Question

justifying steps in the proof of the pythagorean theorem
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}\\)?

the geometric mean (altitude) theorem
the right triangle altitude theorem
the sss theorem
the geometric mean (leg) theorem

Explanation:

To determine what allows stating the triangles are similar for the proportions in the Pythagorean theorem proof using similarity, we analyze the options:

• The geometric mean (altitude) theorem relates to the altitude in a right triangle, but the key here is the right triangle altitude theorem (also known as the geometric mean theorem for right triangles) which states that when an altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. This similarity allows the proportion statements.
• The SSS theorem is for triangle congruence (or similarity via SSS similarity, but here we need the theorem about right triangle altitude creating similar triangles). The geometric mean (leg) theorem is about the leg as a geometric mean, not the similarity justification for the proportions here. The right triangle altitude theorem (also called the geometric mean theorem for the altitude) justifies the similarity of the triangles formed by the altitude to the hypotenuse in a right triangle, enabling the proportion statements. Wait, actually, the correct theorem here is the right triangle altitude theorem (which is part of the geometric mean theorems, but the option "the right triangle altitude theorem" is the one that directly addresses the similarity of the triangles (the original right triangle and the two smaller right triangles formed by the altitude to the hypotenuse) by AA similarity (all right triangles have a right angle, and they share another acute angle, so AA similarity applies, which is what the right triangle altitude theorem is based on). So the right triangle altitude theorem allows us to state the triangles are similar (by AA similarity, since each smaller triangle has a right angle and shares an acute angle with the original triangle) to write the proportions.

Answer:

the right triangle altitude theorem