Algebra For each pair of similar triangles, find the value of x.

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27. proof Given: P Q ¯ ⊥ Q T ¯ , S T ¯ ⊥ T Q ¯ , P Q S T = Q R T V p q bar , up tack , q t bar , comma , s t bar , up tack , t q bar , comma . fraction p q , over s t end fraction . equals . fraction q r , over t v end fraction

    Prove: △ V K R white up pointing triangle v k r  is isosceles.

    
28. proof Given: A B ¯ ‖ C D ¯ , B C ¯ ‖ D G ¯ eh b bar . double vertical bar . c d bar , comma , b c bar . double vertical bar . d g bar

    Prove: A B · C G = C D · A C eh b middle dot c g equals c d middle dot eh c

    
29. Reasoning Does any line that intersects two sides of a triangle and is parallel to the third side of the triangle form two similar triangles? Justify your reasoning.
30. Constructions Draw any △ A B C white up pointing triangle eh b c  with m ∠ C = 30 . m angle , c equals 30 , .  Use a straightedge and compass to construct △ L K J white up pointing triangle l k j  so that △ L K J ∼ △ A B C . white up pointing triangle l k j , tilde operator white up pointing triangle eh b c .

31. Reasoning In the diagram below, △ P M N ∼ △ S R W . white up pointing triangle p m n , tilde operator white up pointing triangle s r w .   M Q ¯ m q bar  and R T ¯ r t bar  are altitudes. The scale factor of △ P M N white up pointing triangle p m n  to △ S R W white up pointing triangle s r w  is 4 : 3. What is the ratio of M Q ¯ m q bar  to R T ¯ ? r t bar , question mark  Explain how you know.

    

32. proof Coordinate Geometry △ A B C white up pointing triangle eh b c  has vertices A(0, 0), B(2, 4), and C(4, 2). △ R S T white up pointing triangle r s t  has vertices R(0, 3), S ( − 1 , 5 ) , s open minus , 1 comma 5 close comma  and T ( − 2 , 4 ) . t open minus , 2 comma 4 close .  Prove that △ A B C ∼ △ R S T . white up pointing triangle eh b c , tilde operator white up pointing triangle r s t .  (Hint: Graph △ A B C white up pointing triangle eh b c  and △ R S T white up pointing triangle r s t  in the coordinate plane.)

C Challenge

33. proof Write a proof of the following: Any two nonvertical parallel lines have equal slopes.

    
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    Given: Nonvertical lines l 1 l sub 1  and l 2 , l 1 ∥ l 2 , E F ¯ l sub 2 , comma . l sub 1 , parallel to , l sub 2 , comma . e f bar  and B C ¯ b c bar  are ⊥ up tack  to the x-axis

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

34. proof Use the diagram in Exercise 33. Prove: Any two nonvertical lines with equal slopes are parallel.

35. proof Prove the Side-Angle-Side Similarity Theorem (Theorem 7-1).

    

    Given: A B Q R = A C Q S , ∠ A ≅ ∠ Q fraction eh b , over q r end fraction . equals . fraction eh c , over q s end fraction . comma . angle , eh approximately equal to , angle q

    Prove: △ A B C ∼ △ Q R S white up pointing triangle eh b c , tilde operator white up pointing triangle q r s

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