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

Question

Responses which answer is correct and why

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 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 given a scale factor, you can determine if triangles are similar.
If given a scale factor, you can determine if triangles are similar.
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Answer 1

The correct application of the SAS (Side-Angle-Side) criterion is as follows:

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

Explanation:

The SAS criterion states that if in two triangles, the lengths of two sides are in proportion and the angle between those sides is equal, then the triangles are similar. This means that all corresponding angles will also be equal, leading to the conclusion that the triangles maintain the same shape, though they may vary in size.

The other responses focus on different criteria for triangle similarity:

1. The SSS (Side-Side-Side) criterion states that if all three sides of one triangle are proportional to the three sides of another triangle, then they are similar.
2. The AA (Angle-Angle) criterion states that if at least two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
3. While a scale factor can imply similarity when comparing dimensions of similar triangles, it is not a criterion itself—it is more of a tool for understanding the relationship between the sizes of similar triangles.

Therefore, the SAS criterion directly addresses the condition of having two proportional sides and the included angle, making it the correct choice for proving triangle similarity in this context.