Exercise 9 Page 455

Let's review the theorems that can help us prove that two triangles are similar.

1. AA (Angle-Angle) ~ Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. SSS (Side-Side-Side) ~ Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
3. SAS (Side-Angle-Side) ~ Theorem: If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

Notice that ∠ S is an angle of both triangles, △ PSQ and △ RST. Also, we know the lengths of the sides that include these angles. Let's check whether these sides are proportional. cccc RS/PS & = & 12 + 24/24 & = & 3/2 [0.8em] TS/QS & = & 8 + 16/16 & = & 3/2 As we can see, the ratios are equal. Therefore, the corresponding sides are proportional, and by the SAS ~ Theorem the given triangles are similar. △ RST ~ △ PSQ