To show two triangles are similar, do you need to check every angle and side? For triangles, two angles are enough, thanks to the angle sum.
Why it works
If two angles of one triangle equal two angles of another, the third angles are equal too, since each is $180^\circ$ minus the other two. Now dilate the first triangle until one side is as long as the matching side of the second. The angles don’t change, and a side with the same two angles at its ends fixes the whole triangle. So a rigid motion can place the scaled copy exactly on the second triangle, and the triangles are similar.
Key idea
Angle-angle (AA) similarity: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar, so their corresponding sides are proportional.
Worked examples
Worked example: Check the angles
One triangle has angles of $40^\circ$ and $75^\circ$. Another has angles of $75^\circ$ and $65^\circ$. Are they similar?
- Third angles: $180 - 40 - 75 = 65^\circ$, and $180 - 75 - 65 = 40^\circ$.
- Both triangles have angles $40^\circ$, $65^\circ$, $75^\circ$.
Answer: Yes, by AA.
Worked example: Shadows
A 6-foot person casts a 4-foot shadow. At the same moment a tree casts a 22-foot shadow. How tall is the tree?
- Both stand straight up (a right angle with the ground), and the sun’s rays hit both at the same angle. Two matching angles: the triangles are similar.
- Height over shadow is the same: $\frac{h}{22} = \frac{6}{4}$.
- $h = 22 \cdot 1.5 = 33$.
Answer: 33 feet
Worked example: A triangle inside a triangle
In $\triangle ABC$, segment $DE$ is parallel to $BC$, with $D$ on $AB$ and $E$ on $AC$. $AD = 3$, $AB = 9$ and $DE = 4$. Find $BC$.
- $\angle A$ is shared, and $\angle ADE = \angle ABC$ (corresponding angles, since $DE \parallel BC$). So $\triangle ADE \sim \triangle ABC$.
- Scale factor: $\frac{AB}{AD} = \frac{9}{3} = 3$.
- $BC = 3 \cdot 4 = 12$.
Answer: $12$
Common mistake
Using only one matching angle. Two triangles can share one angle and still have different shapes. You need two pairs of matching angles (and then the third matches automatically).