SAS (Side-Angle-Side) Similarity 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.
We are asked to determine whether the given triangles are similar.
Notice that ∠ A and ∠ F are both right angles. Therefore, they are congruent angles. In △ ABC, the sides whose lengths are given include ∠ A. Conversely, in △ FDE, the sides whose lengths are given do not include ∠ F. We will use the Pythagorean Theorem to find EF.
FD^2+EF^2=DE^2
Let's do it.
Now we know the lengths of the sides that include ∠ A and ∠ F. Let's check whether these sides are proportional. Note that CA corresponds to EF, because both are the longer legs of the right triangles.
cccc
CA/EF & = & 4/8 & = & 1/2 [0.8em]
AB/FD & = & 3/6 & = & 1/2
As we can see, the ratios are equal. Therefore, the corresponding sides are proportional. Also, the included angles are congruent. By the SAS Similarity Theorem, we can state that the given triangles are similar.
△ ABC ~ △ FDE
