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 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 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.

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 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 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.

Answer 1

The SAS (Side-Angle-Side) criterion can be used to prove the similarity of two triangles based on the following principles:

Proportional Side Lengths: If two triangles have two pairs of corresponding sides that are in proportion (meaning the ratios of the lengths of those sides are equal), and the included angles (the angle formed between the two sides) between those sides are equal, then the triangles are similar.

In simpler terms, this means:

If you have triangle \( ABC \) and triangle \( DEF \), and you know that:
- \( \frac{AB}{DE} = \frac{AC}{DF} \) (the ratios of the lengths of two corresponding sides are equal),
- The included angle \( \angle ABC = \angle DEF \) (the angle between those sides is equal),

then triangle \( ABC \) is similar to triangle \( DEF \) (written as \( \triangle ABC \sim \triangle DEF \)).

Conclusion of Similarity: The similarity of the triangles implies that their corresponding angles are also equal and the lengths of corresponding sides are proportional, which can be used for further calculations or applications involving the triangles.

This criterion is particularly useful in cases where direct measurement of angles or the third side is not feasible, yet the relationship between side lengths and the included angles can be established, confirming the triangles' similarity.