8.3 AA Similarity
Notes
To prove triangles similar, you don't need every part.
- AA Similarity: if two angles of one triangle are congruent to two angles of another, the triangles are similar.
- Look for vertical angles, shared angles and angles formed by parallel lines.
- Example: $\overline{DE} \parallel \overline{BC}$ in $\triangle ABC$ with $D$ on $\overline{AB}$, $E$ on $\overline{AC}$. $\angle A$ is shared and $\angle ADE \cong \angle ABC$ (corresponding), so $\triangle ADE \sim \triangle ABC$.
Practice
$\triangle ABC$ has angles $40^\circ$ and $75^\circ$. $\triangle DEF$ has angles $75^\circ$ and $65^\circ$. Are they similar?
The third angle of $\triangle ABC$ is $65^\circ$, so both have $75^\circ$ and $65^\circ$.
A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 30-foot shadow. How tall is the tree?
The triangles are similar by AA: $\displaystyle \frac{6}{4} = \frac{h}{30}$, so $h = 45$.
Segments $\overline{AD}$ and $\overline{BE}$ cross at $C$, with $\overline{AB} \parallel \overline{DE}$. $AB = 8$, $DE = 12$ and $AC = 6$. Find $CD$.
$\triangle ABC \sim \triangle DEC$ by AA (vertical angles, alternate interior angles). $\displaystyle \frac{AC}{DC} = \frac{AB}{DE}$, so $\displaystyle \frac{6}{CD} = \frac{8}{12}$ and $CD = 9$.
Ready for more? Take the Unit 8 test.