20. Think About a Plan Ship A and Ship B leave from the same point in the ocean. Ship A travels 150 mi due west, turns 65 ° 65 degrees  toward north, and then travels another 100 mi. Ship B travels 150 mi due east, turns 70 ° 70 degrees  toward south, and then travels another 100 mi. Which ship is farther from the starting point? Explain.

    
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    • How can you use the given angle measures?
    • How does the Hinge Theorem help you to solve this problem?

21. Which of the following lists the segment lengths in order from least to greatest?

    

    1. CD, AB, DE, BC, EF
    2. EF, DE, AB, BC, CD
    3. BC, DE, EF, AB, CD
    4. EF, BC, DE, AB, CD
22. Reasoning The legs of a right isosceles triangle are congruent to the legs of an isosceles triangle with an 80 ° 80 degrees  vertex angle. Which triangle has a greater perimeter? How do you know?

23. Proof Use the figure below.

    

    Given: Δ A B E cap delta eh b e  is isosceles with vertex ∠ B , Δ A B E ≅ Δ C B D , m ∠ E B D > m ∠ A B E angle b comma cap delta eh b e approximately equal to cap delta c b d comma m angle e b d greater than m angle eh b e

    Prove: E D > A E e d greater than eh e

C Challenge

24. Coordinate Geometry Δ A B C cap delta eh b c  has vertices A ( 0 , 7 ) , B ( − 1 , − 2 ) , C ( 2 , − 1 ) , eh open 0 comma 7 close comma b open negative 1 comma negative 2 close comma c open 2 comma negative 1 close comma  and O(0, 0). Show that m ∠ A O B > m ∠ A O C . m angle eh o b greater than m angle eh o c .

25. Proof Use the plan below to complete a proof of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is opposite the larger included angle.

    

    Given: A B ¯ ≅ X Y ¯ , B C ¯ ≅ Y Z ¯ , m ∠ B > m ∠ Y eh b bar , approximately equal to . x y bar , comma . b c bar , approximately equal to . y z bar , comma . m angle , b greater than , m angle y

    Prove: A C > X Z eh c greater than x z

Plan for proof:

• Copy Δ A B C . cap delta eh b c .  Locate point D outside Δ A B C cap delta eh b c  so that m ∠ C B D = m ∠ Z Y X m angle c b d equals m angle z y x  and B D = Y X . b d equals y x .  Show that Δ D B C ≅ Δ X Y Z . cap delta d b c approximately equal to cap delta x y z .
• Locate point F on A C ¯ , eh c bar , comma  so that B F ¯ b f bar  bisects ∠ A B D . angle eh b d .
• Show that Δ A B F ≅ Δ D B F cap delta eh b f approximately equal to cap delta d b f  and that A F ¯ ≅ D F ¯ . eh f bar , approximately equal to . d f bar , .
• Show that A C = F C + D F . eh c equals . f c plus d f .
• Use the Triangle Inequality Theorem to write an inequality that relates DC to the lengths of the other sides of Δ F C D . cap delta f c d .
• Relate DC and XZ.

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