Lesson 6-7: Use Angle-Angle Similarity

Property

To prove two triangles are similar ($\sim$), you do not need to check all their side lengths or all three angles. If you can prove that just two angles of one triangle are congruent (equal) to two angles of another triangle, the triangles are guaranteed to be similar. This is the AA Similarity Criterion.

Examples

• Standard AA: $\triangle ABC$ has angles of $45^\circ$ and $60^\circ$. $\triangle DEF$ has angles of $45^\circ$ and $60^\circ$. Because two pairs match ($45^\circ=45^\circ$ and $60^\circ=60^\circ$), $\triangle ABC \sim \triangle DEF$.
• The Hidden Match: $\triangle PQR$ has angles of $30^\circ$ and $80^\circ$. $\triangle XYZ$ has angles of $30^\circ$ and $70^\circ$. Are they similar?
  • Find the missing angle in $\triangle PQR$: $180^\circ - (30^\circ + 80^\circ) = 70^\circ$.
  • Now we see $\triangle PQR$ has a $30^\circ$ and a $70^\circ$ angle, matching $\triangle XYZ$. Yes, they are similar!

Explanation

Why does AA work? Because of the Triangle Angle Sum Theorem. The three interior angles of any triangle must always add up exactly to $180^\circ$. Therefore, if two angles are already matched, the third angle has no choice but to be exactly the same! The AA criterion is the ultimate shortcut in geometry—it saves you from doing unnecessary measurements.

Property

A common mistake is assuming that all right triangles are similar just because they all have a $90^\circ$ angle. A $90^\circ$ angle only gives you ONE matching pair. To use the AA Similarity Criterion, two right triangles must share at least one pair of congruent acute angles.

Examples

• Example 1 (Not Similar): Right $\triangle ABC$ has acute angles of $30^\circ$ and $60^\circ$. Right $\triangle DEF$ has acute angles of $45^\circ$ and $45^\circ$. They both share a $90^\circ$ angle, but their acute angles do not match. They are NOT similar.
• Example 2 (Similar): Right $\triangle GHI$ has an acute angle of $40^\circ$. Right $\triangle JKL$ has an acute angle of $50^\circ$.
  • Find the missing angle in $\triangle GHI$: $180^\circ - 90^\circ - 40^\circ = 50^\circ$.
  • Both triangles have a $90^\circ$ angle and a $50^\circ$ angle. They are similar by AA!

Explanation

Right triangles are tricky. Because the $90^\circ$ angle is always implied, test questions often only give you one other angle. You must always use the $180^\circ$ rule to find the missing third angle before you decide if the triangles are similar or not.

Property

If two angles are known to be congruent (e.g., vertical angles or corresponding angles), you can set their algebraic expressions equal to each other. Solving the resulting equation for the variable allows you to find the angle measures needed to prove two triangles are similar by the Angle-Angle (AA) criterion.

Examples