Angle-Angle Similarity
Recall that when we proved triangles congruent we did not need to prove all six parts were congruent in order to prove the triangles were congruent. All we needed to know was certain parts in a very particular order. Proving triangles are similar to each other is similar to that process.
If we are asked to prove that two triangles are similar, it is not necessary to prove that all angles are congruent and all sides are proportional. Instead, we can use known postulates to prove that triangles are similar. These require that we know specific information about some of the sides and angles.
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The first postulate is called the Angle -Angle Similarity Postulate (AA Similarity Postulate). This states that if two angles in one triangle are congruent to the two corresponding angles in another triangle, then the two triangles are similar. |
Example:
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Triangle ABC and triangle PWT |
Triangle ABC has the following angle measures:
Triangle PWT has the following angle measures:
Because the triangles have two angles that are congruent, we can use the Angle-Angle Similarity Postulate to state that the two triangles are similar. It is worth mentioning here, that if two pairs of angles in triangles are congruent, the third pair of angles will also be congruent.
Notice that the congruent angles are corresponding. The angles written first in each triangle (angle A and angle W) are congruent. The angles written second in each triangle (angle B and angle P) are congruent.
