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Answers 70. equilateral triangle 9. 14; The sum of the interior angle measures of the 71. acute isosceles triangle polygon is 2 (90°) + 3(180°) + 1440° = 2160°. 72. right scalene triangle So, the polygon has 73. obtuse scalene triangle 10. a. 720° 74. 55°; obtuse scalene triangle 2160° + 2 = 14 sides. 180° b. 135 7.1 Enrichment and Extension 75. 76°; acute isosceles triangle 1. 61° 2. 130° 3. 58° 4. 50° 76. 24°; right scalene triangle 5. 84° 6. 85° 7. 146° 8. 145° 77. 95°; obtuse scalene triangle 9. regular decagon 78. 33° 79. 52° 80. 76° Chapter 7 7.1 Start Thinking 1. 540° 2. 720° 3. 900° 2. 70 3. 119 7.1 Warm Up 1. 120 7.1 Cumulative Review Warm Up 2. y = −1 1. x = 1 3. y + 1 = 2 ( x − 5) 3 4. y − 11 14 = ( x + 1) 2 5 7.1 Practice A 2. 19-gon 1. 900° 3. interior: 168°, exterior: 12° 4. 84 5. 125 6. m ∠ X = m ∠Y = 75° 7. m ∠ X = m ∠Y = 135° 8. 76 9. 88 11. 144 people 10. 120° 2. 68 d + e + f = 180°, and g + h + i = 180° because the sum of the interior angles of a triangle equals 180°. You can add those three equations to obtain a + b + c + d + e + f + g + h + i = 540°. m ∠YZV = f + i, m ∠ ZVW = h, m ∠VWX = g + d + a, m ∠WXY = b, and m ∠ XYZ = c + e, so m ∠YZV + m ∠ ZVW + m ∠VWX + m ∠WXY + m ∠ XYZ = 540°. 7.1 Puzzle Time THEY DIDN’T WANT TO WAIT FORTY YEARS FOR A TRAIN 7.2 Start Thinking yes; Sample answer: The scout could use the Pythagorean Theorem to determine the distance that should be between opposite corner posts, the length of the hypotenuse. It 7 inches. should be approximately 15 feet 7 16 Or, the scout could make sure that the distances between the two pairs of opposite corners are the same and not be concerned about the exact measure. This method uses the SSS Congruence Theorem (Thm. 5.8). 7.2 Warm Up REASONS 1. MN ≅ PO , 1. Given NO ≅ MP 3. m ∠ X = m ∠Y = 116° 4. m ∠ X = m ∠Y = 130° 5. 56 11. You know that a + b + c = 180°, 1. STATEMENTS 7.1 Practice B 1. 103 10. 12 sides 6. 55 7. interior: 165°, exterior: 15° 2. NP ≅ NP 2. Reflexive Property of Segment Congruence (Thm. 2.1) 3. PMN ≅ 3. SSS Congruence Theorem (Thm. 5.8) NOP 8. 20 A76 Geometry Answers Copyright © Big Ideas Learning, LLC All rights reserved. Answers 2. STATEMENTS REASONS 11 − x = 9 − 7x 5 11 − x = 5(9 − 7 x) 1. Given 1. AB ≅ CD, AB ⊥ BD, CD ⊥ BD 2. Reflexive Property of Segment Congruence (Thm. 2.1) 2. BD ≅ BD 34 x = 34 4. ∠ ABD ≅ ∠CDB 4. Right Angles Congruence Theorem (Thm. 2.3) x =1 ABD ≅ CDB 5. SAS Congruence Theorem (Thm. 5.5) 6. Corresponding parts of congruent triangles are congruent. 7.2 Cumulative Review Warm Up 13 ; 2 2 x − 8 = 5 + 4 x Write the equation. −13 = 2 x − 2. 13 = x 2 Subtraction Property of Equality 1 (3x + 8) = 2 x − 3 2 3 x + 4 = 2x − 3 2 1 4 = x −3 2 1 7 = x 2 14 = x Addition Property of Equality Subtraction Property of Equality Division Property of Equality 1. x = 14, y = 40 2. a = 10, b = 37 3. u = 62, v = 59 4. s = 9, t = 14 5. ( 2, 1) 6. C ( 2, 0) 9. no; The side lengths of the parallelograms may not be congruent. 10. STATEMENTS REASONS 1. PQRS is a parallelogram. 1. Given 2. PQ ≅ SR 2. Parallelogram Opposite Sides Theorem (Thm. 7.3) 3. QT ≅ TS 3. Parallelogram Diagonals Theorem (Thm. 7.6) 4. PT ≅ TR 4. Parallelogram Diagonals Theorem (Thm. 7.6) Write the equation. Distributive Property Subtraction Property of Equality 5. Addition Property of Equality 7. B (1, 1) 8. Two angles are 50°, and two angles are 130°. Division Property of Equality x = 14; Distributive Property 7.2 Practice A x = − − 8 = 5 + 2 x Subtraction Property of Equality Multiplication Property of Equality 11 + 34 x = 45 3. Perpendicular lines form right angles. 6. AD ≅ BC Write the equation. 11 − x = 45 − 35 x 3. ∠ ABD and ∠CDB are right angles. 5. 1. x = 1; 3. PQT ≅ RST 5. SSS Congruence Theorem (Thm. 5.8) Multiplication Property of Equality Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Answers A77 Answers 7.2 Practice B e. sometimes; when the parallelogram is a square 1. x = 11, y = 8 2. u = 66, v = 38 3. a = 7, b = 42 4. c = 15, d = 48 5. (0, 4 ) 6. C ( 2, − 2 ) 7. STATEMENTS REASONS 1. CEHF is a parallelogram. 1. Given 2. CE ≅ FH 2. Parallelogram Opposite Sides Theorem (Thm. 7.3) 3. CE = FH 3. Definition of segment congruence 4. D bisects CE . G bisects FH . 1 CE 2 1 GH = FH 2 5. CD = f. never; The angles are supplementary by the Consecutive Interior Angles Theorem (Thm. 3.4). 7.2 Enrichment and Extension 1. ( − 9, − 7 ), 2. ( 2, − 2 ), (5, − 7 ), ( −1, 5) (− 4, 6), (8, 4 ) 3. ( a + 2, b + 3), ( ) (a, 2b − b2 ), (2a2 5. 7. CD ≅ GH 7. Definition of segment congruence 8. CF ≅ EH 8. Parallelogram Opposite Sides Theorem (Thm. 7.3) 9. ∠ C ≅ ∠ H 9. Parallelogram Opposite Angles Theorem (Thm. 7.4) 10. CDF ≅ HGE 10. SAS Congruence Theorem (Thm. 5.5) 6. Number of diagonals (d) 3 0 4 2 5 5 6 9 7 14 n( n − 3) − 2, b − 3) ) 8. 21 sides 7. 35; 65 2 (a − a, b2 Number of sides (n) 5. Definition of segment bisector 6. Substitution + 6, b + 3), 4. a, b2 , 4. Given 6. CD = GH (a 9. a. Sample answer: Player 1 Player 2 Player 6 Player 3 Player 5 Player 4 b. 15 total games 8. a. always; Parallelogram Opposite Sides Theorem (Thm. 7.3) b. sometimes; when the parallelogram is a square c. sometimes; when the parallelogram is a square d. always; Parallelogram Opposite Angles Theorem (Thm. 7.4) c. n + n( n − 3) 2 = n( n − 1) 2 7.2 Puzzle Time ON THE BEACH 7.3 Start Thinking 1. no; Sample answer: AB ≅ BC ≅ CD A78 Geometry Answers Copyright © Big Ideas Learning, LLC All rights reserved. Answers 2. no; Sample answer: ∠ E or ∠ H is a right angle. 5. 4 3. no; Sample answer: ∠ X ≅ ∠ Z 1. 4 3 2. 34 4. 45 = 3 5 −4 C −2 2 A 3. 2 5. − 2 B 7.3 Warm Up y 3 5 D −4 6. 5 Because BC = AD = 6, BC ≅ AD. Because both BC and AD are horizontal line segments, their slope is 0, and they are parallel. BC and AD are opposite sides that are both congruent and parallel. So, ABCD is a parallelogram by the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). 7.3 Cumulative Review Warm Up 1. original: If a triangle is right, then it contains two acute angles; converse: If a triangle contains two acute angles, then it is a right triangle; inverse: If a triangle is not right, then it does not contain two acute angles; contrapositive: If a triangle does not contain two acute angles, then it is not a right triangle; The original and contrapositive are true. The converse and inverse are false. 6. y F 4 2. original: If two lines have the same slope, then they are parallel; converse: If two lines are parallel, then they have the same slope; inverse: If two lines do not have the same slope, then they are not parallel; contrapositive: If two lines are not parallel, then they do not have the same slope; All statements are true. 3. original: If there is ice on the road, then I will not go shopping; converse: If I do not go shopping, then there is ice on the road; inverse: If there is not ice on the road, then I will go shopping; contrapositive: If I go shopping, then there is not ice on the road; The original and contrapositive are true. The converse and inverse are false. 7.3 Practice A 1. Parallelogram Opposite Angles Converse Theorem 4 x 2 E −4 G −2 −2 −4 −6 −8 −10 x −2 −4 H −6 Because EF = GH = 5 and EH = FG = 13, EF ≅ GH and EH ≅ FG. Because both pairs of opposite sides are congruent, quadrilateral EFGH is a parallelogram by the Parallelogram Opposite Sides Converse (Thm. 7.7). (Thm. 7.8) 2. Parallelogram Diagonals Converse Theorem (Thm. 7.10) 3. 12 4. 4 Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Answers A79 Answers 7. STATEMENTS 1. ∠ A ≅ ∠ ABE REASONS 5. X 4 y 1. Given 2. AE ≅ BE 2. Base Angles Theorem (Thm. 5.6) 3. AE ≅ CD 3. Given 4. BE ≅ CD 4. Transitive Property of Segment Congruence Theorem (Thm. 2.1) 5. BC ≅ DE 5. Given 6. BCDE is a parallelogram. 6. Parallelogram Opposite Sides Converse Theorem (Thm. 7.7) 8. a. Because AB || CD and AB ≅ CD, ABDE is a parallelogram by the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). −4 −2 2 7.3 Practice B 1. Opposite Sides Parallel and Congruent Theorem 4 x W Z −4 Because WX = YZ = 5, WX ≅ YZ . Because both WX and YZ are vertical line segments, their slope is undefined, and they are parallel. XW and YZ are opposite sides that are both congruent and parallel. So, WXYZ is a parallelogram by the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). 6. 4 2 b. Because ABDC is a parallelogram, CE || DF . From the diagram, you can see that CD || EF . Because the opposite sides are parallel, CDFE is a parallelogram. c. no; You are only given that one pair of opposite sides are parallel, which is not enough information to prove that it is a parallelogram. d. m ∠ ACD = 35°, m ∠ DCE = 145°, m ∠ CEF = 35°, m ∠ EFD = 145° Y 2 y B A −4 2 −2 −2 D 4 x C −4 Because AD = BC = 18 and AB = CD = 40, AD ≅ BC and AB ≅ CD. Because both pairs of opposite sides are congruent, quadrilateral ABCD is a parallelogram by the Parallelogram Opposite Sides Converse (Thm. 7.7). (Thm. 7.9) 2. Parallelogram Opposite Sides Converse Theorem (Thm. 7.7) 3. 11 A80 4. 35 Geometry Answers Copyright © Big Ideas Learning, LLC All rights reserved. Answers 7. STATEMENTS REASONS 7.3 Enrichment and Extension 1. ∠ A ≅ ∠ FDE 1. Given 1. yes 2. yes 3. no 4. yes 2. AB || CD 2. Alternate Interior Angles Converse (Thm. 3.6) 5. no 6. yes 7. yes 8. yes 3. ∠ AFB ≅ ∠ DFE 3. Vertical Angles Congruence Theorem (Thm. 2.6) 4. F is the midpoint of AD. 4. Given 5. AF ≅ DF 5. Definition of midpoint 6. ABF ≅ DEF 6. ASA Congruence Theorem (Thm. 5.10) 7. AB ≅ ED 7. Corresponding sides of congruent triangles are congruent. 8. D is the midpoint of CE. 8. Given 9. ED ≅ CD 9. Definition of midpoint 10. AB ≅ CD 10. Transitive Property of Segment Congruence (Thm. 2.1) 11. ABCD is a parallelogram. 11. Opposite Sides Parallel and Congruent Theorem (Thm. 7.9) 9. 1 11 10. G ( − 4, 1), H (1, 13) 11. yes; It is given that PQRS and QTSU are parallelograms. Because the diagonals of a parallelogram bisect each other, PX = RX and TX = UX . Because PR and TU are diagonals of PTRU that bisect each other, PTRU is a parallelogram. 12. You would need to show that one angle is supplementary to each consecutive angle. 7.3 Puzzle Time FRIENDSHIP 7.4 Start Thinking Sample answer: For both the square and the rhombus: The diagonals bisect each other, the diagonals form right angles, the diagonals form congruent triangles, and the opposite angles are congruent. For the square only: The diagonals are congruent. 7.4 Warm Up 1. 51° 2. 39° 3. 90° 4. 33° 5. 33° 6. 22° 7.4 Cumulative Review Warm Up 1. sometimes; An isosceles triangle could also be an acute or an obtuse triangle. 2. sometimes; A right triangle could also be an isosceles triangle. 3. always; An equilateral triangle will always have 8. no; You cannot determine if a quadrilateral is a parallelogram by only knowing the values of the angles. You must also know the orientation of the angles, and whether the congruent angles are opposite of each other. 9. a. m ∠ FCG = 135°, m ∠ BCF = 45°, m ∠ D = 135° b. Parallelogram Opposite Angles Converse Theorem (Thm. 7.8) c. 8x three 60° angles. 4. never; A right triangle always has one side that is the longest, so it cannot be equilateral. 7.4 Practice A 1. 90° 2. 23° 3. 67° 5. 12 6. 19 7. 22 4. 5 8. rectangle; The sides are perpendicular and not congruent. 9. rectangle, rhombus, square; The diagonals are congruent and perpendicular. Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Answers A81 Answers 10. STATEMENTS REASONS 1. PSUR is a rectangle. 1. Given 2. m∠U = 90° m∠ P = 90° 2. Definition of a rectangle 3. ∠U ≅ ∠ P 3. Transitive Property of Angle Congruence (Thm. 2.2) 4. PS ≅ RU 4. Definition of a rectangle 5. PQ ≅ TU 5. Given 6. PQS ≅ UTR 6. SAS Congruence Theorem (Thm. 5.5) 7. QS ≅ RT 7. Corresponding parts of congruent triangles are congruent. 9. STATEMENTS REASONS 1. ∠ XWY ≅ ∠ XYW 1. Given 2. XW ≅ XY 2. Converse of the Base Angles Theorem (Thm. 5.7) 3. WXYZ is a parallelogram. 3. Given 4. XY ≅ WZ 4. Definition of a parallelogram WX ≅ YZ 5. Transitive Property of Segment Congruence (Thm. 2.1) 5. WX ≅ XY ≅ 11. a. It is a rectangle; By definition, all four angles are right angles. b. It is a rhombus; By definition, all four sides are congruent. c. It is a square; By definition, all four sides are congruent and all four angles are right angles. d. 90° e. 45° 7.4 Practice B 1. rhombus; It has four congruent sides, but it does not have four right angles. YZ ≅ WZ 6. WXYZ is a rhombus. 6. Definition of a rhombus 10. no, Because a similarity transformation maintains the shape of an object, the corresponding angles remain congruent. A rhombus may not have all right angles, but a square always will. 11. yes; The quadrilateral is a rectangle or square, which are both parallelograms. 12. no; Because the quadrilateral is not a rectangle, the other two angles are not 90°. So, the opposite angles are not congruent and the quadrilateral is not a parallelogram by the contrapositive of the Parallelogram Opposite Angles Theorem (Thm. 7.4). 13. yes; If the rectangle is a square, the side lengths of 2. square, rectangle, rhombus; Any square is also a rectangle and a rhombus. 3. 90° 4. 37° 5. 53° 6. 16 7. 24 8. ( 2, 5) the triangle will be congruent. So, the triangles will be isosceles. 7.4 Enrichment and Extension 1. no; Sample answer: If the diagonals of a parallelogram are congruent, then it would have to be a rectangle and have a right angle. 2. yes; Sample answer: If there are congruent diagonals in a parallelogram, it can be a rectangle or square with two opposite sides 2 centimeters long. 3. no; Sample answer: In a parallelogram, consecutive angles must be supplementary, so all angles must be right. This would make it a rectangle. A82 Geometry Answers Copyright © Big Ideas Learning, LLC All rights reserved. Answers 4. 9 5. 10 5 2 7. 7 2 6. 40 2 8. 16° 8. STATEMENTS 1. ABCD is a parallelogram 1. Given 2. AB DC 2. Definition of a parallelogram 3. AE ≅ AD 3. Given 4. ∠ E ≅ ∠ ADE 4. Base Angles Theorem (Thm. 5.6) 5. ∠ ADE ≅ ∠C 5. Definition of a parallelogram 6. ∠ E ≅ ∠C 6. Transitive Property of Angle Congruence (Thm. 2.2) 7. ABCE is an isosceles trapezoid. 7. Isosceles Trapezoid Base Angles Converse (Thm. 7.15) 9. Sample answer: Let parallelogram DFGH have ( ( G − a, − ) b 2 − a 2 , F (b, 0), vertices D a, ) b 2 − a 2 , and H ( − b, 0), respectively. The slope of both HG and DF is b2 − a2 , and a −b the slope of both HD and GF is b2 − a2 . The a +b products of the slopes of the pairs HG and GF , GF and DF , DF and HD , and HD and HG are all equal to −1, making each pair of consecutive segments perpendicular and each angle a right angle. So, parallelogram DFGH is a rectangle. REASONS 7.4 Puzzle Time ALL THE ANGLES 9. a. 90° 7.5 Start Thinking Sample answer: BD is a perpendicular bisector of AC . AB ≅ BC , AD ≅ CD, ∠ BAD ≅ ∠ BCD, and m∠ ABC > m∠ ADC. c. ∠ XWZ 7.5 Practice B 1. Slope of TU = slope of VW and slope of 7.5 Warm Up 1. 120° 2. 60° 3. 90° 4. 45° 5. 135° 6. 109° 1. x = 4, y = 5.5 3. x = 9, y = 2. x = 24, y = 10 370 PQ ≠ slope of RS ; PQ ≅ RS , so PQRS is not an isosceles trapezoid. 3. 17 7.5 Practice A 2. 70 UV ≠ slope of TW ; TW ≅ UV , so TUVW is an isosceles trapezoid. 2. Slope of QR = slope of PS and slope of 7.5 Cumulative Review Warm Up 1. 86 b. 22 in. 4. 61 5. rhombus; ABCD is a quadrilateral with four 3. 6 4. 6 5. isosceles trapezoid; WXYZ has exactly one pair of parallel sides and one pair of congruent base angles. 6. kite; WXYZ has two pairs of consecutive congruent congruent sides. 6. kite; DEFG is a quadrilateral with two pairs of consecutive congruent sides, but opposite sides are not congruent. sides, but opposite sides are not congruent. 7. The quadrilateral is not a kite. Because the opposite sides are congruent, the quadrilateral is a rhombus. Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Answers A83 Answers 7. REASONS 1. VXYZ is a kite. 1. Given 1. x = 5 2. x = −16 3. x = − 5 2. ∠VXY ≅ ∠VZY 2. Kite Opposite Angles Theorem (Thm. 7.19) 4. x = 0 5. x = 15 6. x = − 8 3. ∠WXV ≅ ∠UZV 7. x = 16 8. x = −1 9. x = − 2 10. x = 10 3. Congruent Supplements Theorem (Thm. 2.4) 11. x = 8 12. x = 2 4. VX ≅ VZ 4. Definition of a kite 13. x = − 3 14. x = 3 5. WX ≅ UZ 5. Given 15. x = 4 16. x = − 2 17. equilateral triangle 18. rectangle 19. rhombus 20. parallelogram 21. right triangle 22. square 6. WXV ≅ UZV 6. SAS Congruence Theorem (Thm. 5.5) 8. (6, 0) 9. no; A kite is a quadrilateral and by definition it is a convex polygon. 10. a. The opposite sides are parallel, and all angles are right angles; A = ac bc + ac bc − ac + ac = 4 2 7.5 Enrichment and Extension 1. about 56.6 in.; about 418.3 in. 2. a = 9 3. AD = 7.08 in., AB = CD = 5.08 in., BC = 10.16 in. (x 5. ( a + c, b + c) + y, y + z ) 7. any point of the form ( a, a ), where a is a real number, a > 3.5, and a ≠ 7 7.5 Puzzle Time INCORRECTLY b. 11 units 24. a. P = 2(3 x + 4) + 2( 2 x + 7) c. length = 16 units, width = 15 units c. 2 4. ( a + 3, 3b) 23. a. x = 9 b. 4 bc − ac b. A = 4 6. Cumulative Review STATEMENTS 25. dilation 26. rotation 27. reflection 28. translation 29. rotation 30. dilation 31. A′( −1, 2) 32. B ′(11, 3) 33. C ′(6, −11) 34. A′( 4, 0) 35. B ′( −16, 2) 36. C ( 2, 5) 37. D (7, 5) 38. 139° 39. 105° 40. 115° 41. 143° 42. x = 3, y = 9 43. x = 5, y = 2 44. a. 8 b. 48° c. 42° 45. a. 4 b. 17° c. 73° 46. ∠ DGF A84 Geometry Answers 47. ∠ EFG 48. ∠GDF Copyright © Big Ideas Learning, LLC All rights reserved. Answers 49. ∠ FED 50. ∠ FDE 52. ∠ EDG 53. ∠ DFG 51. ∠ DFE 3. 3 7. 12 54. 11.4; Perpendicular Bisector Theorem (Thm. 6.1) Theorem (Thm. 6.2) 57. 22; Perpendicular Bisector Theorem (Thm. 6.1) b. 36 units 8.1 Practice B Chapter 8 8.1 Start Thinking Sample answer: The three diagrams are the same image, but stretched or shrunk into different sizes or forms; The first resizing is not “similar” to the original in a geometric sense. The proportions of the map were not maintained. The second resizing is “similar” to the original in a geometric sense. It appears to be a dilation of the original in a geometric sense. It appears to be a dilation of the original by a factor less than one and maintains proportionally with the original. 3 ; ∠W ≅ ∠ S , ∠ X ≅ ∠ T , ∠ Y ≅ ∠ U , 2 WX XY YZ ZW = = = ∠ Z ≅ ∠V , ST TU UV VS 3. 6 7. a. 4. 9 5. 7 in. 6. 11 ft 7 4 c. 108° 3 2 2. x = 20 d. about 74.2 units e. about 219.73 square units 9 4. x = − 5 3. x = ± 3 5. x = 2. 3 ; ∠ A ≅ ∠ H , ∠ B ≅ ∠ I , ∠C ≅ ∠ J , 4 AB BC CA = = HI IJ JH b. 7.5 8.1 Warm Up 1. x = 9. 9 12. 336 ft 2 11. 3 1. c. 72 units 8. 60; 540 5.5), ADC ≅ BDC and XWZ ≅ YWZ . Because corresponding parts of congruent triangles are congruent, BC = 13 and YZ = 39. Theorem (Thm. 6.2) 56. 36; Converse of the Perpendicular Bisector 6. 67° 5. 3 10. 13, 39; By the SAS Congruence Theorem (Thm. 55. 1.9; Converse of the Perpendicular Bisector 58. a. 3 4. 22.5 64 7 3 2 6. x = − , x = 4 8.1 Cumulative Review Warm Up 1. 120° 2. 60° 3. 60° 4. 60° 5. 75° 6. 45° f. yes; Because corresponding angles of similar triangles are congruent, ∠ ABC ≅ ∠ D. By the corresponding Angles Converse Theorem (Thm. 3.5), BC || DE. 8.1 Enrichment and Extension 1. Sample answer: 8.1 Practice A 1. 3; ∠ L ≅ ∠ Q , ∠ M ≅ ∠ R , ∠ N ≅ ∠ S , LM MN NL = = QR RS SQ 2. 2. Sample answer: 2 ; ∠ A ≅ ∠ E, ∠ B ≅ ∠ F , ∠ C ≅ ∠ G, 5 AB BC CD DA = = = ∠D ≅ ∠H, EF FG GH HE Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Answers A85