45. Developing Proof The plan suggests a proof of Theorem 6-19. Write a proof that follows the plan.

    Given: Isosceles trapezoid ABCD with A B ¯ ≅ D C ¯ eh b bar , approximately equal to , d c bar

    Prove: ∠ B ≅ ∠ C angle , b approximately equal to , angle c and ∠ B A D ≅ ∠ D angle b eh d approximately equal to angle d

    Plan: Begin by drawing A E ¯ ∥ D C ¯ eh e bar , parallel to , d c bar to form parallelogram AECD so that A E ¯ ≅ D C ¯ ≅ A B ¯ . ∠ B ≅ ∠ C eh e bar , approximately equal to , d c bar , approximately equal to , eh b bar , . . angle , b approximately equal to , angle c because ∠ B ≅ ∠ 1 angle b approximately equal to angle 1 and ∠ 1 ≅ ∠ C . angle , 1 approximately equal to , angle c . Also, ∠ B A D ≅ ∠ D angle b eh d approximately equal to angle d because they are supplements of the congruent angles, ∠ B angle b and ∠ C . angle c .

    

46. Proof Prove the converse of Theorem 6-19: If a trapezoid has a pair of congruent base angles, then the trapezoid is isosceles.

Name each type of special quadrilateral that can meet the given condition. Make sketches to support your answers.

47. exactly one pair of congruent sides
48. two pairs of parallel sides
49. four right angles
50. adjacent sides that are congruent
51. perpendicular diagonals
52. congruent diagonals

53. Proof Prove Theorem 6-20.

    

    Given: Isosceles trapezoid ABCD with A B ¯ ≅ D C ¯ eh b bar , approximately equal to , d c bar

    Prove: A C ¯ ≅ D B ¯ eh c bar , approximately equal to , d b bar

54. Proof Prove the converse of Theorem 6-20: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.
55. Proof Given: Isosceles trapezoid TRAP with T R ¯ ≅ P A ¯ t r bar , approximately equal to , p eh bar

    Prove: ∠ R T A ≅ ∠ A P R angle r t eh approximately equal to angle eh p r

56. Proof Prove that the angles formed by the noncongruent sides of a kite are congruent. (Hint: Draw a diagonal of the kite.)

    

Determine whether each statement is true or false. Justify your response.

57. All squares are rectangles.
58. A trapezoid is a parallelogram.
59. A rhombus can be a kite.
60. Some parallelograms are squares.
61. Every quadrilateral is a parallelogram.
62. All rhombuses are squares.

C Challenge

63. Proof Given: Isosceles trapezoid TRAP with T R ¯ ≅ P A ¯ ; t r bar , approximately equal to , p eh bar . semicolon B I ¯ b i bar is the perpendicular bisector of R A ¯ , r eh bar , comma intersecting R A ¯ r eh bar at B and T P ¯ t p bar at I.

    Prove: B I ¯ b i bar is the perpendicular bisector of T P ¯ . t p bar , .

    

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