To determine which theorem allows us to state that the triangles are similar in a proof of the Pythagorean theorem, let's go through the necessary reasoning and steps.
When proving the Pythagorean theorem using similarity, we often start with a right triangle and draw an altitude from the right angle to the hypotenuse. This altitude divides the right triangle into two smaller right triangles, each of which is similar to the original triangle and to each other.
The similarity of these triangles is established by the AA (Angle-Angle) Similarity Postulate, which states that two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
Therefore, in our case:
1. Each of the triangles has a right angle.
2. Each of the smaller triangles shares one of the acute angles with the original triangle.
Since we have two angles that are the same, the triangles are similar by the AA Similarity Postulate.
This similarity allows us to write the proportions involving the sides of the triangles. Specifically, the proportions [tex]$\frac{c}{a}=\frac{a}{f}$[/tex] and [tex]$\frac{c}{b}=\frac{b}{e}$[/tex] arise from the fact that:
- For [tex]$\frac{c}{a}=\frac{a}{f}$[/tex]: The hypotenuse of the original triangle (c) is to a leg of the original triangle (a) as the same leg of the original triangle (a) is to the length of the segment (f).
- For [tex]$\frac{c}{b}=\frac{b}{e}$[/tex]: Similarly, the hypotenuse of the original triangle (c) is to the other leg of the original triangle (b) as this leg (b) is to the length of the other segment (e).
The theorem that specifically enables these proportions to be written in the context of the altitude drawn to the hypotenuse of a right triangle is the right triangle altitude theorem. This theorem states that the altitude to the hypotenuse of a right triangle creates two segments on the hypotenuse, and the length of the altitude is the geometric mean of the lengths of these two segments.
Thus, the correct answer is the right triangle altitude theorem.
