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Explain how the SAS criterion can be used to prove similarity in two triangles.

A. If at least two angles of the triangles are given and are the same, then the triangles are similar.
B. If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
C. If given a scale factor, you can determine if 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.

1 answer

The SAS (Side-Angle-Side) criterion is a method used to prove that two triangles are similar based on the proportional relationship of their sides and the included angle. Let's break down the options provided to understand how the SAS criterion can be applied to show triangle similarity.

Key Concepts:

  • Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in proportion. This can be established through different criteria, including Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).

Explanation of Options:

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

  • This statement is correct and is an application of the AA criterion, not specifically SAS. Thus, while true, it does not directly relate to the SAS criterion.

B. 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 similarity criterion: if the sides are proportional, the triangles are similar. While true, it does not use the SAS criterion.

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

  • A scale factor relates to similarity but does not constitute a method for proving similarity by itself. It’s a statement about proportionality rather than a use of 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 statement is true and directly aligns with the SAS similarity criterion. If two sides of one triangle are in proportion to two sides of another triangle, and the included angle (the angle between those two sides) is equal, the triangles are similar by the SAS criterion.

Conclusion:
Among the options, D is the correct application of the SAS criterion for proving that two triangles are similar. It combines the requirement of proportional sides with the necessary angle correspondence to establish similarity.