The SAS (Side-Angle-Side) criterion for triangle similarity states that if two triangles have two corresponding sides that are proportional, and the angle included between those two sides is equal, then the two triangles are similar.
To summarize how SAS can be used to prove similarity:
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Proportional Sides: Measure the lengths of two corresponding sides in both triangles. If the ratio of these side lengths is the same (proportional), that is, if \( \frac{AB}{DE} = \frac{AC}{DF} \), where \( AB \) and \( AC \) are sides of the first triangle, and \( DE \) and \( DF \) are the corresponding sides of the second triangle, they fulfill the first condition of the SAS criterion.
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Included Angle: Verify that the angle between these two sides in both triangles is equal. For example, if \( \angle A \) in triangle ABC is equal to \( \angle D \) in triangle DEF, this condition is satisfied.
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Conclusion: If both conditions (the proportionality of the two sides and the equality of the included angle) are met, you can conclude that triangle ABC is similar to triangle DEF by the SAS criterion.
This criterion is useful in various geometric problems to establish the similarity of triangles, which further allows for the comparison of their shapes and ratios of their areas.