Question
Explain how the SAS criterion can be used to prove similarity in two triangles.
A. If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar
B. if given a scale factor, you can determine if triangle are similar
C. if at least two angles of the triangles are given and are the same, then the triangles are similar
D. if two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar
A. If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar
B. if given a scale factor, you can determine if triangle are similar
C. if at least two angles of the triangles are given and are the same, then the triangles are similar
D. if two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar
Answers
GPT-4o mini
The SAS (Side-Angle-Side) criterion is a useful mathematical tool for proving the similarity of two triangles. According to this criterion, two triangles are similar if two sides of one triangle are in proportion to two sides of another triangle, and the included angles between those sides are equal.
Let’s analyze the options provided and see how they relate to the SAS criterion:
**A.** If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
- This describes the SSS (Side-Side-Side) criterion for similarity, not SAS. While true for similarity, it does not specifically invoke the SAS criterion.
**B.** If given a scale factor, you can determine if triangles are similar.
- This likely refers to the SSS criterion as well. A scale factor implies that all corresponding sides are proportional, but it does not directly relate to the SAS criterion, which requires two sides and the included angle.
**C.** If at least two angles of the triangles are given and are the same, then the triangles are similar.
- This describes the AA (Angle-Angle) criterion for similarity. While true, this does not relate to the SAS criterion.
**D.** If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
- This option directly describes the SAS criterion: If two sides of one triangle are proportional to two sides of another triangle and the angle between those sides is congruent, then the two triangles are similar.
Therefore, **the correct answer that appropriately defines the SAS criterion for similarity in triangles is D.**
Let’s analyze the options provided and see how they relate to the SAS criterion:
**A.** If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
- This describes the SSS (Side-Side-Side) criterion for similarity, not SAS. While true for similarity, it does not specifically invoke the SAS criterion.
**B.** If given a scale factor, you can determine if triangles are similar.
- This likely refers to the SSS criterion as well. A scale factor implies that all corresponding sides are proportional, but it does not directly relate to the SAS criterion, which requires two sides and the included angle.
**C.** If at least two angles of the triangles are given and are the same, then the triangles are similar.
- This describes the AA (Angle-Angle) criterion for similarity. While true, this does not relate to the SAS criterion.
**D.** If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
- This option directly describes the SAS criterion: If two sides of one triangle are proportional to two sides of another triangle and the angle between those sides is congruent, then the two triangles are similar.
Therefore, **the correct answer that appropriately defines the SAS criterion for similarity in triangles is D.**