Lesson 4: Using Similar Triangles

Property

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

Examples

• $\triangle ABC$ is similar to $\triangle XYZ$. If side $AB=6$ and corresponding side $XY=2$, the ratio is $\frac{6}{2}=3$. If side $BC=9$, then corresponding side $YZ$ must be $\frac{9}{3}=3$.

• A man who is 6 feet tall casts a 4-foot shadow. A nearby tree casts a 20-foot shadow. Let $h$ be the tree's height. The ratio is $\frac{h}{6} = \frac{20}{4}$, so $4h = 120$, and $h=30$ feet.

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.