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Define the angle-angle (AA) theorem. Identify the measure of at least two angles in one of the triangles. Measure at least two of the angles on the second triangle. Use the angle-angle theorem for similarity.
Two triangles can be proved similar by the angle-angle theorem which states: if two triangles have two congruent angles, then those triangles are similar. This theorem is also called the angle-angle-angle (AAA) theorem because if two angles of the triangle are congruent, the third angle must also be congruent. This is because the angles of a triangle must sum to 180. Using a protractor, measure the degree of at least two angles on the first triangle. Label the angles on the triangle to keep track of them. Choose any two angles on the triangle to measure. Example: Triangle ABC has two angles that measure 30 and 70. Again, use a protractor to measure two of the angles on the second triangle. If both angles are identical on both triangles, then the triangles are similar to each other. Remember, if two angles of a triangle are equal, then all three are equal. Example: The second triangle, DEF, also has two angles that measure 30 and 70. Once you have identified the congruent angles, you can use this theorem to prove that the triangles are similar. State that the measures of the angles between the two triangles are identical and cite the angle-angle theorem as proof of their similarity. It is possible for a triangle with three identical angles to also be congruent, but they would also have to have identical side lengths. Example: Because both triangles have two identical angles, they are similar. Note: If the two triangles did not have identical angles, they would not be similar. For example: Triangle ABC has angles that measure 30 and 70 and triangle DEF has angles that measure 35 and 70. Because 30 does not equal 35, the triangles are not similar.