C Challenge

52. Proof Use the plan and write a proof of Theorem 8-2 (Converse of the Pythagorean Theorem).

    

    Given: Δ A B C cap delta eh b c with sides of length a, b, and c, where a 2 + b 2 = c 2 eh squared , plus , b squared , equals , c squared

    Prove: Δ A B C cap delta eh b c is a right triangle.

    Plan: Draw a right triangle (not Δ A B C cap delta eh b c ) with legs of lengths a and b. Label the hypotenuse x. By the Pythagorean Theorem, a 2 + b 2 = x 2 . eh squared , plus , b squared , equals , x squared , . Use substitution to compare the lengths of the sides of your triangle and Δ A B C . cap delta eh b c , . Then prove the triangles congruent.

53. Proof Use the plan and write a proof of Theorem 8-3.

    

    Given: Δ A B C cap delta eh b c with sides of length a, b, and c, where c 2 > a 2 + b 2 c squared , greater than , eh squared , plus , b squared

    Prove: Δ A B C cap delta eh b c is an obtuse triangle.

    Plan: Draw a right triangle (not Δ A B C cap delta eh b c ) with legs of lengths a and b. Label the hypotenuse x. By the Pythagorean Theorem, a 2 + b 2 = x 2 . eh squared , plus , b squared , equals , x squared , . Use substitution to compare lengths c and x. Then use the Converse of the Hinge Theorem to compare ∠ C angle c to the right angle.

54. Proof Prove Theorem 8-4.

    

    Given: Δ A B C cap delta eh b c with sides of length a, b, and c, where c 2 < a 2 + b 2 c squared , less than , eh squared , plus , b squared

    Prove: Δ A B C cap delta eh b c is an acute triangle.

Standardized Test Prep

GRIDDED RESPONSE

SAT/ACT

55. A 16-ft ladder leans against a building, as shown. To the nearest foot, how far is the base of the ladder from the building?

    

56. What is the measure of the complement of a 67 ° 67 degrees angle?
57. The measure of the vertex angle of an isosceles triangle is 58. What is the measure of one of the base angles?
58. The length of rectangle ABCD is 4 in. The length of similar rectangle DEFG is 6 in. How many times greater than the area of ABCD is the area of DEFG?

Mixed Review

See Lesson 7-5.

59. Δ A B C cap delta eh b c has side lengths A B = 8 , B C = 9 ,  and  A C = 10 . eh b equals 8 comma b c equals 9 comma , and eh c equals 10 . Find the lengths of the segments formed on B C ¯ b c bar by the bisector of ∠ A . angle eh .

Get Ready! To prepare for Lesson 8-2, do Exercises 60–62.

See Review, p. 399.

Simplify each expression.

60. 9 ÷ 3 square root of 9 divides square root of 3
61. 30 ÷ 2 30 divides square root of 2
62. 16 3 fraction 16 , over square root of 3 end fraction

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