Question

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

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

Answer 1

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

Why: Let triangles ABC and A'B'C' satisfy AB/A'B' = AC/A'C' = k and ∠BAC = ∠B'A'C' (the included angle between the two given sides). A dilation centered at A with scale factor k sends B → B' and C → C' because the side lengths scale accordingly and the rays from A through B and C map to the rays from A' through B' and C' (the included angle is preserved). Thus the whole triangle ABC maps to A'B'C', so all corresponding angles are equal and sides are proportional — hence the triangles are similar by SAS similarity.