In our exercise we are asked to determine whether a right triangle and an isosceles triangle are always, sometimes or never similar. Let's recall the definition of similar polygons.
Two polygons are similar if and only if
their corresponding angles are congruent
and corresponding side lengths are proportional.Since similar polygons have congruent corresponding angles, for the two triangles to be similar, they both have to be right and isosceles triangles. Notice that only 45^(∘)-45^(∘)-90^(∘) triangles are both right and isosceles, and these triangles are always similar by the AA Similarity Theorem.
In the case of any other pair of a right triangle and an isosceles triangle, the corresponding angles are not congruent. This means that we cannot say that any right triangle and an isosceles triangle are always similar.
Therefore, a right triangle and an isosceles triangle are sometimes similar polygons.
