Question

Explain how the SAS criterion can be used to prove similarity in two triangles.(1 point)
Responses

If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.

If at least two angles of the triangles are given and are the same, then the triangles are similar.
If at least two angles of the triangles are given and are the same, then the triangles are similar.

If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.

If given a scale factor, you can determine if triangles are similar.
If given a scale factor, you can determine if triangles are similar.

Answer 1

Correct choice: "If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar."

Explanation / proof sketch (SAS similarity):
Let triangles ABC and DEF satisfy AB/DE = AC/DF and ∠A = ∠D (the included angles). Scale triangle DEF by the factor k = AB/DE so that DE·k = AB and DF·k = AC. The scaled triangle now has two sides equal to AB and AC and the included angle ∠D (unchanged), so by SAS congruence the scaled triangle is congruent to triangle ABC. Congruence of the scaled triangle to ABC implies the corresponding angles match, so the original triangles have all corresponding angles equal and their sides are in proportion — hence they are similar.