCABULARY TEKS (6)(E) Prove a quadrilateral is a parallelogram, • Consecutive angles – Consecutive angles of a polygon rectangle, square, or rhombus using opposite sides, share a common side. opposite angles, or diagonals and apply these relationships • Opposite angles – Opposite angles of a quadrilateral are to solve problems. two angles that do not share a side. TEKS (1)(F) Analyze mathematical relationships to • Opposite sides – Opposite sides of a quadrilateral are connect and communicate mathematical ideas. two sides that do not share a vertex. Additional TEKS (1)(G) • Parallelogram – A parallelogram is a quadrilateral with two pairs of parallel sides.

• Analyze – closely examine objects, ideas, or relationships to learn more about their nature

ESSENTIAL UNDERSTANDING Parallelograms have special properties regarding their sides, angles, and diagonals.

Key Concept Parallelograms and Their Parts Term Description Diagram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. You can abbreviate parallelogram with the symbol ▱.

In a quadrilateral, opposite sides do not B C share a vertex and opposite angles do not hsm11gmse_0602_t06469.aiAB and CD ЄA and ЄC share a side. are opposite are opposite sides. angles. A D

Angles of a polygon that share a side are A B consecutive angles. In the diagram, ∠A and ЄB and ЄC ∠B are consecutive angles because they hsm11gmse_0602_t06471.aiare also consecutive share side AB. D C angles.

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PearsonTEXAS.com 255 Theorem 6-3 Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱ AB ≅ CD and BC ≅ DA parallelogram, then its opposite BC B C sides are congruent. A D A D For a proof of Theorem 6-3, see the Reference section on page 683.

Theorem 6-4 hsm11gmse_0603_t06433.ai hsm11gmse_0602_t06473.ai Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱ B C m∠A + m∠B = 180 parallelogram, then its BC m∠B + m∠C = 180 consecutive angles are A D m∠C + m∠D = 180 supplementary. A D m∠D + m∠A = 180 You will prove Theorem 6-4 in Exercise 21.

hsm11gmse_0603_t06432.ai Theorem 6-5 hsm11gmse_0603_t06433.ai Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱. ∠A ≅ ∠C and ∠B ≅ ∠D parallelogram, then its opposite B C BC angles are congruent. A D A D For a proof of Theorem 6-5, see Problem 2.

Theorem 6-6 hsm11gmse_0603_t06434.ai hsm11gmse_0602_t06487.ai Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱ AE ≅ CE and BE ≅ DE parallelogram, then its BC BC diagonals bisect each other. A D A E D You will prove Theorem 6-6 in Exercise 11.

Theorem 6-7 hsm11gmse_0603_t06433.ai hsm11gmse_0603_t06443.ai Theorem If . . . Then . . . < > < > < > If three (or more) parallel lines AB } CD } EF and AC ≅ CE BD ≅ DF cut off congruent segments A B on one transversal, then they A B cut off congruent segments on C D C D every transversal. E F E F

You will prove Theorem 6-7 in Exercise 23.

hsm11gmse_0602_t06483.aihsm11gmse_0602_t06484.ai 256 Lesson 6-2 Properties of Parallelograms Problem 1 Using Consecutive Angles Q What information Multiple Choice What is mjP in ▱PQRS? from the diagram P helps you get 26 116 started? 64 126 R From the diagram, you 64 know m∠PSR and that m∠P + m∠S = 180 Consecutive angles of a ▱ are S ∠P and ∠PSR are supplementary. consecutive angles. So you can write an equation m∠P + 64 = 180 Substitute. and solve for m∠P. m∠P = 116 Subtract 64 from each side. The correct answer is C.

Problem 2 TEKS Process Standard (1)(G)

Proof Using Properties of Parallelograms in a Proof

Given: ▱ABCD B C Prove: ∠A ≅ ∠C and ∠B ≅ ∠D

A D

ABCD is a ▱. Given hsm11gmse_0602_t06478.ai

∠A and ∠B are ∠B and ∠C are ∠C and ∠D are consecutive ⦞. consecutive ⦞. consecutive ⦞. Def. of consecutive ⦞ Def. of consecutive ⦞ Def. of consecutive ⦞

∠A and ∠B are ∠B and ∠C are ∠C and ∠D are supplementary. supplementary. supplementary. Consecutive ⦞ Consecutive ⦞ Consecutive ⦞ Why is a flow proof are supplementary. are supplementary. are supplementary. useful here? A flow proof allows you to see how the pairing of two statements leads to ∠A ≅ ∠C ∠B ≅ ∠D a conclusion. Supplements of the Supplements of the same ∠ are ≅. same ∠ are ≅.

hsm11gmse_0602_t06480.ai PearsonTEXAS.com 257 Problem 3 Using Algebra to Find Lengths L M x Solve a system of linear equations to find the values of x and y 2x Ϫ 8 y ϩ 2 in ▱KLMN. What are KM and LN? y ϩ 10 P K N

The diagonals of a hsm11gmse_0602_t06481.ai parallelogram bisect each KP ≅ M P other. LP ≅ N P

Set up a system of linear ① y  10  2x  8 equations by substituting ② x  y  2 the algebraic expressions for each segment length. y  10  2(y  2)  8 y  10  2y  4  8 Substitute (y + 2) for x in equation ①. Then solve y  10  2y  4 for y. 10  y  4 14  y

Substitute 14 for y in x  14  2 equation ②. Then solve  16 for x.

KM  2(KP) LN  2(LP) Use the values of x and y  2(y  10)  2(x) to find KM and LN.  2(14  10)  2(16)  48  32

Problem 4 TEKS Process Standard (1)(F)

Using Parallel Lines and Transversals < > < > < > < > In the figure at the right, AE } BF } CG } DH , What information do AB = BC = CD = 2, and EF = 2.25. What is EH? A BCD you need? You know the length of EF = FG = GH Since } lines divide AD E EF. To find EH, you need into equal parts, they also the lengths of FG and divide EH into equal parts. F GH. G EH = EF + FG + GH Segment Addition Postulate H EH = 2.25 + 2.25 + 2.25 = 6.75 Substitute.

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258 Lesson 6-2 Properties of Parallelograms NLINE O

H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O

1. What are the values of x and y in the parallelogram? y Њ 2. The perimeter of ▱ABCD is 92 cm. AD is 7 cm more 3y Њ 3x Њ For additional support when than twice AB. Find the lengths of all four sides of ▱ABCD. completing your homework, In the figure, PQ QR RS. Find each length. go to PearsonTEXAS.com. = = W S U 3. ZU 4. XZ 2.25hsm11gmse_0602_t06078.ai 5. TU 6. XV Y Z R 7. YX 8. YV Q 9. WX 10. WV 3 X T P 11. Justify Mathematical Arguments (1)(G) V Proof Complete this two-column proof of Theorem 6-6. B C Given: ▱ABCD 2 4 Prove: AC and BD bisect each other at E. hsm11gmse_0602_t06077.ai1 E 3 A D Statements Reasons 1) ABCD is a parallelogram. 1) Given 2) AB } DC 2) a. ? 3) ∠1 ≅ ∠4; ∠2 ≅ ∠3 3) b. ? hsm11gmse_0602_t06075.ai 4) AB ≅ DC 4) c. ? 5) d. ? 5) ASA 6) AE ≅ CE; BE ≅ DE 6) e. ? 7) f. ? 7) Definition of bisector Find the values of x and y in ▱PQRS. Q R 12. PT = 2x, TR = y + 4, QT = x + 2, TS = y 13. PT = x + 2, TR = y, QT = 2x, TS = y + 3 T 14. PT = y, TR = x + 3, QT = 2y, TS = 3x - 1 P S

Use the diagram at the right for each proof. S Y T 15. Given: ▱RSTW and ▱XYTZ Proof hsm11gmse_0602_t06076.ai Prove: ∠R ≅ ∠X X Z 16. Given: ▱RSTW and ▱XYTZ R W Proof Prove: XY } RS Find the measures of the numbered angles for each parallelogram. 17. 3 18. 28Њ 19. 3 81Њ hsm11gmse_0602_t06087.ai 38Њ 3 1 110Њ 1 2 85Њ 2 1 48Њ 2

hsm11gmse_0602_t06132.ai PearsonTEXAS.com 259 hsm11gmse_0602_t06131.ai hsm11gmse_0602_t06134.ai 20. Apply Mathematics (1)(A) A pantograph is an expandable device, shown at the right. Pantographs are used in the television industry in positioning lighting and other equipment. In the photo, points D, E, F, and G are the vertices of a parallelogram. ▱DEFG is one of many parallelograms that change shape as the pantograph extends and retracts. E a. If DE 2.5 ft, what is FG? b. If m∠E 129, what is m∠G? = = D F c. What happens to m D as m E increases or decreases? ∠ ∠ G Explain. 21. Prove Theorem 6-4. B C Proof Given: ▱ABCD Prove: ∠A is supplementary to ∠B. A D ∠A is supplementary to ∠D. 22. Explain Mathematical Ideas (1)(G) Is there an SSSS congruence theorem for parallelograms? Explain. 23. Prove Theorem 6-7. Use the diagram at the hsm11gmse_0602_t06084.airight. A B Proof < > < > < > Given: AB } CD } EF , AC ≅ CE 3 C 1 D Prove: BD ≅ DF G 2 6 24. Explain Mathematical Ideas (1)(G) Explain how to separate E 4 F a blank card into three strips that are the same height by H 5 using lined paper, a straightedge, and Theorem 6-7.

TEXAS Test Practice hsm11gmse_0602_t06136.ai 25. PQRS is a parallelogram with m∠Q = 4x and m∠R = x + 10. Which statement P Q explains why you can use the equation 4x + (x + 10) = 180 to solve for x? A. The measures of the interior angles of a quadrilateral have a sum of 360. B. Opposite sides of a parallelogram are congruent. C. Opposite angles of a parallelogram are congruent. D. Consecutive angles of a parallelogram are supplementary. S R 26. In the figure of DEFG at the right, DE } GF. Which statement must be true? DE F. m∠D + m∠E = 180 H. DE ≅ GF G. m∠D m∠G 180 J. DG ≅ EF hsm11gmse_0602_t12794 + = FG 27. An obtuse triangle has side lengths of 5 cm, 9 cm, and 12 cm. What is the length of the side opposite the obtuse angle? A. 5 cm B. 9 cm C. 12 cm D. not enough information hsm11gmse_0602_t06139.ai 28. Find the measure of one exterior angle of a regular hexagon. Explain your method.

260 Lesson 6-2 Properties of Parallelograms Proving That a Quadrilateral 6-3 Is a Parallelogram

TEKS FOCUS VOCABULARY TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, • Analyze – closely examine objects, ideas, or rhombus using opposite sides, opposite angles, or diagonals and apply or relationships to learn more about their these relationships to solve problems. nature

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(G)

ESSENTIAL UNDERSTANDING You can decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties.

Theorem 6-8 Theorem If . . . Then . . . If both pairs of opposite sides B C AB ≅ CD ABCD is a ▱ of a quadrilateral are congruent, BC ≅ DA BC then the quadrilateral is a A D parallelogram. A D For a proof of Theorem 6-8, see Problem 1.

Theorem 6-9 hsm11gmse_0602_t06473.ai hsm11gmse_0603_t06433.ai Theorem If . . . Then . . . If an angle of a quadrilateral is B C m∠A + m∠B = 180 ABCD is a ▱ supplementary to both of its m∠A + m∠D = 180 BC consecutive angles, then the A D quadrilateral is a parallelogram. A D You will prove Theorem 6-9 in Exercise 17.

hsm11gmse_0603_t06432.ai Theorem 6-10 hsm11gmse_0603_t06433.ai Theorem If . . . Then . . . If both pairs of opposite angles BC ∠A ≅ ∠C ABCD is a ▱ of a quadrilateral are congruent, ∠B ≅ ∠D BC then the quadrilateral is a A D parallelogram. A D

For a proof of Theorem 6-10, see Problem 2. hsm11gmse_0603_t06434.ai hsm11gmse_0603_t06433.ai

PearsonTEXAS.com 261 Theorem 6-11 Theorem If . . . Then . . . If the diagonals of a BC AE ≅ CE ABCD is a ▱ quadrilateral bisect each other, BE ≅ DE BC then the quadrilateral is a A E D parallelogram. A D For a proof of Theorem 6-11, see Problem 3.

hsm11gmse_0603_t06443.ai hsm11gmse_0603_t06433.ai Theorem 6-12 Theorem If . . . Then . . . If one pair of opposite sides B C BC ≅ DA ABCD is a ▱ of a quadrilateral is both BC } DA BC congruent and parallel, then the A D quadrilateral is a parallelogram. A D You will prove Theorem 6-12 in Exercise 16.

hsm11gmse_0603_t06448.ai hsm11gmse_0603_t06433.ai Concept Summary Proving That a Quadrilateral Is a Parallelogram Method Source Diagram Prove that both pairs of opposite sides are parallel. Definition of parallelogram

Prove that both pairs of opposite sides are congruent. Theorem 6-8

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Prove that an angle is supplementary to both of its Theorem 6-9 75Њ consecutive angles. 75Њ 105Њ hsm11gmse_0603_t06463.ai Prove that both pairs of opposite angles are Theorem 6-10 congruent. hsm11gmse_0603_t12047 Prove that the diagonals bisect each other. Theorem 6-11

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Prove that one pair of opposite sides is congruent Theorem 6-12 and parallel. hsm11gmse_0603_t06467.ai

262 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram hsm11gmse_0603_t06468.ai Problem 1

Proof Proving Theorem 6-8 B C Given: AB ≅ CD, BC ≅ DA

Prove: ABCD is a parallelogram. A D Statements Reasons Why do you start by drawing BD? 1) Draw BD. 1) Construction In many proofs about 2) AB CD and BC DA 2) hsm11gmse_0603_t06430.aiGiven parallelograms, it is ≅ ≅ convenient to have a pair 3) BD ≅ BD 3) Reflexive Property of Congruence of triangles that you can 4) △ABD ≅ △CDB 4) SSS show to be congruent. Draw a diagonal to form 5) ∠ADB ≅ ∠CBD and 5) Corresponding parts of congruent two triangles. ∠CDB ≅ ∠ABD triangles are congruent. 6) AB } DC and BC } AD 6) Converse of the Alternate Interior Angles Theorem 7) ABCD is a parallelogram. 7) Definition of parallelogram

Problem 2 TEKS Process Standard (1)(G)

Proof Proving Theorem 6-10 B C x Њ y Њ How do you get Given: ∠A ≅ ∠C, ∠B ≅ ∠D started with the proof? Prove: ABCD is a parallelogram. A D Since the goal is to show that opposite sides are Statements Reasons parallel, you can label the 1) ∠A ≅ ∠C, ∠B ≅ ∠D 1) Given angle measures as in the diagram and show that 2) x + y + x + y = 360 2) Thehsm11gmse_0603_t06158.ai sum of the measures of the same-side interior angles angles of a quadrilateral is 360. are supplementary. 3) 2(x + y) = 360 3) Distributive Property 4) x + y = 180 4) Division Property of Equality 5) ∠A and ∠B are supplementary. 5) Definition of supplementary angles ∠A and ∠D are supplementary. 6) AD } BC, AB } DC 6) Converse of the Same-Side Interior Angles Postulate 7) ABCD is a parallelogram. 7) Definition of parallelogram

PearsonTEXAS.com 263 Problem 3

Proof Proving Theorem 6-11 B C How can you get Given: AC and BD bisect each other at E. started? E Notice that in the Prove: ABCD is a parallelogram. A D diagram there are several pairs of triangles. Use AC and BD bisect each other at E. the given information to prove pairs of triangles Given congruent. Then use their corresponding parts to hsm11gmse_0603_t06445.ai AE ≅ CE ∠AEB ≅ ∠CED ∠BEC ≅ ∠DEA show that ABCD is a BE ≅ DE parallelogram. Vertical ⦞ are ≅. Def. of segment bisector Vertical ⦞ are ≅.

△AEB ≅ △CED △BEC ≅ △DEA SAS SAS

∠BAE ≅ ∠DCE ∠ECB ≅ ∠EAD Corresp. parts of ≅ are ≅. Corresp. parts of ≅ are ≅.

AB ʈ CD BC ʈ AD If alternate interior ⦞ ≅, If alternate interior ⦞ ≅, then lines are ʈ. then lines are ʈ.

ABCD is a parallelogram. Def. of parallelogram

Problem 4 Finding Values for Parallelograms

A For what value of y must PQRS be a parallelogram? Which theorem P 3x Ϫ 5 Q should you use? Step 1 Find x. The diagram gives you x ϩ 2 y information about sides. 3x - 5 = 2x + 1 If opp. sides are ≅, then the quad. Use Theorem 6-8 because is a ▱. it uses sides to conclude S 2x ϩ 1 R that a quadrilateral is a x - 5 = 1 Subtract 2x from each side. parallelogram. 8 x = 6 Add 5 to each side...... 0 0 0 0 0 0 0 1 1 1 1 1 1 1 Step 2 Find y. 2 2 2 2 2 2 2 hsm11gmse_0603_t06439.ai3 3 3 3 3 3 3 y x 2 If opp. sides are ≅, then the quad. is a ▱. 4 4 4 4 4 4 4 = + 5 5 5 5 5 5 5 6 6 6 6 6 6 6 = 6 + 2 Substitute 6 for x. 7 7 7 7 7 7 7 8 8 8 8 8 8 8 = 8 Simplify. 9 9 9 9 9 9 9 For PQRS to be a parallelogram, the value of y must be 8.

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264 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram Problem 4 continued

B For what values of w and z must ABCD be a parallelogram? (7z 1 5)8 Step 1 Find w. A B (5w 2 30)8 5w - 30 = 3w + 10 If opp. angles are ≅, then the quad. is a ▱. 2w - 30 = 10 Subtract 3w from each side. (3w 1 10)8 2w = 40 Add 30 to each side. D C (8z 2 10)8 w = 20 Divide each side by 2. Step 2 Find z.

8z - 10 = 7z + 5 If opp. angles are ≅, then the quad. is a ▱. z - 10 = 5 Subtract 7z from each side. z = 15 Add 10 to each side. For ABCD to be a parallelogram, the value of w must be 20 and the value of z must be 15.

Problem 5 TEKS Process Standard (1)(F)

Deciding Whether a Quadrilateral Is a Parallelogram

How do you decide Can you prove that the quadrilateral is a parallelogram based on the given if you have enough information? If so, write a paragraph proof. If not, explain. information? If you can satisfy every A Given: AB = 5, CD = 5, B Given: HI ≅ HK, JI ≅ JK m∠A 50, m∠D 130 condition of a theorem = = Prove: HIJK is a parallelogram. about parallelograms, Prove: ABCD is a parallelogram. then you have enough H information. I A 5 B 50Њ

130Њ D 5 C K J

Yes. Proof: Because it is given that No. By Theorem 6-8, you need to m∠A = 50 and m∠D = 130, show that both pairs of opposite same-side interior angles A and D sides, nothsm11gmse_0603_t06453.ai consecutive sides, are hsm11gmse_0603_t06451.ai are supplementary. So AB } CD. congruent. It is given that AB = 5 and CD = 5, so AB ≅ CD. Therefore, ABCD is a parallelogram by Theorem 6-12.

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PearsonTEXAS.com 265 Problem 5 continued

C Given: m∠N = m∠Q = 39, D Given: GE = 24, GH = 12, m∠P = 141 DF = 32, HF = 16 Prove: MNPQ is a parallelogram. Prove: DEFG is a parallelogram. M N D 398 E 398 1418 H Q P 12 16 G Yes. Proof: It is given that F m∠N = m∠Q = 39 and m∠P = 141. Since the sum of the Yes. Proof: It is given that GE = 24, angle measures of a quadrilateral is GH = 12, DF = 32, and HF = 16. By 360, m M 141. Since m N m Q ∠ = ∠ = ∠ the Segment Addition Postulate, and m∠P = m∠M, ∠N ≅ ∠Q and HE = 12 and DH = 16, so the M P. Therefore, MNPQ is a ∠ ≅ ∠ diagonals of the quadrilateral bisect parallelogram by Theorem 6-10. each other. DEFG is a parallelogram by Theorem 6-11.

Problem 6

Identifying Parallelograms

As the arms of the lift Vehicle Lifts A truck sits on the platform of a vehicle lift. Two moving arms move, what changes raise the platform until a mechanic can fit underneath. Why will the truck and what stays the always remain parallel to the ground as it is lifted? Explain. same? The angles the arms form with the ground and the platform change, but the lengths of the arms and the platform stay the same. Q R Q R 26 ft 26 ft 6 ft 6 ft 6 ft 6 ft P 26 ft S P 26 ft S

The angles of PQRS change as platform QR rises, but its side lengths remain the same. Both pairs of opposite sides are congruent, so PQRS is a parallelogram by Theorem 6-8. By the definition of a parallelogram, PS } QR. Since the base of the lift PS lies along the ground, platform QR, and therefore the truck, will always be parallel to the ground.

266 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram NLINE O

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1. Given: AB ≅ CD, DE ≅ FC, EA ≅ BF 2. Given: ∠M ≅ ∠P, ∠MNQ ≅ ∠PQN, Proof Proof Prove: ABCD is a parallelogram. ∠MQN ≅ ∠PNQ For additional support when A B Prove: MNPQ is a parallelogram. completing your homework, M N go to PearsonTEXAS.com. E F D C

Q P 3. Given: M is the midpoint of HK and JL. 4. Given: ∠A and ∠C are right angles, Proof Proof Prove: HJKL is a parallelogram. AD ≅ CB H J Prove: ABCD is a parallelogram. M A L K B

D

C Analyze Mathematical Relationships (1)(F) For what values of x and y must ABCD be a parallelogram?

5. B C 6. D C 7. D 5x Ϫ 8 C 7 3x Њ (y ϩ 78)Њ 2x Ϫ 2y 1 4 3y Њ (4x Ϫ 21)Њ y Ϫ A D B A B A 2x ϩ 7

8. 9. 10. B C A B A 2y ϩ 2 B (3x ϩ 10)Њ hsm11gmse_0603_t06150.ai (2x ϩ 15)Њ (8x ϩ 5)Њ 3hsm11gmse_0603_t06149.aix ϩ 6 y ϩ 4 hsm11gmse_0603_t06152.ai(4x Ϫ 33)Њ 5yЊ A D D C D 3y Ϫ 9 C

11. Display Mathematical Ideas (1)(G) Sketch two noncongruent parallelograms ABCD and EFGH so that AB ≅ EF and BC ≅ FG. hsm11gmse_0603_t06162.ai hsm11gmse_0603_t06160.ai hsm11gmse_0603_t06161.ai Can you prove that the quadrilateral is a parallelogram based on the given information? Explain. 12. 13. 14.

PearsonTEXAS.com 267 hsm11gmse_0603_t06154.ai hsm11gmse_0603_t06155.ai hsm11gmse_0603_t06157.ai 15. Apply Mathematics (1)(A) Quadrilaterals are AB formed on the side of this fishing tackle box by the adjustable shelves and connecting pieces. Explain why the shelves are always parallel to each other no matter what their position is. CD

16. Justify Mathematical 17. Prove Theorem 6-9. Proof Proof Arguments (1)(G) Prove Given: ∠A is supplementary to ∠B. Theorem 6-12. ∠A is supplementary to ∠D. Given: BC } DA, BC ≅ DA Prove: ABCD is a parallelogram. Prove: ABCD is a parallelogram. B C B C

A D A D

TEXAS Test Practice hsm11gmse_0603_t06159.ai hsm11gmse_0603_t06581.ai

18. Which piece of additional information would allow you P Q to prove that PQRS is a parallelogram? T A. PQ ≅ RS C. ∠PTQ ≅ ∠RTS B. QR ≅ SP D. ∠QPR ≅ ∠SRP S R 19. In quadrilateral ABCD, m∠A = 3x + 2, m∠B = x - 22, and m∠C = 2x + 52. Which value of x allows you to conclude that ABCD is a parallelogram? F. 50 H. 28

G. 34 J. -12

20. Quadrilateral JKLM is a parallelogram. Which of the following J N K does NOT guarantee that JNPM is a parallelogram? A. N is the midpoint of JK and P is the midpoint of ML. B. JM NP ≅ M P L C. JM } NP D. ∠JMP ≅ ∠NPL

21. Write a proof using the diagram. N T P Given: △NRJ ≅ △CPT, JN } CT Prove: JNTC is a parallelogram. R J C

268 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram hsm11gmse_0603_t06167.ai Properties of Rhombuses, Rectangles, 6-4 and Squares

TEKS FOCUS VOCABULARY TEKS (5)(A) Investigate patterns to make conjectures about • Rectangle – A rectangle is a parallelogram with geometric relationships, including angles formed by parallel lines four right angles. cut by a transversal, criteria required for triangle congruence, • Rhombus – A rhombus is a parallelogram with four special segments of triangles, diagonals of quadrilaterals, interior congruent sides. and exterior angles of polygons, and special segments and angles Square – A square is a parallelogram with four of circles choosing from a variety of tools. • congruent sides and four right angles. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, • Number sense – the understanding of what including mental math, estimation, and number sense as numbers mean and how they are related appropriate, to solve problems.

Additional TEKS (1)(F), (6)(E)

ESSENTIAL UNDERSTANDING The parallelograms in the Take Note box below have basic properties about their sides and angles that help identify them. The diagonals of these parallelograms also have certain properties.

Key Concept Special Parallelograms

A rhombus is a parallelogram A rectangle is a parallelogram A square is a parallelogram with four congruent sides. with four right angles. with four congruent sides and four right angles.

hsm11gmse_0604_t06018 hsm11gmse_0604_t06019 Theorem 6-13 hsm11gmse_0604_t06020 Theorem If . . . Then . . . If a parallelogram is a ABCD is a rhombus AC # BD rhombus, then its diagonals A D A D are perpendicular.

B C B C For a proof of Theorem 6-13, see Lesson 7-3.

hsm11gmse_0604_t06022 hsm11gmse_0604_t06023PearsonTEXAS.com 269 Theorem 6-14 Theorem If . . . Then . . . If a parallelogram is a rhombus, ABCD is a rhombus A D ∠1 ≅ ∠2 then each diagonal bisects a A D 2 3 ∠3 ≅ ∠4 1 4 pair of opposite angles. ∠5 ≅ ∠6 ∠7 ≅ ∠8 7 5 8 6 B C B C You will prove Theorem 6-14 in Exercise 10.

hsm11gmse_0604_t06024 Theorem 6-15 hsm11gmse_0604_t06022 Theorem If . . . Then . . . If a parallelogram is a ABCD is a rectangle AC ≅ BD rectangle, then its A D A D diagonals are congruent.

B C B C

You will prove Theorem 6-15 in Exercise 13.

hsm11gmse_0604_t06028 hsm11gmse_0604_t06031 Problem 1 Classifying Special Parallelograms Is ▱ABCD a rhombus, a rectangle, or a square? How do you decide Explain. whether ABCD is a ▱ABCD is a rectangle. Opposite angles of a A B rhombus, a rectangle, E or a square? parallelogram are congruent, so m∠D is 90. By the Use the definitions of Same-Side Interior Angles Theorem, m∠A = 90 F rhombus, rectangle, and and m∠C = 90. Since ▱ABCD has four right H square along with the angles, it is a rectangle. You cannot conclude that markings on the figure. ABCD is a square because you do not know its G side lengths. D C

270 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares Problem 2 TEKS Process Standards (1)(C) Investigating Diagonals of Quadrilaterals

A Choose from a variety of tools (such as a protractor, a ruler, a compass, or a geoboard) to investigate patterns in the diagonals of quadrilaterals. Explain your choice. A manipulative such as a geoboard makes it easy to make different types of quadrilaterals and their diagonals.

B Make several parallelograms, rectangles, and rhombuses. Then make a conjecture about the diagonals of each type of quadrilateral. Parallelogram Rectangle Rhombus

How can you measure distances on a geoboard? You can use the grid of pegs to indicate horizontal and vertical units.

Conjecture: The diagonals Conjecture: The diagonals Conjecture: The diagonals of a parallelogram bisect of a rectangle are congruent. of a rhombus are each other. perpendicular.

Problem 3 TEKS Process Standard (1)(F)

Finding Angle Measures How are the What are the measures of the numbered angles in rhombus ABCD? numbered angles B C formed? m∠1 = 90 The diagonals of a rhombus are #. The angles are formed 58Њ by diagonals. Use what m 2 58 Alternate Interior Angles Theorem ∠ = 1 2 you know about the 4 3 diagonals of a rhombus Each diagonal of a rhombus bisects a A m∠3 = 58 D to find the angle pair of opposite angles. measures. m∠1 + m∠3 + m∠4 = 180 Triangle Angle-Sum Theorem 90 + 58 + m∠4 = 180 Substitute. hsm11gmse_0604_t06026 148 + m∠4 = 180 Simplify. m∠4 = 32 Subtract 148 from each side.

PearsonTEXAS.com 271 Problem 4 Finding Diagonal Length

How can you find the Multiple Choice In rectangle RSBF, SF 2x  15 and RB 5x  12. S B length of a diagonal? What is the length of a diagonal? Since RSBF is a rectangle 1 9 18 33 and its diagonals are congruent, use the expressions to write an R F equation. You know that the diagonals of a rectangle are congruent, so SF RB their lengths are equal. hsm11gmse_0604_t06032

Set the algebraic expressions for SF and RB equal to each other 2x  15 5x  12 and find the value of x. 15 3x  12 27 3x 9 x

Substitute 9 for x in the RB 5x  12 expression for RB. 5(9)  12 33 The correct answer is D.

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Find the measures of the numbered angles in each rhombus. 1. 2. 3. 3 3 1 2 For additional support when 4 2 2 35Њ completing your homework, 3 go to PearsonTEXAS.com. 1 35Њ 1 60Њ

LMNP is a rectangle. Find the value of x and the length of each diagonal.

4. LN = x and MP = 2x - 4 5. LN = 5x - 8 and MP = 2x + 1 hsm11gmse_0604_t05920.aihsm11gmse_0604_t05921.aihsm11gmse_0604_t05935.ai 6. LN = 3x + 1 and MP = 8x - 4 7. LN = 9x - 14 and MP = 7x + 4 8. LN = 7x - 2 and MP = 4x + 3 9. LN = 3x + 5 and MP = 9x - 10 10. Prove Theorem 6-14. A D Proof 4 Given: ABCD is a rhombus. 3 Prove: AC bisects ∠BAD and ∠BCD. 2 1 B C

272 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares hsm11gmse_0604_t06250.ai Decide whether the parallelogram is a rhombus, a rectangle, or a square. Explain. 11. 12.

13. Justify Mathematical Arguments (1)(G) Complete the flow A D Proof proof of Theorem 6-15.

Given: ABCD is a rectangle. B C Prove: AC ≅ BD

ABCD is a ▱. e. b. Opposite sideshsm11gmse_0604_t06239.ai of a ▱ are ≅.

ABCD is BC ≅ BC f. AC ≅ BD a rectangle. c. SAS h. a.

∠ABC and ∠DCB ∠ABC ≅ ∠DCB are right ⦞. g. d.

14. Connect Mathematical Ideas (1)(F) Summarize the properties of squares that follow from a square being (a) a parallelogram, (b) a rhombus, and (c) a rectangle. hsm11gmse_0604_t06243.ai15. Analyze Mathematical Relationships (1)(F) Find the angle K 4b Ϫ 6r J measures and the side lengths of the rhombus at the right. r ϩ 1 b Ϫ 3 16. Create Representations to Communicate Mathematical Ideas xЊ (2x ϩ 6)Њ (1)(E) On graph paper, draw a parallelogram that is neither a H 2r Ϫ 4 G rectangle nor a rhombus. ABCD is a rectangle. Find the length of each diagonal. 17. AC = 2(x - 3) and BD = x + 5 18. AC = 2(5a + 1) and BD = 2(a + 1) 3y 3c hsm11gmse_0604_t06254.ai 19. AC = 5 and BD = 3y - 4 20. AC = 9 and BD = 4 - c

PearsonTEXAS.com 273 Find the values of the variables. Then find the side lengths.

21. rhombus 15 22. square 2x Ϫ 7

3y 5x y Ϫ 1 2y Ϫ 5

4x ϩ 3 3y Ϫ 9

23. Justify Mathematical Arguments (1)(G) Write a proof. P L Proof Given: Rectangle PLAN T hsm11gmse_0604_t05942.ai hsm11gmse_0604_t05943.ai Prove: △LTP ≅ △NTA N A 24. a. Select Tools to Solve Problems (1)(C) To investigate the diagonals and the interior angles of rhombuses, choose from the following tools: ruler, paper folding, or graphing calculator. Explain your choice. b. Make several rhombuses with their diagonals. Observe anyhsm11gmse_0604_t06262.ai patterns. Make a conjecture about the diagonals and the interior angles of rhombuses. Find the value of x in the rhombus.

25. (6x 2 Ϫ 3x)Њ 26. (7x 2 Ϫ 10)Њ (2x 2 Ϫ 25x)Њ

(3x 2 ϩ 60)Њ

TEXAS Test Practicehsm11gmse_0604_t05944.ai hsm11gmse_0604_t05945.ai

27. A part of a design for a quilting pattern consists of a regular pentagon and five isosceles triangles, as shown. What is m∠1? 1 A. 18 C. 72 B. 36 D. 108 28. Which statement is true for some, but not all, rectangles? F. Opposite sides are parallel. H. Adjacent sides are perpendicular. G. It is a parallelogram. J. All sides are congruent. 29. Which term best describes AD in △ABC? hsm11gmse_0604_t12846A A. altitude C. median

B. angle bisector D. perpendicular bisector B D C 30. Write the first step of an indirect proof that △PQR is not a right triangle.

hsm11gmse_0604_t05947.ai

274 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares Conditions for Rhombuses, Rectangles, 6-5 and Squares

TEKS FOCUS VOCABULARY TEKS (6)(E) Prove a quadrilateral is a parallelogram, • Analyze – closely examine objects, ideas, or relationships rectangle, square, or rhombus using opposite sides, to learn more about their nature opposite angles, or diagonals and apply these relationships to solve problems.

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(G)

ESSENTIAL UNDERSTANDING You can determine whether a parallelogram is a rhombus or a rectangle based on the properties of its diagonals.

Theorem 6-16 Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱ and AC # BD ABCD is a rhombus parallelogram with A D A D perpendicular diagonals, then the quadrilateral is a rhombus.

B C B C For a proof of Theorem 6-16, see Problem 1.

Theorem 6-17hsm11gmse_0605_t06034.ai hsm11gmse_0605_t06274.ai Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱, ∠1 ≅ ∠2, and ABCD is a rhombus parallelogram with a ∠3 ≅ ∠4 A D diagonal that bisects a pair A D of opposite angles, then the 3 quadrilateral is a rhombus. 4 1 2 B C B C You will prove Theorem 6-17 in Exercise 16.

hsm11gmse_0605_t06274.ai hsm11gmse_0605_t06036.ai

PearsonTEXAS.com 275 Theorem 6-18 Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱, and AC ≅ BD ABCD is a rectangle parallelogram with congruent A D A D diagonals, then the quadrilateral is a rectangle.

B C B C You will prove Theorem 6-18 in Exercise 17.

Theorem 6-19hsm11gmse_0605_t06037.ai hsm11gmse_0604_t06028 Theorem If . . . Then . . . If a quadrilateral is a ABCD is a ▱, AC # BD, and ABCD is a square parallelogram with AC ≅ BD A D perpendicular, congruent A D diagonals, then the quadrilateral is a square. E

B C B C For a proof of Theorem 6-19, see Problem 2.

Problem 1 TEKS Process Standard (1)(G)

Proof Proving Theorem 6-16 A D How can knowing Given: ABCD is a parallelogram, AC # BD that the quadrilateral is a parallelogram Prove: ABCD is a rhombus. E help you prove the Since ABCD is a parallelogram, AC and BD bisect each B theorem? C You can use any of other, so BE ≅ DE. Since AC # BD, ∠AED and ∠AEB are the properties of congruent right angles. By the Reflexive Property of parallelograms to Congruence, AE ≅ AE. So △AEB ≅ △AED by SAS. help you. Corresponding parts of congruent triangles are congruent, hsm11gmse_0605_t06035.ai so AB ≅ AD. Since opposite sides of a parallelogram are congruent, AB ≅ DC ≅ BC ≅ AD. By definition, ABCD is a rhombus.

276 Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares Problem 2

Proof Proving Theorem 6-19 A D Write a two-column proof to prove Theorem 6-19. How can knowing the E figure is a rectangle Given: ABCD is a parallelogram, AC # DB, and help you prove it is a AC ≅ DB square? B C A rectangle has four 90° Prove: ABCD is a square. angles. If you know the figure is a rectangle, you Statements Reasons only need to show all 1) ABCD is a parallelogram, AC # DB, and 1) Given sides are congruent to prove it is a square. AC ≅ DB 2) ABCD is a rectangle. 2) Theorem 6-18 3) ∠DAB, ∠ABC, ∠BCD, and ∠CDA are right angles. 3) Def. of a rectangle 4) ABCD is a rhombus. 4) Theorem 6-16 5) AB ≅ BC ≅ CD ≅ DA 5) Def. of a rhombus 6) ABCD is a square. 6) Def. of a square

Problem 3 TEKS Process Standard (1)(F)

Identifying Rhombuses, Rectangles, and Squares Can you conclude that quadrilateral ABCD is a rhombus, a rectangle, or a square? If so, write a paragraph proof. If not, explain. How do you get started? A Given: Quadrilateral ABCD with B Given: Quadrilateral ABCD with Use the properties of AE ≅ BE ≅ CE ≅ DE AE ≅ CE, BE ≅ DE rhombuses, rectangles, and squares and the Prove: ABCD is a rhombus, a rectangle, or a Prove: ABCD is a rhombus, theorems you learned square. a rectangle, or a square. to help you determine whether each figure is a A D A D rhombus, a rectangle, or a square. E E

B C B C Yes. Proof: It is given that No. The diagonals bisect each AE ≅ BE ≅ CE ≅ DE in quadrilateral ABCD. other, so by Theorem 6-11, By the definition of segment bisector, AC quadrilateral ABCD is a and DB bisect each other. By Theorem 6-11, parallelogram. The diagonals ABCD is a parallelogram. By the definition of are not perpendicular, so congruent segments, AE BE CE DE. By = = = ABCD is not a rhombus or a the Segment Addition Postulate, AC DB. = square. The diagonals are not So AC ≅ DB by the definition of congruent congruent, so ABCD is not a segments. Therefore, by Theorem 6-18, ABCD rectangle or a square. is a rectangle.

PearsonTEXAS.com 277 Problem 4 Using Properties of Special Parallelograms Algebra For what value of x is ▱ABCD a rhombus? A D

(6x Ϫ 2)Њ

For ▱ABCD to be a rhombus, its diagonals m∠ABD = m∠CBD B C must bisect a pair of (4x ϩ 8)Њ opposite angles.

Set the expressions for 6x 2 4x 8 m∠ABD and m∠CBD - = + hsm11gmse_0605_t06040.ai equal to each other.

2x - 2 = 8 Solve for x. 2x = 10 x = 5

Problem 5

Using Properties of Parallelograms Community Service Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio for a youth center.

A How can you use properties of diagonals to locate the four corners? You can use two theorems. • Theorem 6-11: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. • Theorem 6-18: If a quadrilateral is a parallelogram with congruent diagonals, If a quadrilateral is then the quadrilateral is a rectangle. both a rectangle and Step 1 Cut two pieces of rope that will be the diagonals of the foundation a rhombus, why is it a square? rectangle. Cut them the same length because of Theorem 6-18. If a quadrilateral is Step 2 Join the two pieces of rope at their midpoints because of Theorem 6-11. a rectangle, then its diagonals are congruent Step 3 Pull the ropes straight and taut. The ends of the ropes will be the corners bisectors. If it is a rhombus, of a rectangle. then its diagonals are perpendicular bisectors. B Can you adapt this method slightly to stake off a square play area? Explain. So, by Theorem 6-19, the Yes, you can if you make the diagonals perpendicular. The result will be a quadrilateral is a square. rectangle and a rhombus, so the play area will be square.

278 Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares NLINE O

H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O

For what value of x is the figure the given special parallelogram? 1. rhombus 2. rectangle 3. rectangle For additional support when L O completing your homework, (6x Ϫ 9)Њ 4x ϩ ϭ Ϫ go to PearsonTEXAS.com. ϩ 3 7 LN 4x 7 8x MO ϭ 2x ϩ 13 4 4 (2x ϩ 39)Њ M N 4. rectangle 5. rhombus 6. rectangle (5x ϩ 2)Њ (4x Ϫ 12)Њ hsm11gmse_0605_t05965.ai hsm11gmse_0605_t05964.ai hsm11gmse_0605_t05966.ai(3x ϩ 4)Њ (3x ϩ 6)Њ 3xЊ (8x ϩ 7)Њ

7. Analyze Mathematical Relationships (1)(F) Decide whether D the given information is sufficient to show the hsm11gmse_0605_t05969 quadrilateralhsm11gmse_0605_t05967.ai is a rectangle. Explain. C hsm11gmse_0605_t05968.ai E a. AE ≅ CE and DE ≅ BE

b. AD ≅ BC, AB ≅ DC, and m∠DAB = 90 A c. AB } CD, AD } BC, and AC ≅ DB B d. AE ≅ CE ≅ DE ≅ BE

STEM 8. Apply Mathematics (1)(A) You can use a simple device called a turnbuckle to “square up” structures that are parallelograms. For the gate pictured at the right, you tighten or loosen the turnbuckle on the diagonal cable so that the rectangular frame will keep the shape of a parallelogram when it sags. What are two ways you can make sure that the turnbuckle works? Explain. 9. Explain Mathematical Ideas (1)(G) Suppose the diagonals of a parallelogram are both perpendicular and congruent. What type of special quadrilateral is it? Explain your reasoning. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. 10. 11. 12.

hsm11gmse_0605_t05961.ai PearsonTEXAS.com 279 hsm11gmse_0605_t05962.ai hsm11gmse_0605_t05963.ai Create Representations to Communicate Mathematical Ideas (1)(E) Given two segments with lengths a and b (a b), what special parallelograms meet the given conditions? Show each sketch. 13. Both diagonals have length a. 14. The two diagonals have lengths a and b. 15. One diagonal has length a, and one side of the quadrilateral has length b.

16. Prove Theorem 6-17. A D Proof 4 Given: ABCD is a parallelogram. 3 AC bisects ∠BAD and ∠BCD. 1 Prove: ABCD is a rhombus. 2 A B C 17. Prove Theorem 6-18. Proof Given: ABCD, AC BD ▱ ≅ D B Prove: ABCD is a rectangle. hsm11gmse_0605_t05970C Explain Mathematical Ideas (1)(G) Explain how to construct each figure given its diagonals. 18. parallelogram 19. rectangle 20. rhombus hsm11gmse_0605_t05971 Determine whether the quadrilateral can be a parallelogram. Explain. 21. The diagonals are congruent, but the quadrilateral has no right angles. 22. Each diagonal is 3 cm long, and two opposite sides are 2 cm long. 23. Two opposite angles are right angles, but the quadrilateral is not a rectangle.

24. Justify Mathematical Arguments (1)(G) In Theorem 6-17, replace “a pair Proof of opposite angles” with “one angle.” Write a paragraph that proves this new statement to be true, or give a counterexample to prove it to be false.

TEXAS Test Practice

25. Each diagonal of a quadrilateral bisects a pair of opposite angles of the quadrilateral. What is the most precise name for the quadrilateral? A. parallelogram B. rhombus C. rectangle D. not enough information 26. Given a triangle with side lengths 7 and 11, which value could NOT be the length of the third side of the triangle? F. 13 G. 7 H. 5 J. 2 27. What is the sum of the measures of the exterior angles, one at each vertex, in a pentagon? A. 180 B. 360 C. 540 D. 108

28. The midpoint of PQ is (-1, 4). One endpoint is P(-7, 10). What are the coordinates of endpoint Q? Explain your work.

280 Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares 6-6 Trapezoids and Kites

TEKS FOCUS VOCABULARY TEKS (5)(A) Investigate patterns to make • Base angles of a trapezoid – The • Legs of a trapezoid – The legs of conjectures about geometric relationships, base angles of a trapezoid are a trapezoid are the nonparallel including angles formed by parallel lines the two angles that share a base sides of the trapezoid. cut by a transversal, criteria required for of the trapezoid. • Midsegment of a trapezoid – triangle congruence, special segments of • Bases of a trapezoid – The bases The midsegment of a trapezoid triangles, diagonals of quadrilaterals, interior of a trapezoid are the parallel is the segment that joins the and exterior angles of polygons, and special sides of the trapezoid. midpoints of its legs. segments and angles of circles choosing from Isosceles trapezoid – An isosceles Trapezoid – A trapezoid is a a variety of tools. • • trapezoid is a trapezoid with quadrilateral with exactly one TEKS (1)(F) Analyze mathematical legs that are congruent. pair of parallel sides. relationships to connect and communicate • Kite – A kite is a quadrilateral mathematical ideas. with two pairs of consecutive • Analyze – closely examine Additional TEKS (1)(C) sides congruent and no opposite objects, ideas, or relationships to sides congruent. learn more about their nature

ESSENTIAL UNDERSTANDING The angles, sides, and diagonals of a trapezoid have certain properties. The angles, sides, and diagonals of a kite have certain properties.

Key Concept Trapezoids and Their Parts Term Description Diagram

A trapezoid is a quadrilateral with exactly one pair of parallel base sides. The parallel sides of a trapezoid are called bases. The leg leg nonparallel sides are called legs. The two angles that share a base angles base angles base of a trapezoid are called base angles. A trapezoid has two pairs of base angles. base

An isosceles trapezoid is a trapezoid with legs that are B C congruent. hsm11gmse_0606_t06314 A D

A midsegment of a trapezoid is the segment that joins the R A midpoints of its legs. M N hsm11gmse_0606_t06315 T P

PearsonTEXAS.com 281 hsm11gmse_0606_t06325 Theorem 6-20

Theorem If . . . Then . . . If a quadrilateral is an isosceles TRAP is an isosceles trapezoid ∠T ≅ ∠P, ∠R ≅ ∠A trapezoid, then each pair of with bases RA and TP R A base angles is congruent. R A

T P T P

You will prove Theorem 6-20 in Exercise 1.

hsm11gmse_0606_t06317 hsm11gmse_0606_t06317 Theorem 6-21

Theorem If . . . Then . . . If a quadrilateral is an isosceles ABCD is an isosceles trapezoid AC ≅ BD trapezoid, then its diagonals are B C B C congruent.

A D A D

You will prove Theorem 6-21 in Exercise 16.

hsm11gmse_0606_t06321 hsm11gmse_0606_t06324 Theorem 6-22 Trapezoid Midsegment Theorem

Theorem If . . . Then . . . If a quadrilateral is a trapezoid, TRAP is a trapezoid with (1) MN } TP, MN } RA, and then midsegment MN 1 (2) MN = 2(TP + RA) (1) the midsegment is parallel R A to the bases, and (2) the length of the M N midsegment is half the sum of T P the lengths of the bases. You will prove Theorem 6-22 in Lesson 7-3.

hsm11gmse_0606_t06325 Key Concept Kites Term Description Diagram A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

282 Lesson 6-6 Trapezoids and Kites hsm11gmse_0606_t06333 Theorem 6-23

Theorem If . . . Then . . . If a quadrilateral is a kite, then ABCD is a kite AC # BD its diagonals are perpendicular. B B

A C A C

D D For a proof of Theorem 6-23, see the Reference section on page 683.

Concept Summaryhsm11gmse_0606_t06335 Relationships Amonghsm11gmse_0606_t06336 Quadrilaterals

Quadrilateral 2 pairs of parallel sides No pairs of Only 1 pair of parallel sides parallel sides Parallelogram Kite Trapezoid Rectangle Rhombus

Isosceles trapezoid Square

Problemhsm11gmse_0606_t06342.ai 1 TEKS Process Standard (1)(F) Finding Angle Measures in Trapezoids D E What do you know CDEF is an isosceles trapezoid and mjC 65. What are mjD, about the angles mjE, and mjF? of an isosceles m∠C + m∠D = 180 Two angles that form same-side interior trapezoid? 65Њ You know that each angles along one leg are supplementary. C F pair of base angles is 65 m D 180 Substitute. congruent. Because the + ∠ = bases of a trapezoid m∠D = 115 Subtract 65 from each side. are parallel, you also know that two angles Since each pair of base angles of an isosceles trapezoid is congruent, m C m F 65 and ∠ = hsm11gmse_0606_t06318∠ = that share a leg are m∠D = m∠E = 115. supplementary.

PearsonTEXAS.com 283 Problem 2 Finding Angle Measures in Isosceles Trapezoids Paper Fans The second ring of the paper fan shown at the right consists of 20 congruent isosceles trapezoids that appear to form circles. What are the measures of the base angles of these trapezoids? What do you notice about the diagram? Step 1 Find the measure of each angle at the center Each trapezoid is part of of the fan. This is the measure of the vertex angle an isosceles triangle with of an isosceles triangle. base angles that are the 360 acute base angles of the m∠1 = = 18 trapezoid. 20 Step 2 Find the measure of each acute base angle of an isosceles triangle.

18 + x + x = 180 Triangle Angle-Sum Theorem 18 + 2x = 180 Combine like terms. 2x = 162 Subtract 18 from each side. x = 81 Divide each side by 2. Step 3 Find the measure of each obtuse base angle of the isosceles trapezoid. Two angles that form same-side interior 81 + y = 180 angles along one leg are supplementary.

y = 99 Subtract 81 from each side. Each acute base angle measures 81. Each obtuse base angle measures 99.

Problem 3 TEKS Process Standard (1)(C)

Investigating the Diagonals of Isosceles Trapezoids

How is an isosceles A Choose from a variety of tools (such as a protractor, a ruler, or a compass) to trapezoid different investigate patterns in the diagonals of the three given isosceles trapezoids. from other Explain your choice. trapezoids? An isosceles trapezoid A ruler is useful for measuring segments. is a trapezoid whose nonparallel legs are B Make a conjecture about the diagonals of isosceles trapezoids. congruent. Isosceles Trapezoid ABCD

AB AC = 3 cm and BD = 3 cm. So AC = BD and AC ≅ BD.

D C

continued on next page ▶

284 Lesson 6-6 Trapezoids and Kites Problem 3 continued Isosceles Trapezoid EFGH Isosceles Trapezoid JKLM E F J

EG = 2.5 cm and FH = 2.5 cm. JL = 2 cm and KM = 2 cm. M So EG = FH and EG ≅ FH. So JL = KM and JL ≅ KM.

L K H G Conjecture: If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Problem 4 Using the Midsegment of a Trapezoid Algebra QR is the midsegment of trapezoid LMNP. 4x Ϫ 10 What is x? M 1 L QR = 2 (LM + PN) Trapezoid Midsegment Theorem 1 x ϩ 2 x + 2 = [(4x - 10) + 8] Substitute. Q R How can you check 2 1 your answer? x + 2 = (4x - 2) Simplify. Find LM and QR. Then 2 see if QR equals half x + 2 = 2x - 1 Distributive Property P 8 N of the sum of the base lengths. 3 = x Subtract x and add 1 to each side.

Problem 5 hsm11gmse_0606_t06328

Finding Angle Measures in Kites D 52Њ Quadrilateral DEFG is a kite. What are mj1, mj2, and mj3? 3 1 2 How are the triangles m∠1 = 90 Diagonals of a kite are #. E G congruent by SSS? 90 m∠2 52 180 Triangle Angle-Sum Theorem DE ≅ DG and FE ≅ FG + + = because a kite has 142 + m∠2 = 180 Simplify. congruent consecutive sides. DF ≅ DF by the m∠2 = 38 Subtract 142 from each side. F Reflexive Property of Congruence. △DEF ≅ △DGF by SSS. Since corresponding parts of congruent triangles are congruent, m∠3 = m∠GDF = 52.

hsm11gmse_0606_t06340.ai

PearsonTEXAS.com 285 NLINE O

H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O

1. Justify Mathematical Arguments (1)(G) The plan suggests a proof of A D Proof Theorem 6-20. Write a proof that follows the plan. Given: Isosceles trapezoid ABCD with AB ≅ DC For additional support when 1 completing your homework, Prove: ∠B ≅ ∠C and ∠BAD ≅ ∠D B C go to PearsonTEXAS.com. E Plan: Begin by drawing AE } DC to form parallelogram AECD so that AE ≅ DC ≅ AB. ∠B ≅ ∠C because ∠B ≅ ∠1 and ∠1 ≅ ∠C. Also, ∠BAD ≅ ∠D because they are supplements of the congruent angles, ∠B and ∠C. Analyze Mathematical Relationships (1)(F) Find the value(s) of the variable(s) hsm11gmse_0606_t06008in each isosceles trapezoid or kite.

2. Q R 3. 4. (3x ϩ 5)Њ

yЊ (x ϩ 6)Њ (2y Ϫ 20)Њ P S (2x Ϫ 4)Њ QS ϭ x ϩ 5 2xЊ RP ϭ 3x ϩ 3 (4x Ϫ 30)Њ

5. Explain Mathematical Ideas (1)(G) If KLMN is an isosceles trapezoid, is it possible for KM to bisect ∠LMN and ∠LKN? Explain. STEM Applyhsm11gmse_0606_t06001 Mathematics (1)(A) The beamshsm11gmse_0606_t06005 of the hsm11gmse_0606_t06006 bridge at the right form quadrilateral ABCD. △AED @ △CDE @ △BEC and mjDCB  120.

6. Classify the quadrilateral. Explain your reasoning. A E B 7. Find the measures of the other interior angles of the quadrilateral. D C 8. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less than twice the length of another. Find the length of each side of the kite. 9. Prove the converse of Theorem 6-20: If a trapezoid has a pair of congruent base Proof angles, then the trapezoid is isosceles. Name each type of special quadrilateral that can meet the given condition. Make sketches to support your answers. 10. exactly one pair of congruent sides 11. two pairs of parallel sides 12. four right angles 13. adjacent sides that are congruent 14. perpendicular diagonals 15. congruent diagonals B C 16. Prove Theorem 6-21. Proof Given: Isosceles trapezoid ABCD with AB ≅ DC

Prove: AC ≅ DB A D

286 Lesson 6-6 Trapezoids and Kites hsm11gmse_0606_t06010 17. Prove the converse of Theorem 6-21: If the diagonals of a trapezoid are congruent, Proof then the trapezoid is isosceles. T P 18. Given: Isosceles trapezoid TRAP with TR ≅ PA Proof Prove: ∠RTA ≅ ∠APR 19. Prove that the angles formed by the noncongruent sides of a R A Proof kite are congruent. Determine whether each statement is true or false. Justify your response. 20. All squares are rectangles. 21. A trapezoid is a parallelogram. 22. A rhombus can be a kite. 23. Some parallelograms arehsm11gmse_0606_t06009 squares. 24. Every quadrilateral is a parallelogram. 25. All rhombuses are squares. 26. Select Tools to Solve Problems (1)(C) A wallpaper border pattern consists of isosceles trapezoids, each with two diagonals separating it into four triangles as shown. To investigate the trapezoids, choose from the following tools: protractor, ruler, compass, or graphing calculator. Explain your choice. Then observe any patterns. Make a conjecture about the triangles that are formed by the diagonals.

27. Given: Isosceles trapezoid TRAP with TR ≅ PA; T P Proof BI is the perpendicular bisector of RA, intersecting RA at B and TP at I. Prove: BI is the perpendicular bisector of TP. R A < > 28. BN is the perpendicular bisector of AC at N. Describe the set of points, D, for which ABCD is a kite. B For a trapezoid, consider the segment joining the midpoints of the two A N C given segments. How are its length and the lengths of thehsm11gmse_0606_t15810 two parallel sides of the trapezoid related? Justify your answer. 29. the two nonparallel sides 30. the diagonals

hsm11gmse_0606_t06011 TEXAS Test Practice

31. Which statement is never true? A. Square ABCD is a rhombus. C. Parallelogram PQRS is a square. B. Trapezoid GHJK is a parallelogram. D. Square WXYZ is a parallelogram. 32. A quadrilateral has four congruent sides. Which name best describes the figure? F. trapezoid H. rhombus D G G. parallelogram J. kite 33. Given DE is congruent to FG and EF is congruent to GD, prove ∠E ≅ ∠G. E F

PearsonTEXAS.com 287 hsm11gmse_0606_t06013 Topic 6 Review

TOPIC VOCABULARY

• base angles of a trapezoid, • equilateral polygon, p. 249 • midsegment of a trapezoid, • rectangle, p. 269 p. 281 • isosceles trapezoid, p. 281 p. 281 • regular polygon, p. 249 • bases of a trapezoid, p. 281 • kite, p. 282 • opposite angles, p. 255 • rhombus, p. 269 • consecutive angles, p. 255 • legs of a trapezoid, p. 281 • opposite sides, p. 255 • square, p. 269 • equiangular polygon, • parallelogram, p. 255 • trapezoid, p. 281 p. 249

Check Your Understanding Choose the vocabulary term that correctly completes the sentence. 1. A parallelogram with four congruent sides is a(n) ? . 2. A polygon with all angles congruent is a(n) ? . 3. Angles of a polygon that share a side are ? . 4. A(n) ? is a quadrilateral with exactly one pair of parallel sides.

6-1 The Polygon Angle-Sum Theorems

Quick Review Exercises The sum of the measures of the interior angles of an n-gon Find the measure of an interior angle and an exterior is (n - 2)180. The measure of one interior angle of a regular angle of each regular polygon. (n - 2)180 n-gon is n . The sum of the measures of the exterior 5. hexagon 6. 16-gon 7. pentagon angles of a polygon, one at each vertex, is 360. 8. What is the sum of the exterior angles for each polygon in Exercises 5–7? Example Find the measure of an interior angle of a regular 20‑gon. Find the measure of the missing angle. (n - 2)180 Measure = n Corollary to the Polygon Angle-Sum 9. xЊ 83Њ 10. 122Њ Theorem 89Њ (20 - 2)180 Substitute. zЊ = 20 79Њ 119Њ 18 # 180 = 20 Simplify. = 162 The measure of an interior angle is 162. hsm11gmse_06cr_t06359 hsm11gmse_06cr_t06357

288 Topic 6 Review 6-2 Properties of Parallelograms

Quick Review Exercises Opposite sides and opposite angles of a parallelogram Find the measures of the numbered angles for each are congruent. Consecutive angles in a parallelogram are parallelogram. supplementary. The diagonals of a parallelogram bisect 11. 12. 3 38Њ 1 2 each other. If three (or more) parallel lines cut off congruent 2 segments on one transversal, then they cut off congruent Њ segments on every transversal. 1 99 79Њ 3 Example Find the measures of the numbered angles in 13. hsm11gmse_06cr_t063613 1 14. 2 the parallelogram. 1 3 63Њ hsm11gmse_06cr_t06363 37Њ 2 2 3 1 56Њ Find the values of x and y in ▱ABCD. Since consecutive angles are supplementary, 15. AB = 2y, BC = y + 3, CD = 5x - 1, DA = 2x + 4 m∠1 = 180 - 56, or 124. Since opposite angles are hsm11gmse_06cr_t06365hsm11gmse_06cr_t06366 congruent, m∠2 = 56 and m∠3 = 124. 16. AB = 2y + 1, BC = y + 1, CD = 7x - 3, DA = 3x hsm11gmse_06cr_t06360

6-3 Proving That a Quadrilateral Is a Parallelogram

Quick Review Exercises A quadrilateral is a parallelogram if any one of the following Determine whether the quadrilateral must be a is true. parallelogram. • Both pairs of opposite sides are parallel. 17. 18. • Both pairs of opposite sides are congruent. • Consecutive angles are supplementary. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. Find the values of the variables for which ABCD must be a parallelogram. • One pair of opposite sides is both congruent and hsm11gmse_06cr_t06372 parallel. 19. Bhsm11gmse_06cr_t06370C 20. B C 4 (3y Ϫ 20)Њ x Ϫ 2 3x 3 Example (4y ϩ 4)Њ y Ϫ 1 y Ϫ 3 Must the quadrilateral be a parallelogram? 3 4xЊ (2x ϩ 6)Њ A D Yes, both pairs of opposite angles are A D congruent.

hsm11gmse_06cr_t06375 hsm11gmse_06cr_t06373 hsm11gmse_06cr_t06368

PearsonTEXAS.com 289 6-4 Properties of Rhombuses, Rectangles, and Squares

Quick Review Exercises A rhombus is a parallelogram with four congruent sides. Find the measures of the numbered angles in each A rectangle is a parallelogram with four right angles. special parallelogram. A square is a parallelogram with four congruent sides and 21. 22. 1 2 1 2 3 four right angles. 56Њ The diagonals of a rhombus are perpendicular. Each 3 diagonal bisects a pair of opposite angles. 32Њ The diagonals of a rectangle are congruent. Determine whether each statement is always, sometimes, or never true. hsm11gmse_06cr_t06381.ai Example 2 23. hsm11gmse_06cr_t06380.aiA rhombus is a square. What are the measures of the numbered 1 angles in the rhombus? 3 24. A square is a rectangle. Each diagonal of a rhombus 60Њ m∠1 = 60 25. A rhombus is a rectangle. bisects a pair of opposite angles. 26. The diagonals of a parallelogram are perpendicular. m∠2 = 90 The diagonals of a rhombus are #. 60 + m∠2 + m∠3 = 180 Triangle Angle-Sum Thm. 27. The diagonals of a parallelogram are congruent. hsm11gmse_06cr_t06379.ai 60 + 90 + m∠3 = 180 Substitute. 28. Opposite angles of a parallelogram are congruent. m∠3 = 30 Simplify.

6-5 Conditions for Rhombuses, Rectangles, and Squares

Quick Review Exercises If a quadrilateral is a parallelogram with a diagonal Can you conclude that the parallelogram is a rhombus, that bisects two angles of the parallelogram, then a rectangle, or a square? Explain. the quadrilateral is a rhombus. If a quadrilateral is 29. 30. a parallelogram with perpendicular diagonals, then the quadrilateral is a rhombus. If a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle.

For what value of x is the figure the given parallelogram? Example Justify your answer. Can you conclude that the parallelogram is a rhombus, 31. hsm11gmse_06cr_t06383.ai Rhombus 32. hsm11gmse_06cr_t06384.aiRectangle a rectangle, or a square? Explain. Yes, the diagonals are perpendicular, x Ϫ 1 ϩ 3 2x so the parallelogram is a rhombus. 2 2 (5x Ϫ 30)Њ (3x ϩ 6)Њ

hsm11gmse_06cr_t06382.aihsm11gmse_06cr_t06386.aihsm11gmse_06cr_t06385.ai 290 Topic 6 Review 6-6 Trapezoids and Kites

Quick Review Exercises The parallel sides of a trapezoid are its bases, and the Find the measures of the numbered angles in each nonparallel sides are its legs. Two angles that share a isosceles trapezoid. base of a trapezoid are base angles of the trapezoid. 33. 34. The midsegment of a trapezoid joins the midpoints 1 2 1 of its legs. 2 45Њ 3 3 The base angles of an isosceles trapezoid are congruent. 80Њ The diagonals of an isosceles trapezoid are congruent. The diagonals of a kite are perpendicular. Find the measures of the numbered angles in each kite. hsm11gmse_06cr_t06388.ai 35. 36. 34Њ 38Њ Example 1 1 B C ABCD is an isosceles trapezoid. 2 hsm11gmse_06cr_t06389.ai2 What is m∠C? 65Њ 60Њ Since BC } AD, ∠C and ∠D are A D same-side interior angles. 37. A trapezoid has base lengths of (6x - 1) units and Same-side interior angles are 3 units. Its midsegment has a length of (5x - 3) units. m C m D 180 hsm11gmse_06cr_t06391.ai ∠ + ∠ = supplementary. What is the value of x? m∠C + 60 = 180 Substitute. hsm11gmse_06cr_t06387.ai hsm11gmse_06cr_t06390.ai m∠C = 120 Subtract 60 from each side.

PearsonTEXAS.com 291 Topic 6 TEKS Cumulative Practice

Multiple Choice 5. FGHJ is a quadrilateral. If at least one pair of opposite angles in quadrilateral FGHJ is congruent, which Read each question. Then write the letter of the correct statement is false? answer on your paper. A. Quadrilateral FGHJ is a trapezoid. 1. Which list could represent the lengths of the sides of a triangle? B. Quadrilateral FGHJ is a rhombus. A. 7 cm, 10 cm, 25 cm C. Quadrilateral FGHJ is a kite. B. 4 in., 6 in., 10 in. D. Quadrilateral FGHJ is a parallelogram. C. 1 ft, 2 ft, 4 ft 6. For which value of x are lines g and h parallel? D. 3 m, 5 m, 7 m (2x ϩ 10)Њ g 2. Which quadrilateral CANNOT contain four right angles? (5x Ϫ 5)Њ F. square H. trapezoid h G. rhombus J. rectangle F. 12 H. 18 3. What is the circumcenter of △ABC with vertices G. 15 J. 25 A(-7, 0), B(-3, 8), and C(-3, 0)? 7. In △GHJ, GH ≅ HJ. Using the indirect proof method, A. (-7, -3) C. (-4, 3) you attempthsm11gmse_06cu_t06104 to derive a contradiction by .aiproving that B. (-5, 4) D. (-3, 4) ∠G and ∠J are right angles. Which theorem will contradict this claim? 4. ABCD is a rhombus. To prove that the diagonals of a rhombus are perpendicular, which pair of angles below A. Triangle Angle-Sum Theorem must you prove congruent by using corresponding B. Side-Angle-Side Theorem parts of congruent triangles? C. Converse of the Isosceles Triangle Theorem A B D. Angle-Angle-Side Theorem

E 8. Which quadrilateral must have congruent diagonals? F. kite G. rectangle D C H. parallelogram F. ∠AEB and ∠DEC J. rhombus G. ∠AEB and ∠AED H. ∠BEC and ∠AED hsm11gmse_06cu_t06102.ai J. ∠DAB and ∠ABC

292 Topic 6 TEKS Cumulative Practice 9. What values of x and y make the quadrilateral below a 14. The outer walls of the Pentagon parallelogram? in Arlington, Virginia, are formed by two regular pentagons, as xЊ y shown at the right. What is the ϩ 6 1 Ϫ 5x value of x?

6 Ϫ 2 y 3x Ϫ

4 Constructed Response 15. What are the possible values for n to make ABC a valid A. x = 2, y = 1 C. x = 1, y = 2 triangle? Show your work. hsm11gmse_06cu_t06110.ai 9 B. x = 3, y = 5 D. x = 2, y = 7 C 10. Whichhsm11gmse_06cu_t06105 is the most valid conclusion based .ai on the statements below? 1 2 ϩ n n If a triangle is equilateral, then it is isosceles. △ABC is not equilateral. A B 5n Ϫ 4 F. △ABC is not isosceles. 16. The pattern of a soccer G. △ABC is isosceles. ball contains regular H. △ABC may or may not be isosceles. hexagons and regular pentagons. The figure J. △ABC is equilateral. at the righthsm11gmse_06cu_t06112.ai shows what a section of the pattern Gridded Response would look like on a flat 11. What is m∠1 in the figure below? surface. Use the fact that there are 360° in a circle 31Њ to explain why there are gaps between the hexagons. 38Њ Does the information help you A B 1 prove that ABCD is a 69Њ parallelogram? Explain. hsm11gmse_06cu_t06113.ai 17. AC bisects BD. D C 12. ∠ABE and ∠CBD are vertical angles, and both are 18. AB ≅ DC , AB } DC complementary with ∠FGH. If m∠ABE = (3x - 1), and m∠FGH = 4x, what is m∠CBD? 19. AB ≅ DC , BC ≅ AD hsm11gmse_06ct_t05844.ai 13. What is thehsm11gmse_06cu_t06108 value of x in the kite below? .ai 20. ∠DAB ≅ ∠BCD , ∠ABC ≅ ∠CDA 21. CD has endpoints C(5, 7) and D(10, -5). What are the coordinates of the midpoint of CD? What is CD? Show your work.

22Њ xЊ

hsm11gmse_06cu_t06109.ai PearsonTEXAS.com 293