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Topic 6 Polygons and Quadrilaterals TOPIC OVERVIEW VOCABULARY 6-1 The Polygon Angle-Sum Theorems English/Spanish Vocabulary Audio Online: EnglishSpanish equiangular polygon, p. 249 polígono equiángulo equilateral polygon, p. 249 polígono equilátero isosceles trapezoid, p. 281 trapecio isósceles kite, p. 282cometa midsegment of a trapezoid, p. 281segmento medio de un trapecio parallelogram, p. 255paralelogramo rectangle, p. 269rectángulo regular polygon, p. 249 polígono regular rhombus, p. 269rombo square, p. 269cuadrado trapezoid, p. 281trapecio 6-2 Properties of Parallelograms 6-3 Proving That a Quadrilateral Is a Parallelogram 6-4 Properties of Rhombuses, Rectangles, and Squares 6-5 Conditions for Rhombuses, Rectangles, and Squares 6-6 Trapezoids and Kites DIGITAL APPS PRINT and eBook Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. Your Digital Resources PearsonTEXAS.com Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 246 Topic 6 Polygons and Quadrilaterals 3--Act Math The Mystery Sides Have you every looked closely at honeycombs? What shape are they? How do you know? Most often the cells in the honeycombs look like hexagons, but they might also look like circles. Scientists now believe that the bees make circular cells that become hexagonal due to the bees’ body heat and natural physical forces. What are some strategies you use to identify shapes? Think about this as you watch the 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support. Learning Animations You can also access all of the stepped-out learning animations that you studied in class. Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device. Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 247 Technology Lab Use With Lesson 6-1 Exterior Angles of Polygons 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 2 4 4 5 3 1 2 4 5 3 2 1 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 gaps or overlaps. hsm11gmse_0601a_t06179 hsm11gmse_0601a_t06180 hsm11gmse_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. 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 hsm11gmse_0601a_t06182 with another regular polygon. 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 geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. •Equiangular polygon – An equiangular TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. a polygon that is both equilateral and equiangular. polygon is a polygon with all angles congruent. •Equilateral polygon – An equilateral polygon is a polygon with all sides congruent. •Regular polygon – A regular polygon is •Number sense – the understanding of what Additional TEKS (1)(E), (1)(F) 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 is a polygon with all sides congruent. An equiangular polygon is a polygon with all angles congruent. A regular polygon is a polygon that is both equilateral and equiangular. Theorem 6-1 Polygon Angle-Sum Theorem hsm11gmse_0601_t06300 hsm11gmse_0601_t06299 The sum of the measures of the interior angles of an n-gon is (n - 2)180. hsm11gmse_0601_t06301 For a proof of Theorem 6-1, see the Reference section on page 683. Corollary to the Polygon Angle-Sum Theorem The measure of each interior angle of a regular n-gon is (n - 2)180 . 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 each vertex, is 360. For the pentagon, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360. 3 2 4 1 5 You will prove Theorem 6-2 in Exercise 9. hsm11gmse_0601_t06313.ai TEKS Process Standard (1)(C) Problem 1 Investigating Interior Angles of Polygons Choose from among a variety of tools (such as a ruler, a compass, or geometry A 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. How can recording data in a table help you make a conjecture? Recording data in a table is an organized way to present and analyze information. You can look for patterns in the data and make a conjecture. 250 Sum of Interior Angle Measures Polygon Triangle 1 Sum of Interior Angle Measures 180 Polygon Pentagon 1 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 3 720 540 C Use the data in the table in part B to make a conjecture about the sum of the measures of the interior angles of a polygon. Notice that the numbers in the table are all multiples of 180. Look at the patterns: # 180 = 180Pentagon3# 180 = 540 Quadrilateral2 # 180 = 360 Hexagon 4 # 180 = 720 Triangle 1 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. Lesson 6-1 The Polygon Angle-Sum Theorems Problem 2 Finding a Polygon Angle Sum How many sides does a heptagon have? A heptagon has 7 sides. What is the sum of the interior angle measures of a heptagon? Sum = (n - 2)180 = (7 - 2)180 =5 = 900 # 180 Polygon Angle-Sum Theorem Substitute 7 for n. Simplify. The sum of the interior angle measures of a heptagon is 900. Problem 3 Using the Polygon Angle-Sum Theorem How does the word regular help you answer the question? The word regular tells you that each angle has the same measure. STEM Biology The common housefly, Musca domestica, has eyes that consist of approximately 4000 facets. Each facet is a regular hexagon. What is the measure of each interior angle in one hexagonal facet? (n - 2)180 n (6 - 2)180 = 6 4 180 = 6 = 120 Measure of an angle = # Corollary to the Polygon Angle-Sum Theorem Substitute 6 for n. Simplify. The measure of each interior angle in one hexagonal facet is 120. PearsonTEXAS.com 251 Problem 4 Using the Polygon Angle-Sum Theorem How does the diagram help you? You know the number of sides and four of the five angle measures. What is mjY in pentagon TODAY? T Use the Polygon Angle-Sum Theorem for n = 5. m∠T + m∠O + m∠D + m∠A + m∠Y = (5 - 2)180 110 + 90 + 120 + 150 + m∠Y = 3 # 180 470 + m∠Y = 540 110 O 120 150 Substitute. D Simplify. A m∠Y = 70 Subtract 470 from each side. hsm11gmse_0601_t06302 Problem 5 Investigating Exterior Angles of Polygons Choose from a variety of tools (such as a ruler, a protractor, or a graphing A 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 investigate patterns? If you draw a triangle, a quadrilateral, and a pentagon, you can investigate patterns for different numbers of exterior angles in each polygon. Step 1Draw three different polygons. Then draw the exterior angles at each vertex of the polygons as shown. 120° 135° 58° 63° 90° 79° 88° 105° 64° 130° 58° 90° Step 2Use 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 Y 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? By the Polygon Exterior Angle-Sum Theorem, the sum of the exterior angle measures is 360. Since the octagon is regular, the interior angles are congruent. So their supplements, the exterior angles, are also congruent. What kind of angle is j1? Looking at the diagram, you know that ∠1 is an exterior angle. 360 m∠1 = 8 NLINE HO ME RK O = 45 WO 7 6 2 Divide 360 by 8, the number of sides in an octagon. 3 Simplify. PRACTICE and APPLICATION EXERCISES 8 1 5 4 Scan page for a Virtual Nerd™ tutorial video. 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. y 110 z 100 87 7. z x (z 13) w y (z 10) 8. 3x 2x 4x x 9.a. A polygon has n sides. An interior angle of the polygon and an adjacent exterior form a straight angle. What is the sum of the measures of the hsm 11gm angle se_0601_t06058.ai hsm 11gm se_0601_t06060.ai n straight angles? Of the nhsm interior 11gmangles? se_0601_t06059.ai 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? hsm 11gm se_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) and 180 - 360 n 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 Shr ubs 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? Maple Street C A 18.The Public Garden is located between two parallel streets: 64 Maple Street and Oak Street. The garden faces Maple Street Public Sh and is bordered by rows of shrubs that intersect Oak Street at Garden rub s point B. What is m∠ABC, the angle formed by the shrubs? 19.△ABC ≅ △DEF . If m∠A = 3x + 4, m∠C = 2x, and m∠E = 4x + 5, what is m∠B? 254 Lesson 6-1 The Polygon Angle-Sum Theorems 37 Oak Street B hsm11gmse_0601_t12841.ai 6-2 Properties of Parallelograms TEKS FOCUS VOCABULARY TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. •Consecutive angles – Consecutive angles of a polygon TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. •Opposite sides – Opposite sides of a quadrilateral are Additional TEKS (1)(G) •Parallelogram – A parallelogram is a quadrilateral with share a common side. •Opposite angles – Opposite angles of a quadrilateral are two angles that do not share a side. two sides that do not share a vertex. 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 share a vertex and opposite angles do not share a side. B C AB and CD hsm11gmse_0602_t06469.ai are opposite sides. A D Angles of a polygon that share a side are consecutive angles. In the diagram, ∠A and ∠B are consecutive angles because they share side AB. A A and C are opposite angles. B B and C are also hsm11gmse_0602_t06471.ai consecutive angles. C D hsm11gmse_0602_t06477.ai PearsonTEXAS.com 255 Theorem 6-3 Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. If . . . ABCD is a ▱ B C Then . . . AB ≅ CD and BC ≅ DA B C D A A D For a proof of Theorem 6-3, see the Reference section on page 683. Theorem 6-4 Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. hsm11gmse_0603_t06433.ai hsm11gmse_0602_t06473.ai If . . . ABCD is a ▱ B C Then . . . B C m∠A + m∠B = 180 m∠B + m∠C = 180 D A m∠C + m∠D = 180 m∠D + m∠A = 180 D A You will prove Theorem 6-4 in Exercise 21. Theorem 6-5 Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. hsm11gmse_0603_t06432.ai hsm11gmse_0603_t06433.ai If . . . Then . . . ABCD is a ▱. ∠A ≅ ∠C and ∠B ≅ ∠D B B C C A D A D For a proof of Theorem 6-5, see Problem 2. Theorem 6-6 Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. hsm11gmse_0603_t06434.ai hsm11gmse_0602_t06487.ai If ... Then . . . ABCD is a ▱ AE ≅ CE and BE ≅ DE B C B C D A A D E You will prove Theorem 6-6 in Exercise 11. Theorem 6-7 Theorem If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. hsm11gmse_0603_t06433.ai If < .> . .< > < > AB } CD } EF and AC ≅ CE A C E hsm11gmse_0603_t06443.ai Then . . . BD ≅ DF A B C D F E B D F You will prove Theorem 6-7 in Exercise 23. 256 Lesson 6-2 Properties of Parallelograms hsm11gmse_0602_t06483.aihsm11gmse_0602_t06484.ai Problem 1 Using Consecutive Angles Q Multiple Choice What is mjP in ▱PQRS? What information from the diagram helps you get started? From the diagram, you know m∠PSR and that ∠P and ∠PSR are consecutive angles. So you can write an equation and solve for m∠P. 26 116 64 126 P R 64 m∠P + m∠S = 180 Consecutive angles of a ▱ are supplementary. m∠P + 64 = 180 S Substitute. m∠P = 116 Subtract 64 from each side. The correct answer is C. Problem 2 Proof TEKS Process Standard (1)(G) Using Properties of Parallelograms in a Proof Given: ▱ABCD B C Prove: ∠A ≅ ∠C and ∠B ≅ ∠D A D ABCD is a ▱. Given Why is a flow proof useful here? A flow proof allows you to see how the pairing of two statements leads to a conclusion. hsm11gmse_0602_t06478.ai ∠A and ∠B are consecutive ⦞. ∠B and ∠C are consecutive ⦞. ∠C and ∠D are consecutive ⦞. Def. of consecutive ⦞ Def. of consecutive ⦞ Def. of consecutive ⦞ ∠A and ∠B are supplementary. ∠B and ∠C are supplementary. ∠C and ∠D are supplementary. Consecutive ⦞ are supplementary. Consecutive ⦞ Consecutive ⦞ are supplementary. are supplementary. ∠A ≅ ∠C ∠B ≅ ∠D Supplements of the same ∠ are ≅. Supplements of the same ∠ are ≅. hsm11gmse_0602_t06480.ai PearsonTEXAS.com 257 Problem 3 Using Algebra to Find Lengths L Solve a system of linear equations to find the values of x and y in ▱KLMN . What are KM and LN? y x 10 K The diagonals of a parallelogram bisect each other. M 8 2x y P 2 N hsm11gmse_0602_t06481.ai KP ≅ M P LP ≅ N P ① y + 10 = 2x − 8 ② x=y+2 Set up a system of linear equations by substituting the algebraic expressions for each segment length. Substitute ( y + 2) for x in equation ①. Then solve for y. y + 10 = 2(y + 2) − 8 y + 10 = 2y + 4 − 8 y + 10 = 2y − 4 10 = y − 4 14 = y Substitute 14 for y in equation ②. Then solve for x. x = 14 + 2 = 16 Use the values of x and y to find KM and LN. KM = 2(KP) = 2(y + 10) = 2(14 + 10) = 48 LN = 2(LP) = 2(x) = 2(16) = 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 you need? You know the length of EF. To find EH, you need the lengths of FG and GH. AB = BC = CD = 2, and EF = 2.25. What is EH? Since } lines divide AD into equal parts, they also divide EH into equal parts. EF = FG = GH EH = EF + FG + GH EH = 2.25 + 2.25 + 2.25 = 6.75 Segment Addition Postulate A B C D E F G H Substitute. hsm11gmse_0602_t06485.ai 258 Lesson 6-2 Properties of Parallelograms HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. 1. What are the values of x and y in the parallelogram? For additional support when completing your homework, go to PearsonTEXAS.com. y 2. The perimeter of ▱ABCD is 92 cm. AD is 7 cm more than twice AB. Find the lengths of all four sides of ▱ABCD. In the figure, PQ = QR = RS. Find each length. 3. ZU 4.XZ 5. TU 6.XV 7. YX 8.YV 9. WX 10.WV 3y W 3x S U 2.25 Y hsm11gmse_0602_t06078.ai Z R 3 X T 11.Justify Mathematical Arguments (1)(G) Proof Complete this two-column proof of Theorem 6-6. Q P V B Given: ▱ABCD 2 4 C E 1 3 hsm11gmse_0602_t06077.ai A D Reasons Prove: AC and BD bisect each other at E. Statements 1) ABCD is a parallelogram. 1) Given 2) AB } DC 2) a. ? 3) ∠1 ≅ ∠4; ∠2 ≅ ∠3 3) b. ? 4) AB ≅ DC 4) c. ? 5) d. 5) ASA ? hsm11gmse_0602_t06075.ai 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 T 13.PT = x + 2, TR = y, QT = 2x, TS = y + 3 P 14.PT = y, TR = x + 3, QT = 2y, TS = 3x - 1 Use the diagram at the right for each proof. S S Y T 15.Given: ▱RSTW and ▱XYTZ Proof hsm11gmse_0602_t06076.ai X Z R W Prove: ∠R ≅ ∠X 16.Given: ▱RSTW and ▱XYTZ Proof Prove: XY } RS Find the measures of the numbered angles for each parallelogram. 17.18. 19. 28 3 3 1 38 2 81 110 1 2 hsm11gmse_0602_t06087.ai 3 85 1 48 2 PearsonTEXAS.com 259 hsm11gmse_0602_t06132.ai 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. a.If DE = 2.5 ft, what is FG? b.If m∠E = 129, what is m∠G? c.What happens to m∠D as m∠E increases or decreases? Explain. 21.Prove Theorem 6-4. B Proof E D F G C Given: ▱ABCD Prove: ∠A is supplementary to ∠B. ∠A is supplementary to ∠D. A 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.ai right. B A < > < > < > Proof Given: AB } CD } EF , AC ≅ CE 3 D C 1 Prove: BD ≅ DF G 2 6 F E 4 24.Explain Mathematical Ideas (1)(G) Explain how to separate H 5 a blank card into three strips that are the same height by using lined paper, a straightedge, and Theorem 6-7. TEXAS Test Practice hsm11gmse_0602_t06136.ai P 25.PQRS is a parallelogram with m∠Q = 4x and m∠R = x + 10. Which statement explains why you can use the equation 4x + (x + 10) = 180 to solve for x? Q 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 26.In the figure of DEFG at the right, DE } GF . Which statement must be true? F. m∠D + m∠E = 180 H.DE ≅ GF G.m∠D + m∠G = 180 J.DG ≅ EF R D E hsm11gmse_0602_t12794 G F 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 D.not enough information hsm11gmse_0602_t06139.ai 28.Find the measure of one exterior angle of a regular hexagon. Explain your method. 260 B.9 cm Lesson 6-2 Properties of Parallelograms C.12 cm 6-3 Proving That a Quadrilateral Is a Parallelogram TEKS FOCUS VOCABULARY TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. •Analyze – closely examine objects, ideas, or relationships to learn more about their 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 . . . If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B A Then . . . C ABCD is a ▱ B C AB ≅ CD BC ≅ DA D A D For a proof of Theorem 6-8, see Problem 1. Theorem 6-9 Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. hsm11gmse_0602_t06473.ai If . . . B A C D m∠A + m∠B = 180 m∠A + m∠D = 180 hsm11gmse_0603_t06433.ai Then . . . ABCD is a ▱ B C A D You will prove Theorem 6-9 in Exercise 17. Theorem 6-10 Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. hsm11gmse_0603_t06432.ai If . . . B A C ∠A ≅ ∠C ∠B ≅ ∠D D hsm11gmse_0603_t06433.ai Then . . . ABCD is a ▱ B C 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 the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If . . . B A E Then . . . C D ABCD is a ▱ B C AE ≅ CE BE ≅ DE A D For a proof of Theorem 6-11, see Problem 3. hsm11gmse_0603_t06443.ai Theorem 6-12 Theorem If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. hsm11gmse_0603_t06433.ai If . . . Then . . . B A ABCD is a ▱ B C C D BC ≅ DA BC } DA A D You will prove Theorem 6-12 in Exercise 16. hsm11gmse_0603_t06448.ai hsm11gmse_0603_t06433.ai Concept Summary P roving That a Quadrilateral Is a Parallelogram Method Source Prove that both pairs of opposite sides are parallel. Definition of parallelogram Prove that both pairs of opposite sides are congruent. Theorem 6-8 Diagram hsm11gmse_0603_t06461.ai Prove that an angle is supplementary to both of its consecutive angles. Theorem 6-9 75 75 105 hsm11gmse_0603_t06463.ai Prove that both pairs of opposite angles are congruent. Theorem 6-10 hsm11gmse_0603_t12047 Prove that the diagonals bisect each other. Theorem 6-11 hsm11gmse_0603_t06465.ai Prove that one pair of opposite sides is congruent and parallel. 262 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram Theorem 6-12 hsm11gmse_0603_t06467.ai hsm11gmse_0603_t06468.ai Problem 1 B Proof Proving Theorem 6-8 C Given: AB ≅ CD, BC ≅ DA Prove: ABCD is a parallelogram. Why do you start by drawing BD? In many proofs about parallelograms, it is convenient to have a pair of triangles that you can show to be congruent. Draw a diagonal to form two triangles. Statements A D Reasons 1) Draw BD. 1) Construction 2) AB ≅ CD and BC ≅ DA 2) Given hsm11gmse_0603_t06430.ai 3) BD ≅ BD 3) Reflexive Property of Congruence 4) △ABD ≅ △CDB 4) SSS 5) ∠ADB ≅ ∠CBD and ∠CDB ≅ ∠ABD 5) Corresponding parts of congruent 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) B Proof Proving Theorem 6-10 How do you get started with the proof? Since the goal is to show that opposite sides are parallel, you can label the angle measures as in the diagram and show that same-side interior angles are supplementary. x Given: ∠A ≅ ∠C, ∠B ≅ ∠D Prove: ABCD is a parallelogram. Statements C A y D Reasons 1) ∠A ≅ ∠C, ∠B ≅ ∠D 1) Given 2) x + y + x + y = 360 2) The sum of the measures of the hsm11gmse_0603_t06158.ai angles of a quadrilateral is 360. 3) 2(x + y) = 360 3) Distributive Property 4) x + y = 180 4) Division Property of Equality 5) ∠A and ∠B are supplementary. ∠A and ∠D are supplementary. 5) Definition of supplementary angles 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 How can you get started? Notice that in the diagram there are several pairs of triangles. Use the given information to prove pairs of triangles congruent. Then use their corresponding parts to show that ABCD is a parallelogram. B C Given: AC and BD bisect each other at E. E Prove: ABCD is a parallelogram. A D AC and BD bisect each other at E. Given hsm11gmse_0603_t06445.ai ∠AEB ≅ ∠CED AE ≅ CE BE ≅ DE ∠BEC ≅ ∠DEA 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 CD BC AD If alternate interior ⦞ ≅, then lines are . If alternate interior ⦞ ≅, then lines are . ABCD is a parallelogram. Def. of parallelogram Problem 4 Finding Values for Parallelograms Which theorem should you use? The diagram gives you information about sides. Use Theorem 6-8 because it uses sides to conclude that a quadrilateral is a parallelogram. A For what value of y must PQRS be a parallelogram? 3x 5 P Q Step 1Find x. If opp. sides are ≅ , then the quad. 3x - 5 = 2x + 1 is a ▱ . x-5=1 x=6 Subtract 2x from each side. Add 5 to each side. Step 2Find y. y x2 S 2x 1 R 8 . . . . . . . 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 hsm11gmse_0603_t06439.ai y = x + 2 If opp. sides are ≅ , then the quad. is a ▱ . = 6 + 2 Substitute 6 for x. =8 Simplify. For PQRS to be a parallelogram, the value of y must be 8. continued on next page ▶ 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 A Step 1Find w. 5w - 30 = 3w + 10 If opp. angles are ≅, then the quad. is a ▱ . 2w - 30 = 10 2w = 40 Add 30 to each side. w = 20 Divide each side by 2. Subtract 3w from each side. (5w 2 30)8 (3w 1 10)8 D B C (8z 2 10)8 Step 2Find 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 if you have enough information? If you can satisfy every condition of a theorem about parallelograms, then you have enough information. Can you prove that the quadrilateral is a parallelogram based on the given information? If so, write a paragraph proof. If not, explain. A Given: AB = 5, CD = 5, B Given: HI ≅ HK, JI ≅ JK m∠A = 50, m∠D = 130 Prove: HIJK is a parallelogram. Prove: ABCD is a parallelogram. A 5 H I B 50 130 D 5 C Yes. Proof: Because it is given that m∠A = 50 and m∠D = 130, same-side interior angles A and D hsm11gmse_0603_t06451.ai are supplementary. So AB } CD. It is given that AB = 5 and CD = 5, so AB ≅ CD. Therefore, ABCD is a parallelogram by Theorem 6-12. K J No. By Theorem 6-8, you need to show that both pairs of opposite sides, not consecutive sides, are hsm11gmse_0603_t06453.ai congruent. continued on next page ▶ 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. M Prove: DEFG is a parallelogram. N D 398 398 P Q E H 1418 G Yes. Proof: It is given that m∠N = m∠Q = 39 and m∠P = 141. Since the sum of the angle measures of a quadrilateral is 360, m∠M = 141. Since m∠N = m∠Q and m∠P = m∠M, ∠N ≅ ∠Q and ∠M ≅ ∠P. Therefore, MNPQ is a parallelogram by Theorem 6-10. 16 12 F Yes. Proof: It is given that GE = 24, GH = 12, DF = 32, and HF = 16. By the Segment Addition Postulate, HE = 12 and DH = 16, so the diagonals of the quadrilateral bisect each other. DEFG is a parallelogram by Theorem 6-11. Problem 6 Identifying Parallelograms As the arms of the lift move, what changes and what stays the 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. Vehicle Lifts A truck sits on the platform of a vehicle lift. Two moving arms raise the platform until a mechanic can fit underneath. Why will the truck always remain parallel to the ground as it is lifted? Explain. Q Q R R 26 ft 6 ft P 26 ft 6 ft 6 ft 26 ft S 6 ft P 26 ft 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 S HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. 1.Given: AB ≅ CD, DE ≅ FC, EA ≅ BF 2.Given: ∠M ≅ ∠P, ∠MNQ ≅ ∠PQN, Proof Proof ∠MQN ≅ ∠PNQ Prove: ABCD is a parallelogram. For additional support when completing your homework, go to PearsonTEXAS.com. A Prove: MNPQ is a parallelogram. B M E N 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 AD ≅ CB Prove: HJKL is a parallelogram. H Prove: ABCD is a parallelogram. J M A L B K D C Analyze Mathematical Relationships (1)(F) For what values of x and y must ABCD be a parallelogram? 5. B (y 78) 3y 2x (4x 21) A 8. A 6. C 3x D D 2y 1 4 y A A 2y 2 B 5y D D 3y 9 B C (2x 15) hsm11gmse_0603_t06152.ai (4x 33) 6 3x y4 hsm11gmse_0603_t06149.ai C C 2x 7 10. B (3x 10) hsm11gmse_0603_t06150.ai (8x 5) 5x 8 A B 9. B 7. D C 7 A C D 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. hsm11gmse_0603_t06154.ai 13. hsm11gmse_0603_t06155.ai 14. PearsonTEXAS.com 267 hsm11gmse_0603_t06157.ai 15.Apply Mathematics (1)(A) Quadrilaterals are 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. A B D C 16.Justify Mathematical 17.Prove Theorem 6-9. Proof Arguments (1)(G) Prove Proof 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 A TEXAS Test Practice B C C A D D hsm11gmse_0603_t06581.ai hsm11gmse_0603_t06159.ai 18.Which piece of additional information would allow you to prove that PQRS is a parallelogram? A.PQ ≅ RS C.∠PTQ ≅ ∠RTS B.QR ≅ SP D.∠QPR ≅ ∠SRP P Q T 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 G.34 H.28 J. -12 20.Quadrilateral JKLM is a parallelogram. Which of the following does NOT guarantee that JNPM is a parallelogram? N J K A.N is the midpoint of JK and P is the midpoint of ML. B.JM ≅ NP M L P C.JM } NP D.∠JMP ≅ ∠NPL 21.Write a proof using the diagram. N Prove: JNTC is a parallelogram. 268 Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram T P Given: △NRJ ≅ △CPT, JN } CT J R C hsm11gmse_0603_t06167.ai 6-4 Properties of Rhombuses, Rectangles, and Squares TEKS FOCUS VOCABULARY •Rectangle – A rectangle is a parallelogram with TEKS (5)(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. four right angles. •Rhombus – A rhombus is a parallelogram with four congruent sides. •Square – A square is a parallelogram with four congruent sides and four right angles. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. •Number sense – the understanding of what numbers mean and how they are related 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 with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. hsm11gmse_0604_t06019 Theorem 6-13 hsm11gmse_0604_t06018 hsm11gmse_0604_t06020 Theorem If a parallelogram is a rhombus, then its diagonals are perpendicular. If . . . ABCD is a rhombus A D Then . . . AC # BD B B C A D C For a proof of Theorem 6-13, see Lesson 7-3. hsm11gmse_0604_t06022 PearsonTEXAS.com 269 hsm11gmse_0604_t06023 Theorem 6-14 Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If . . . ABCD is a rhombus A D B C Then . . . A D 2 3 1 4 7 5 8 6 B C ∠1 ≅ ∠2 ∠3 ≅ ∠4 ∠5 ≅ ∠6 ∠7 ≅ ∠8 You will prove Theorem 6-14 in Exercise 10. hsm11gmse_0604_t06024 Theorem Theorem If a parallelogram is a rectangle, then its diagonals are congruent. 6-15 hsm11gmse_0604_t06022 If . . . ABCD is a rectangle D A B C Then . . . AC ≅ BD A D B C You will prove Theorem 6-15 in Exercise 13. Problem 1 hsm11gmse_0604_t06028 hsm11gmse_0604_t06031 Classifying Special Parallelograms How do you decide whether ABCD is a rhombus, a rectangle, or a square? Use the definitions of rhombus, rectangle, and square along with the markings on the figure. 270 Is ▱ABCD a rhombus, a rectangle, or a square? Explain. ▱ABCD is a rectangle. Opposite angles of a parallelogram are congruent, so m∠D is 90. By the Same-Side Interior Angles Theorem, m∠A = 90 and m∠C = 90. Since ▱ABCD has four right angles, it is a rectangle. You cannot conclude that ABCD is a square because you do not know its side lengths. Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares A B E F H D G C Problem 2 TEKS Process Standards (1)(C) Investigating Diagonals of Quadrilaterals Choose from a variety of tools (such as a protractor, a ruler, a compass, A 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. Make several parallelograms, rectangles, and rhombuses. Then make a B conjecture about the diagonals of each type of quadrilateral. Parallelogram Rectangle Rhombus Conjecture: The diagonals of a parallelogram bisect each other. Conjecture: The diagonals of a rectangle are congruent. Conjecture: The diagonals of a rhombus are perpendicular. How can you measure distances on a geoboard? You can use the grid of pegs to indicate horizontal and vertical units. Problem 3 How are the numbered angles formed? The angles are formed by diagonals. Use what you know about the diagonals of a rhombus to find the angle measures. TEKS Process Standard (1)(F) Finding Angle Measures What are the measures of the numbered angles in rhombus ABCD? m∠1 = 90 The diagonals of a rhombus are #. m∠2 = 58 Alternate Interior Angles Theorem m∠3 = 58 m∠1 + m∠3 + m∠4 = 180 90 + 58 + m∠4 = 180 148 + m∠4 = 180 m∠4 = 32 E ach diagonal of a rhombus bisects a pair of opposite angles. B C 58 A 4 1 2 3 D Triangle Angle-Sum Theorem Substitute. Simplify. hsm11gmse_0604_t06026 Subtract 148 from each side. PearsonTEXAS.com 271 Problem 4 Finding Diagonal Length Multiple Choice In rectangle RSBF, SF = 2x + 15 and RB = 5x − 12. What is the length of a diagonal? How can you find the length of a diagonal? Since RSBF is a rectangle and its diagonals are congruent, use the expressions to write an equation. 1 9 18 You know that the diagonals of a rectangle are congruent, so their lengths are equal. RK O HO WO R F 33 hsm11gmse_0604_t06032 2x + 15 = 5x − 12 15 = 3x − 12 27 = 3x 9=x RB = 5x − 12 = 5(9) − 12 = 33 The correct answer is D. Substitute 9 for x in the expression for RB. ME B SF = RB Set the algebraic expressions for SF and RB equal to each other and find the value of x. NLINE S PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Find the measures of the numbered angles in each rhombus. 1. For additional support when completing your homework, go to PearsonTEXAS.com. 3 2. 4 2 35 1 3 3. 2 1 3 1 2 35 60 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 hsm 11gm se_0604_t05921.aihsm 11gm se_0604_t05935.ai hsm 11gm se_0604_t05920.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 A 10.Prove Theorem 6-14. Proof 3 Given: ABCD is a rhombus. Prove: AC bisects ∠BAD and ∠BCD. B 272 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares 4 D 2 1 C 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 Proof proof of Theorem 6-15. A D Given: ABCD is a rectangle. B C Prove: AC ≅ BD ABCD is a ▱. e. b. ABCD is a rectangle. Opposite sides hsmof11gm se_0604_t06239.ai a ▱ are ≅. f. BC ≅ BC a. AC ≅ BD SAS c. ∠ABC and ∠DCB are right ⦞. h. ∠ABC ≅ ∠DCB 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.ai K 4b 6r J 15. Analyze Mathematical Relationships (1)(F) Find the angle measures and the side lengths of the rhombus at the right. 16.Create Representations to Communicate Mathematical Ideas (1)(E) On graph paper, draw a parallelogram that is neither a rectangle nor a rhombus. r1 H x 2r 4 G b3 (2x 6) ABCD is a rectangle. Find the length of each diagonal. 17.AC = 2(x - 3) and BD = x + 5 3y 19.AC = 5 and BD = 3y - 4 18. AC = 2(5a + 1) and BD = 2(a + 1) 3c 20. AC = 9 and BD = 4 - c hsm11gmse_0604_t06254.ai PearsonTEXAS.com 273 Find the values of the variables. Then find the side lengths. 21.rhombus 22.square 15 3y 2x 7 y1 5x 4x 3 2y 5 3y 9 23.Justify Mathematical Arguments (1)(G) Write a proof. P L Proof Given: Rectangle PLAN hsm 11gm se_0604_t05942.ai Prove: △LTP ≅ △NTA T hsm 11gm se_0604_t05943.ai 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. 2 (7x 10) (6x 2 3x) 26. (2x 2 25x) (3x 2 60) TEXAS Test Practice hsm 11gm se_0604_t05944.ai hsm 11gm se_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? A.18 C.72 B.36 D.108 1 28.Which statement is true for some, but not all, rectangles? F. Opposite sides are parallel. G.It is a parallelogram. H.Adjacent sides are perpendicular. J.All sides are congruent. 29.Which term best describes AD in △ABC? A.altitude C.median B.angle bisector D.perpendicular bisector A hsm11gmse_0604_t12846 B D C 30.Write the first step of an indirect proof that △PQR is not a right triangle. hsm 11gm se_0604_t05947.ai 274 Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares 6-5 Conditions for Rhombuses, Rectangles, and Squares TEKS FOCUS VOCABULARY •Analyze – closely examine objects, ideas, or relationships TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. to learn more about their nature 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 a quadrilateral is a parallelogram with perpendicular diagonals, then the quadrilateral is a rhombus. If . . . ABCD is a ▱ and AC # BD A D Then . . . ABCD is a rhombus A D B B C C For a proof of Theorem 6-16, see Problem 1. hsm11gmse_0605_t06034.ai Theorem 6-17 Theorem If a quadrilateral is a parallelogram with a diagonal that bisects a pair of opposite angles, then the quadrilateral is a rhombus. If . . . ABCD is a ▱, ∠1 ≅ ∠2, and ∠3 ≅ ∠4 A D 3 4 1 B hsm11gmse_0605_t06274.ai Then . . . ABCD is a rhombus A D B 2 C 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 a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle. If . . . ABCD is a ▱, and AC ≅ BD A D C B Then . . . ABCD is a rectangle D A C B You will prove Theorem 6-18 in Exercise 17. hsm11gmse_0605_t06037.ai Theorem 6-19 Theorem If a quadrilateral is a parallelogram with perpendicular, congruent diagonals, then the quadrilateral is a square. If . . . ABCD is a ▱, AC # BD, and AC ≅ BD A D hsm11gmse_0604_t06028 Then . . . ABCD is a square A D B C E B C For a proof of Theorem 6-19, see Problem 2. Problem 1 TEKS Process Standard (1)(G) Proof Proving Theorem 6-16 How can knowing that the quadrilateral is a parallelogram help you prove the theorem? You can use any of the properties of parallelograms to help you. 276 A D Given: ABCD is a parallelogram, AC # BD Prove: ABCD is a rhombus. Since ABCD is a parallelogram, AC and BD bisect each other, so BE ≅ DE. Since AC # BD, ∠AED and ∠AEB are congruent right angles. By the Reflexive Property of Congruence, AE ≅ AE. So △AEB ≅ △AED by SAS. Corresponding parts of congruent triangles are congruent, so AB ≅ AD. Since opposite sides of a parallelogram are congruent, AB ≅ DC ≅ BC ≅ AD. By definition, ABCD is a rhombus. Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares E B C hsm11gmse_0605_t06035.ai Problem 2 Proof Proving Theorem 6-19 How can knowing the figure is a rectangle help you prove it is a square? A rectangle has four 90° angles. If you know the figure is a rectangle, you only need to show all sides are congruent to prove it is a square. A Write a two-column proof to prove Theorem 6-19. Given: ABCD is a parallelogram, AC # DB, and AC ≅ DB Prove: ABCD is a square. D E B Statements C Reasons 1) ABCD is a parallelogram, AC # DB, and AC ≅ DB 1) Given 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 How do you get started? Use the properties of rhombuses, rectangles, and squares and the theorems you learned to help you determine whether each figure is a rhombus, a rectangle, or a square. Can you conclude that quadrilateral ABCD is a rhombus, a rectangle, or a square? If so, write a paragraph proof. If not, explain. Given: Quadrilateral ABCD with A Given: Quadrilateral ABCD with B AE ≅ BE ≅ CE ≅ DE AE ≅ CE, BE ≅ DE Prove: ABCD is a rhombus, a rectangle, or a square. A D Prove: ABCD is a rhombus, a rectangle, or a square. A E B D E C Yes. Proof: It is given that AE ≅ BE ≅ CE ≅ DE in quadrilateral ABCD. By the definition of segment bisector, AC and DB bisect each other. By Theorem 6-11, ABCD is a parallelogram. By the definition of congruent segments, AE = BE = CE = DE. By the Segment Addition Postulate, AC = DB. So AC ≅ DB by the definition of congruent segments. Therefore, by Theorem 6-18, ABCD is a rectangle. B C No. The diagonals bisect each other, so by Theorem 6-11, quadrilateral ABCD is a parallelogram. The diagonals are not perpendicular, so ABCD is not a rhombus or a square. The diagonals are not congruent, so ABCD is not a rectangle or a square. PearsonTEXAS.com 277 Problem 4 Using Properties of Special Parallelograms A Algebra For what value of x is ▱ABCD a rhombus? D (6x 2) For ▱ABCD to be a rhombus, its diagonals must bisect a pair of opposite angles. Set the expressions for m∠ABD and m∠CBD equal to each other. Solve for x. m∠ABD = m∠CBD 6x - 2 = 4x + 8 B (4x 8) hsm11gmse_0605_t06040.ai 2x - 2 = 8 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. How can you use properties of diagonals to locate the A 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. If a quadrilateral is both a rectangle and a rhombus, why is it a square? If a quadrilateral is a rectangle, then its diagonals are congruent bisectors. If it is a rhombus, then its diagonals are perpendicular bisectors. So, by Theorem 6-19, the quadrilateral is a square. 278 • Theorem 6-18: If a quadrilateral is a parallelogram with congruent diagonals, then the quadrilateral is a rectangle. Step 1Cut two pieces of rope that will be the diagonals of the foundation rectangle. Cut them the same length because of Theorem 6-18. Step 2 Join the two pieces of rope at their midpoints because of Theorem 6-11. Step 3Pull the ropes straight and taut. The ends of the ropes will be the corners of a rectangle. B Can you adapt this method slightly to stake off a square play area? Explain. Yes, you can if you make the diagonals perpendicular. The result will be a rectangle and a rhombus, so the play area will be square. Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares C HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. For what value of x is the figure the given special parallelogram? 1. rhombus For additional support when completing your homework, go to PearsonTEXAS.com. 2. rectangle 3. rectangle L O (6x 9) (2x 39) 4. rectangle 8x 3 4 4 4x LN 4x 7 MO 2x 13 7 N M 6. rectangle 5. rhombus (5x 2) hsm11gmse_0605_t05964.ai 3x (4x 12) hsm11gmse_0605_t05965.ai (3x 6) (8x 7) (3x 4) hsm11gmse_0605_t05966.ai Analyze Mathematical Relationships (1)(F) Decide whether 7. the given information is sufficient to show the quadrilateral is a rectangle. Explain. hsm11gmse_0605_t05967.ai a.AE ≅ CE and DE ≅ BE hsm11gmse_0605_t05968.ai b.AD ≅ BC, AB ≅ DC, and m∠DAB = 90 D hsm11gmse_0605_t05969 C E A c.AB } CD, AD } BC, and AC ≅ DB d.AE ≅ CE ≅ DE ≅ BE B Apply Mathematics (1)(A) STEM8. 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. hsm11gmse_0605_t05961.ai hsm11gmse_0605_t05962.ai 12. PearsonTEXAS.com hsm11gmse_0605_t05963.ai 279 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. Proof Given: ABCD is a parallelogram. A 3 AC bisects ∠BAD and ∠BCD. Prove: ABCD is a rhombus. A 17.Prove Theorem 6-18. Proof Given: ▱ABCD, AC ≅ BD B D Prove: ABCD is a rectangle. D 4 2 1 C B hsm11gmse_0605_t05970 C 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 conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(C) •Base angles of a trapezoid – The •Legs of a trapezoid – The legs of base angles of a trapezoid are the two angles that share a base of the trapezoid. a trapezoid are the nonparallel sides of the trapezoid. •Midsegment of a trapezoid – •Bases of a trapezoid – The bases The midsegment of a trapezoid is the segment that joins the midpoints of its legs. of a trapezoid are the parallel sides of the trapezoid. •Isosceles trapezoid – An isosceles •Trapezoid – A trapezoid is a trapezoid is a trapezoid with legs that are congruent. quadrilateral with exactly one pair of parallel sides. •Kite – A kite is a quadrilateral •Analyze – closely examine with two pairs of consecutive sides congruent and no opposite sides congruent. objects, ideas, or relationships to 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 A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base angles. Diagram base leg leg base angles base angles base An isosceles trapezoid is a trapezoid with legs that are congruent. B C hsm11gmse_0606_t06314 A A midsegment of a trapezoid is the segment that joins the midpoints of its legs. D R M A N hsm11gmse_0606_t06315 T P PearsonTEXAS.com hsm11gmse_0606_t06325 281 Theorem 6-20 Theorem If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. If . . . TRAP is an isosceles trapezoid with bases RA and TP Then . . . ∠T ≅ ∠P, ∠R ≅ ∠A A R A R T T P P You will prove Theorem 6-20 in Exercise 1. hsm11gmse_0606_t06317 hsm11gmse_0606_t06317 Theorem 6-21 Theorem If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent. If . . . ABCD is an isosceles trapezoid B Then . . . AC ≅ BD B C A C A D D You will prove Theorem 6-21 in Exercise 16. hsm11gmse_0606_t06321 hsm11gmse_0606_t06324 If . . . TRAP is a trapezoid with midsegment MN Then . . . (1) MN } TP, MN } RA, and Theorem 6-22 Trapezoid Midsegment Theorem Theorem If a quadrilateral is a trapezoid, then (1) the midsegment is parallel to the bases, and (2) the length of the midsegment is half the sum of the lengths of the bases. R M T A ( (2) MN = 12 TP + RA ) N P You will prove Theorem 6-22 in Lesson 7-3. hsm11gmse_0606_t06325 Key Concept Kites Term Description A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent. 282 Lesson 6-6 Trapezoids and Kites Diagram hsm11gmse_0606_t06333 Theorem 6-23 Theorem If a quadrilateral is a kite, then its diagonals are perpendicular. If . . . ABCD is a kite Then . . . AC # BD B B A A C D C D For a proof of Theorem 6-23, see the Reference section on page 683. hsm11gmse_0606_t06335 hsm11gmse_0606_t06336 Concept Summary Relationships Among Quadrilaterals s of pair o es N l sid e l l a par Kite Quadrilateral Only 1 pair of parallel sides 2 pair s of l sides paralle Parallelogram Trapezoid Rhombus Rectangle Isosceles trapezoid hsm11gmse_0606_t06342.ai Problem 1 Square TEKS Process Standard (1)(F) Finding Angle Measures in Trapezoids What do you know about the angles of an isosceles trapezoid? You know that each pair of base angles is congruent. Because the bases of a trapezoid are parallel, you also know that two angles that share a leg are supplementary. D CDEF is an isosceles trapezoid and mjC = 65. What are mjD, mjE, and mjF ? m∠C + m∠D = 180 Two angles that form same-side interior angles along one leg are supplementary. 65 + m∠D = 180 m∠D = 115 E 65 C F Substitute. Subtract 65 from each side. Since each pair of base angles of an isosceles trapezoid is congruent, m∠C = m∠F = 65 and hsm11gmse_0606_t06318 m∠D = m∠E = 115. PearsonTEXAS.com 283 Problem 2 Finding Angle Measures in Isosceles Trapezoids What do you notice about the diagram? Each trapezoid is part of an isosceles triangle with base angles that are the acute base angles of the trapezoid. 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? Step 1Find the measure of each angle at the center of the fan. This is the measure of the vertex angle of an isosceles triangle. 360 m∠1 = 20 = 18 Step 2Find the measure of each acute base angle of an isosceles triangle. 18 + x + x = 180 18 + 2x = 180 2x = 162 x = 81 Triangle Angle-Sum Theorem Combine like terms. Subtract 18 from each side. Divide each side by 2. Step 3Find the measure of each obtuse base angle of the isosceles trapezoid. 81 + y = 180 y = 99 Two angles that form same-side interior angles along one leg are supplementary. 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 trapezoid different from other trapezoids? An isosceles trapezoid is a trapezoid whose nonparallel legs are congruent. Choose from a variety of tools (such as a protractor, a ruler, or a compass) to A investigate patterns in the diagonals of the three given isosceles trapezoids. Explain your choice. A ruler is useful for measuring segments. B Make a conjecture about the diagonals of isosceles trapezoids. Isosceles Trapezoid ABCD A AC = 3 cm and BD = 3 cm. B 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 E Isosceles Trapezoid JKLM 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. K L G H 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. What is x? QR = 12 (LM + PN) How can you check your answer? Find LM and QR. Then see if QR equals half of the sum of the base lengths. 4x 10 Trapezoid Midsegment Theorem x + 2 = 12 [(4x - 10) + 8] Substitute. x + 2 = 12 (4x - 2) Simplify. x + 2 = 2x - 1 Distributive Property 3 = x M L Q x2 P R N 8 Subtract x and add 1 to each side. Problem 5 hsm11gmse_0606_t06328 Finding Angle Measures in Kites D Quadrilateral DEFG is a kite. What are mj1, mj2, and mj3? How are the triangles congruent by SSS? DE ≅ DG and FE ≅ FG because a kite has congruent consecutive sides. DF ≅ DF by the Reflexive Property of Congruence. m∠1 = 90 90 + m∠2 + 52 = 180 142 + m∠2 = 180 m∠2 = 38 Diagonals of a kite are #. 3 E 52 1 2 G Triangle Angle-Sum Theorem Simplify. Subtract 142 from each side. F △DEF ≅ △DGF by SSS. Since corresponding parts of congruent triangles are congruent, m∠3 = m∠GDF = 52. hsm11gmse_0606_t06340.ai PearsonTEXAS.com 285 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. 1.Justify Mathematical Arguments (1)(G) The plan suggests a proof of Proof Theorem 6-20. Write a proof that follows the plan. For additional support when completing your homework, go to PearsonTEXAS.com. A Given: Isosceles trapezoid ABCD with AB ≅ DC Prove: ∠B ≅ ∠C and ∠BAD ≅ ∠D D 1 B 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. C Analyze Mathematical Relationships (1)(F) Find the value(s) of the variable(s)hsm11gmse_0606_t06008 in each isosceles trapezoid or kite. 2. 3. 4. Q (3x 5) R y P S QS x 5 RP 3x 3 (x 6) 2x (2y 20) (2x 4) (4x 30) 5. Explain Mathematical Ideas (1)(G) If KLMN is an isosceles trapezoid, is it possible for KM to bisect ∠LMN and ∠LKN ? Explain. STEM pplyhsm11gmse_0606_t06001 A Mathematics (1)(A) The beams of the hsm11gmse_0606_t06005 bridge at the right form quadrilateral ABCD. △AED @ △CDE @ △BEC and mjDCB = 120. 6. Classify the quadrilateral. Explain your reasoning. hsm11gmse_0606_t06006 A B E 7. Find the measures of the other interior angles of the quadrilateral. D 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. C 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 16.Prove Theorem 6-21. C Proof Given: Isosceles trapezoid ABCD with AB ≅ DC Prove: AC ≅ DB 286 A D 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 Proof kite are congruent. A R 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; Proof BI is the perpendicular bisector of RA, intersecting RA at B and TP at I. T P Prove: BI is the perpendicular bisector of TP. A R < > 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 N A given segments. How are its length and the lengths of the two parallel sides hsm11gmse_0606_t15810 of the trapezoid related? Justify your answer. 29.the two nonparallel sides C 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 G.parallelogram H.rhombus G D J.kite 33.Given DE is congruent to FG and EF is congruent to GD, prove ∠E ≅ ∠G. E PearsonTEXAS.com F 287 hsm11gmse_0606_t06013 Topic 6 Review TOPIC VOCABULARY • base angles of a trapezoid, p. 281 • equilateral polygon, p. 249 • isosceles trapezoid, p. 281 • bases of a trapezoid, p. 281 • consecutive angles, p. 255 • midsegment of a trapezoid, • rectangle, p. 269 p. 281 • regular polygon, p. 249 • kite, p. 282 • opposite angles, p. 255 • rhombus, p. 269 • legs of a trapezoid, p. 281 • opposite sides, p. 255 • square, p. 269 • parallelogram, p. 255 • trapezoid, p. 281 • equiangular polygon, 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 is (n - 2)180. The measure of one interior angle of a regular Find the measure of an interior angle and an exterior angle of each regular polygon. n-gon is . The sum of the measures of the exterior n angles of a polygon, one at each vertex, is 360. 5.hexagon (n - 2)180 Example Find the measure of an interior angle of a regular 20‑gon. (n - 2)180 Measure = n = Corollary to the Polygon Angle-Sum Theorem (20 - 2)180 20 Substitute. 18 Simplify. # 180 20 = = 162 The measure of an interior angle is 162. 288 Topic 6 Review 6.16-gon 7.pentagon 8.What is the sum of the exterior angles for each polygon in Exercises 5–7? Find the measure of the missing angle. 9. 89 x 119 83 10. 122 79 z hsm11gmse_06cr_t06359 hsm11gmse_06cr_t06357 6-2 Properties of Parallelograms Quick Review Exercises Opposite sides and opposite angles of a parallelogram are congruent. Consecutive angles in a parallelogram are supplementary. The diagonals of a parallelogram bisect each other. If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Find the measures of the numbered angles for each parallelogram. 11. 3 38 12. 1 2 2 1 99 79 3 Example 13. 14. 3 1 2 hsm11gmse_06cr_t06361 1 3 63 hsm11gmse_06cr_t06363 Find the measures of the numbered angles in the parallelogram. 2 1 37 2 3 56 Find the values of x and y in ▱ABCD. Since consecutive angles are supplementary, m∠1 = 180 - 56, or 124. Since opposite angles are congruent, m∠2 = 56 and m∠3 = 124. hsm11gmse_06cr_t06360 15. AB = 2y, BC = y + 3, CD = 5x - 1, DA = 2x + 4 hsm11gmse_06cr_t06365hsm11gmse_06cr_t06366 16. AB = 2y + 1, BC = y + 1, CD = 7x - 3, DA = 3x 6-3 Proving That a Quadrilateral Is a Parallelogram Quick Review Exercises A quadrilateral is a parallelogram if any one of the following is true. Determine whether the quadrilateral must be a parallelogram. 17. • Both pairs of opposite sides are parallel. 18. • Both pairs of opposite sides are congruent. • Consecutive angles are supplementary. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • One pair of opposite sides is both congruent and parallel. Find the values of the variables for which ABCD must be a parallelogram. hsm11gmse_06cr_t06372 19. Bhsm11gmse_06cr_t06370 20. B C 4x (3y 20) Example (4y 4) Must the quadrilateral be a parallelogram? Yes, both pairs of opposite angles are congruent. 4x (2x 6) A 3y D 3 C 2 3x 3y 1 A D 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 special parallelogram. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. 21. The diagonals of a rhombus are perpendicular. Each diagonal bisects a pair of opposite angles. 2 1 32 22. 12 3 56 3 The diagonals of a rectangle are congruent. Example Determine whether each statement is always, sometimes, or never true. hsm11gmse_06cr_t06381.ai 2 What are the measures of the numbered angles in the rhombus? 1 m∠1 = 60 Each diagonal of a rhombus bisects a pair of opposite angles. 23. hsm11gmse_06cr_t06380.ai A rhombus is a square. 3 60 24. A square is a rectangle. 25. A rhombus is a rectangle. m∠2 = 90 The diagonals of a rhombus are #. 26. The diagonals of a parallelogram are perpendicular. 60 + m∠2 + m∠3 = 180 Triangle Angle-Sum Thm. 27. The diagonals of a parallelogram are congruent. 60 + 90 + m∠3 = 180 Substitute. hsm11gmse_06cr_t06379.ai 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 that bisects two angles of the parallelogram, then the quadrilateral is a rhombus. If a quadrilateral is 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. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. 29. 30. Example For what value of x is the figure the given parallelogram? Justify your answer. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. 31. hsm11gmse_06cr_t06383.ai Rhombus 32. hsm11gmse_06cr_t06384.ai Rectangle Yes, the diagonals are perpendicular, so the parallelogram is a rhombus. (5x 30) (3x 6) 1 x 2 2 2 x 3 hsm11gmse_06cr_t06385.ai hsm11gmse_06cr_t06386.ai hsm11gmse_06cr_t06382.ai 290 Topic 6 Review 6-6 Trapezoids and Kites Quick Review Exercises The parallel sides of a trapezoid are its bases, and the nonparallel sides are its legs. Two angles that share a base of a trapezoid are base angles of the trapezoid. The midsegment of a trapezoid joins the midpoints of its legs. Find the measures of the numbered angles in each isosceles trapezoid. 33. 34. 1 2 1 The base angles of an isosceles trapezoid are congruent. The diagonals of an isosceles trapezoid are congruent. 80 The diagonals of a kite are perpendicular. Find the measures of the numbered angles in each kite. hsm11gmse_06cr_t06388.ai 36. 35. Example B ABCD is an isosceles trapezoid. What is m∠C? Since BC } AD, ∠C and ∠D are same-side interior angles. m∠C + m∠D = 180 m∠C + 60 = 180 m∠C = 120 C 34 1 2 1 38 hsm11gmse_06cr_t06389.ai 2 65 A S ame-side interior angles are supplementary. Substitute. 2 3 3 45 60 D 37. A trapezoid has base lengths of (6x - 1) units and 3 units. Its midsegment has a length of (5x - 3) units. hsm11gmse_06cr_t06391.ai What is the value of x? hsm11gmse_06cr_t06387.ai hsm11gmse_06cr_t06390.ai Subtract 60 from each side. PearsonTEXAS.com 291 Topic 6 TEKS Cumulative Practice Multiple Choice Read each question. Then write the letter of the correct answer on your paper. 1.Which list could represent the lengths of the sides of a triangle? 5.FGHJ is a quadrilateral. If at least one pair of opposite angles in quadrilateral FGHJ is congruent, which statement is false? A. Quadrilateral FGHJ is a trapezoid. 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. 6.For which value of x are lines g and h parallel? C. 1 ft, 2 ft, 4 ft D. 3 m, 5 m, 7 m (2x 10) (5x 5) 2.Which quadrilateral CANNOT contain four right angles? F. squareH. trapezoid h G. rhombusJ. rectangle F. 12H. 18 3.What is the circumcenter of △ABC with vertices A( -7, 0), B( -3, 8), and C( -3, 0)? A. ( -7, -3)C. ( -4, 3) B. ( -5, 4)D. ( -3, 4) 4.ABCD is a rhombus. To prove that the diagonals of a rhombus are perpendicular, which pair of angles below must you prove congruent by using corresponding parts of congruent triangles? A B G. 15J. 25 7.In △GHJ, GH ≅ HJ . Using the indirect proof method, hsm11gmse_06cu_t06104 you attempt to derive a contradiction by.ai proving that ∠G and ∠J are right angles. Which theorem will contradict this claim? A. Triangle Angle-Sum Theorem B. Side-Angle-Side Theorem C. Converse of the Isosceles Triangle Theorem D. Angle-Angle-Side Theorem 8.Which quadrilateral must have congruent diagonals? E F. kite D G. rectangle C F. ∠AEB and ∠DEC G. ∠AEB and ∠AED H. ∠BEC and ∠AED hsm11gmse_06cu_t06102.ai J. ∠DAB and ∠ABC 292 g Topic 6 TEKS Cumulative Practice H. parallelogram J. rhombus 9.What values of x and y make the quadrilateral below a parallelogram? y 1 5x 2 4 Constructed Response 15. What are the possible values for n to make ABC a valid hsm11gmse_06cu_t06110.ai triangle? Show your work. x = 2, y = 1C. x = 1, y = 2 A. 9 x = 3, y = 5D. B. x = 2, y = 7 C F. △ABC is not isosceles. G. △ABC is isosceles. H. △ABC may or may not be isosceles. J. △ABC is equilateral. Gridded Response 11. What is m∠1 in the figure below? 31 38 1 69 n If a triangle is equilateral, then it is isosceles. △ABC is not equilateral. 2n 1 10. Which is the most valid conclusion based hsm11gmse_06cu_t06105 .ai on the statements below? x 6y 3x 6 14. The outer walls of the Pentagon in Arlington, Virginia, are formed by two regular pentagons, as shown at the right. What is the value of x? A B 5n 4 16.The pattern of a soccer ball contains regular hexagons and regular pentagons. The figure at the righthsm11gmse_06cu_t06112.ai shows what a section of the pattern would look like on a flat surface. Use the fact that there are 360° in a circle to explain why there are gaps between the hexagons. Does the information help you A B prove that ABCD is a parallelogram? Explain. hsm11gmse_06cu_t06113.ai 17. AC bisects BD. D C 12. ∠ABE and ∠CBD are vertical angles, and both are complementary with ∠FGH. If m∠ABE = (3x - 1), and m∠FGH = 4x, what is m∠CBD? 18. AB ≅ DC , AB } DC 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. 22 19. AB ≅ DC , BC ≅ AD hsm11gmse_06ct_t05844.ai x hsm11gmse_06cu_t06109.ai PearsonTEXAS.com 293