The SAS (Side-Angle-Side) criterion for triangle similarity states that if two triangles have two corresponding sides that are in proportion and the included angle between those sides is congruent, then the two triangles are similar.
To prove that two triangles are similar using the SAS criterion, you would follow these steps:
-
Identify the Corresponding Sides: Measure or be given the lengths of two corresponding sides of each triangle.
-
Check the Proportion: Ensure that the ratio of the lengths of the first pair of sides is equal to the ratio of the lengths of the second pair of sides. That is, if triangle ABC and triangle DEF are such that: \[ \frac{AB}{DE} = \frac{AC}{DF} \] this condition must hold true.
-
Verify the Included Angle: Check that the angle between the two sides in one triangle is equal to the angle between the two corresponding sides in the other triangle. For our triangles, confirm that: \[ \angle A \cong \angle D \]
If both the proportionality of the side lengths and the congruence of the included angles are verified, then by the SAS criterion, you can conclude that the two triangles are similar.
In summary, the key points for using the SAS criterion are:
- Two pairs of corresponding sides are proportional.
- The included angle between those sides is congruent.
Therefore, your response emphasizing that “if two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar” is correct.
1 answer