35. Writing You and a friend compete in a scavenger hunt at a museum. The two of you walk from the Picasso exhibit to the Native American gallery along the dashed red line. When he sees that another team is ahead of you, your friend says, “They must have cut through the courtyard.” Explain what your friend means.

    

36. Error Analysis Your family drives across Kansas on Interstate 70. A sign reads, “Wichita 90 mi, Topeka 110 mi.” Your little brother says, “I didn't know that it was only 20 miles from Wichita to Topeka.” Explain why the distance between the two cities does not have to be 20 mi.

Reasoning Determine which segment is shortest in each diagram.

37. 
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39. 

40. Developing Proof Fill in the blanks for a proof of Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

    Given: Δ T O Y , cap delta t o y comma  with Y O > Y T y o greater than y t

    Prove: a. ? ¯ > modified question mark with under bar below , greater than  b. ? ¯ modified question mark with under bar below

    Mark P on Y O ¯ y o bar  so that Y P ¯ ≅ Y T ¯ . y p bar , approximately equal to . y t bar , .  Draw T P ¯ . t p bar , .

    

    Statements	Reasons
    1) Y P ¯ ≅ Y T ¯ y p bar , approximately equal to , y t bar	1) Ruler Postulate
    2) m ∠ 1 = m ∠ 2 m angle , 1 equals , m angle 2	2) c. ? ¯ modified question mark with under bar below
    3) m ∠ O T Y = m ∠ 4 + m ∠ 2 m angle o t y equals m angle , 4 plus , m angle 2	3) d. ? ¯ modified question mark with under bar below
    4) m ∠ O T Y > m ∠ 2 m angle o t y greater than m angle 2	4) e. ? ¯ modified question mark with under bar below
    5) m ∠ O T Y > m ∠ 1 m angle o t y greater than m angle 1	5) f. ? ¯ modified question mark with under bar below
    6) m ∠ 1 > m ∠ 3 m angle , 1 greater than , m angle 3	6) g. ? ¯ modified question mark with under bar below
    7) m ∠ O T Y > m ∠ 3 m angle o t y greater than m angle 3	7) h. ? ¯ modified question mark with under bar below

41. Proof Prove this corollary to Theorem 5-11: The perpendicular segment from a point to a line is the shortest segment from the point to the line.

    

    Given: P T ¯ ⊥ T A ¯ p t bar , up tack , t eh bar

    Prove: P A > P T p eh greater than p t

C Challenge

42. Probability A student has two straws. One is 6 cm long and the other is 9 cm long. She picks a third straw at random from a group of four straws whose lengths are 3 cm, 5 cm, 11 cm, and 15 cm. What is the probability that the straw she picks will allow her to form a triangle? Justify your answer.

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