Determine whether the red segments are parallel. Explain each answer. You can use the theorem proved in Exercise 37.

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41. An angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 3 cm long. A second side of the triangle is 7.5 cm long. Find all possible lengths for the third side of the triangle.
42. Open-Ended In a triangle, the bisector of an angle divides the opposite side into two segments with lengths 6 cm and 9 cm. How long could the other two sides of the triangle be? (Hint: Make sure the three sides satisfy the Triangle Inequality Theorem.)
43. Reasoning In △ A B C , white up pointing triangle eh b c comma  the bisector of ∠ C angle c  bisects the opposite side. What type of triangle is △ A B C ? white up pointing triangle eh b c question mark  Explain your reasoning.

Algebra Solve for x.

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46. proof Prove the Corollary to the Side-Splitter Theorem. In the diagram from page 473, draw the auxiliary line C W ↔ modified c w with left right arrow above  and label its intersection with line b as point P.

    

    Given: a ∥ b ∥ c eh parallel to , b parallel to , c

    Prove: A B B C = W X X Y fraction eh b , over b c end fraction . equals . fraction w x , over x y end fraction

47. proof Prove the Triangle-Angle-Bisector Theorem. In the diagram from page 473, draw the auxiliary line B E ↔ modified b e with left right arrow above  so that B E ↔ ‖ D A ¯ . modified b e with left right arrow above . double vertical bar , d eh bar . .  Extend C A ¯ c eh bar  to meet B E ↔ modified b e with left right arrow above  at point F.

    Given: A D ↔ modified eh d with left right arrow above  bisects ∠ C A B . angle c eh b .

    Prove: C D D B = C A B A fraction c d , over d b end fraction . equals . fraction c eh , over b eh end fraction

C Challenge

48. Use the definition in part (a) to prove the statements in parts (b) and (c).

    

    1. Write a definition for a midsegment of a parallelogram.
    2. A parallelogram midsegment is parallel to two sides of the parallelogram.
    3. A parallelogram midsegment bisects the diagonals of a parallelogram.

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