An acute angle in one right triangle has the same measure as an acute angle in another right triangle. Name the theorem or postulate that is the most direct way to prove the triangles similar.

If one of the acute angles of a right triangle is congruent to an acute angle of another right triangle, then by Angle-Angle Similarity the triangles are similar.

In the figure, ∠M≅∠Y , since both are right angles, and ∠N≅∠Z .

So, ΔLMN∼ΔXYZ .

Leg-Leg Similarity

If the lengths of the corresponding legs of two right triangles are proportional, then by Side-Angle-Side Similarity the triangles are similar.

In the figure, ABPQ=BCQR .

So, ΔABC∼ΔPQR .

Hypotenuse-Leg Similarity

If the lengths of the hypotenuse and a leg of a right triangle are proportional to the corresponding parts of another right triangle, then the triangles are similar. (You can prove this by using the Pythagorean Theorem to show that the third pair of sides is also proportional.)

In the figure, DFST=DESR .

So, ΔDEF∼ΔSRT .

Taking Leg-Leg Similarity and Hypotenus-Leg Similarity together, we can say that if any two sides of a right triangle are proportional to the corresponding sides of another right triangle, then the triangles are similar.