In previous sections, you have defined similarity and used similar figures to identify corresponding parts. With that experience, you are ready to use corresponding sides to solve problems!

Consider the following problem.

Pause and Reflect

What is another proportion, that Joaquin could use to solve the palm tree problems?

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Instead of keeping the tree dimensions in the numerator, he could have kept the heights in the numerator. Then, the shadow lengths would have been in the denominator.
n over 4 n 4 = 5 over 1.5 5 1.5

Suppose you had two similar isosceles triangles. What would you know about the base angles, the lengths of the legs of each triangle, and the ratios of the lengths of the corresponding legs?

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In an isosceles triangle, base angles are congruent, and the two legs are congruent. For a pair of similar isosceles triangles, the ratios of the lengths of the corresponding legs would be the same.

Practice

1. In the figure below, triangle ABC is similar to triangle JKL. Write a proportion that you could use to determine the length of KL.
   
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   Write two equivalent ratios, where each ratio contains the lengths of corresponding sides. Be careful to keep the side lengths from triangle ABC in the numerator of both ratios.
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   AB over JK AB JK = BC over KL BC KL
   
   6 over 9 6 9 = 4 over n 4 n
2. In the figure below, triangle FGM is similar to triangle JHM.
   

   Which of the following statements must be true?

   1. GM is twice as long as HM.
   2. m∠FGM + m∠FMG + m∠HJM = 180°
   3. m∠FMG = m∠HMJ

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   As you analyze the figure, consider the following questions.
   • Which pairs of angles are corresponding?
   • What is the scale factor between the two triangles?
   • What do you know about the sum of the measures of the interior angles of a triangle?
   
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   II and III only.
3. In the figure below, triangle ABC is similar to triangle FCD.
   

   Which of the following statements must be true?

   1. m∠BCA + m∠DCF = 180°
   2. m∠BAC + m∠CDF = 90°
   3. AC is 8 inches long.

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   Need a hint?

   As you analyze the figure, consider the following questions.
   • Which pairs of angles are corresponding?
   • What is the scale factor between the two triangles?
   • What do you know about the sum of the measures of the interior angles of a triangle?
   
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   Check Your Answer

   I and II only.