Question

in a proof of the pythagorean theorem using similarity, what allows you to state that the triangles are similar in order to write the true proportions \\( \frac { c } { a } = \frac { a } { f } \\) and \\( \frac { c } { b } = \frac { b } { e } \\)?
\\( \times \\) the geometric mean (altitude) theorem
\\( \bigcirc \\) the geometric mean (leg) theorem
\\( \bigcirc \\) the right triangle altitude theorem
\\( \bigcirc \\) the sss theorem

Explanation:

Step1: Recall triangle similarity conditions

In a right - triangle, when proving similarity for the purpose of writing proportions like \(\frac{c}{a}=\frac{a}{f}\) and \(\frac{c}{b}=\frac{b}{e}\), we use the fact that if two right - triangles have an acute angle in common, they are similar. The right - triangle altitude theorem (also known as the geometric mean (altitude) theorem) is based on the similarity of right - triangles formed by an altitude drawn to the hypotenuse of a right - triangle.

The SSS (Side - Side - Side) theorem is used to prove triangle congruence (not similarity in the context of these proportions for the Pythagorean theorem proof). The geometric mean (leg) theorem gives a relationship between the leg of a right - triangle and the segments of the hypotenuse, but it is not the basis for the initial similarity - based proportions \(\frac{c}{a}=\frac{a}{f}\) and \(\frac{c}{b}=\frac{b}{e}\).

Answer:

the right triangle altitude theorem