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

(1 point)
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 two corresponding side lengths of each triangle are given and are
• proportional, with a proven included angle measure, the triangles are similar.
• 'í given a scale factor, you can determine if triangles are similar.

3 answers

The SAS (Side-Angle-Side) criterion is a method for proving the similarity of two triangles based on the relationships between their sides and an included angle. According to the SAS criterion:

  1. Proportional Sides: If two sides of one triangle are proportional to two sides of another triangle, and the included angles (the angle between those two sides) are equal, then the triangles are similar.

This relationship can be summarized as follows:

  • Let triangle \( ABC \) and triangle \( DEF \).
  • If \( \frac{AB}{DE} = \frac{AC}{DF} \) and \( \angle A = \angle D \), then triangle \( ABC \sim triangle DEF \).

Using this criterion allows us to establish that the corresponding angles in the two triangles are congruent, which is a key characteristic of similar triangles. Thus, SAS is a reliable way to demonstrate similarity when dealing with triangles.

In summary, if you can show that two pairs of sides are in proportion and the included angle is equal, the triangles are similar according to the SAS criterion.

What’s the answer

The answer involves correctly stating how the SAS criterion proves the similarity of two triangles. Here it is:

SAS Criterion for Triangle Similarity: If two corresponding sides of each triangle are proportional and the included angle between those sides is equal, then the triangles are similar.

In simpler terms:

  • If \( \frac{AB}{DE} = \frac{AC}{DF} \) (the sides are proportional) and \( \angle A = \angle D \) (the angles are equal), then triangles \( ABC \) and \( DEF \) are similar, denoted as \( ABC \sim DEF \).

This indicates that the triangles maintain the same shape but may differ in size.