SAS Similarity (Side-Angle-Side Similarity, SAS~)

SAS Similarity (Side-Angle-Side Similarity, SAS~)

How to solve SAS similarity problems: theorem, example, and its solution.

Theorem

If two corresponding sides of two triangles are proportional and if the included angles of those two triangles are congruent, then those two triangles are similar.

If two corresponding sides of two triangles are proportional
and if the included angles of those two triangles are congruent,
then those two triangles are similar.

Example

Which of the following would be sufficient to prove that triangle APQ ~ triangle ABC? A AP/AB = 2/3, B PQ/BC = 2/3, C AP/AB = 2/5, D AQ/AC = 2/5

Draw the two triangles to prove its similarity:
APQ and △ABC.

APQ and △ABC have the same angle (red).

And AQ/AC = 2/5.

Then, to use the SAS similarity,
AP/AB has to be 2/5.
AP/AB = AQ/AC = 2/5.

So the answer is
C AP/AB = 2/5.