Proving Two Triangles are Similar

Definition: 

There are three helpful methods for determining if two triangles are similar: AA, SAS and SSS

AA (Angle-Angle): If triangles have two of the same angles, then the triangles are similar.

SAS (Side-Angle-Side): If triangles have two pairs of proportional sides and equal included angles, then the triangles are similar.

SSS (Side-Side-Side): If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar

Example: Determine the triangle similarity rule for proving these triangles similar:

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Solution: 

Use the figure to determine the most suitable triangle similarity rule.

Step 1: We can see that these triangles share at least one pair of congruent angles, their 81° angle measures. These triangles do not share any congruent pairs of sides.

Step 2: Because they triangles are formed by two intersecting lines, a pair of vertical angles is formed in the center of the figure. Vertical angles are always congruent, which means the triangles share another pair of congruent angles.

Step 3: Because these triangles have two pairs of congruent angles, we can use the AA (Angle-Angle) rule to determine that they are similar.

Because they both have 81° angles and share a pair of vertical angles, we can prove that these triangles are congruent by the AA rule.

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