EXAMPLE 3 Use the SAS Similarity Theorem

You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown?

Lean-to Shelter

EXAMPLE 3 Use the SAS Similarity Theorem

Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F.

~

SOLUTION

Shorter sides Longer sides

ABFG

32

96 ==

ACFH

32

1510 ==

The lengths of the sides that include A and F are proportional.

EXAMPLE 3 Use the SAS Similarity Theorem

ANSWER

So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

EXAMPLE 4 Choose a method

Tell what method you would use to show that the triangles are similar.

Find the ratios of the lengths of the corresponding sides.

Shorter sides Longer sides

SOLUTION

CACD

35

1830 ==

BCEC

35

915 ==

The corresponding side lengths are proportional. The included angles ACB and DCE are congruent because they are vertical angles. So, ACB ~ DCE by the SAS Similarity Theorem.

GUIDED PRACTICE for Examples 3 and 4

3. SRT ~ PNQ

Explain how to show that the indicated triangles are similar.

Find the ratios of the lengths of the corresponding sides.

Shortest sides Longer sides

SOLUTION

RTNQ

43

2821 ==

SRPN

43==

2418

GUIDED PRACTICE for Examples 3 and 4

Explain how to show that the indicated triangles are similar.

The corresponding side lengths are proportional. The included angles R and N are right angles. So, SRI ~ PNQ by the SAS Similarity Theorem.

ANSWER

GUIDED PRACTICE for Examples 3 and 4

4. XZW ~ YZX

Explain how to show that the indicated triangles are similar.

Find the ratios of the lengths of the corresponding sides.

SOLUTION

XZYZ

129=

WZXZ

43=

43==

1612

GUIDED PRACTICE for Examples 3 and 4

Explain how to show that the indicated triangles are similar.

WXXY

43=

2015=

WXY XZY = 90°

The corresponding side lengths are proportional. The angles WZX and XZY are right angles.

ANSWER

So, XZW ~ YZX by the SAS and SSS Similarity Theorem.