Similar Triangles: AA, SSS, SAS Similarity
Define similarity as equal corresponding angles and proportional corresponding sides. Apply AA, SSS~, and SAS~ similarity criteria to prove triangles similar and find unknown side lengths using proportions.
A lesson of Geometry, in Triangle Relationships, about 10 minutes.
By the end of this lesson, you will be able to prove triangles similar using the AA, SSS~, and SAS~ similarity postulates/theorems, find missing sides in similar triangles using proportions, and apply similarity to real-world problems.
The Big Idea
Similar figures have the same shape but not necessarily the same size. For triangles, similarity means corresponding angles are equal and corresponding sides are proportional. Once two triangles are proven similar, their side ratios immediately give you the means to compute any unknown length.
Similarity is the foundation of trigonometry, scale models, maps, and photography.
Definition
$\triangle ABC \sim \triangle DEF$ (read "triangle $ABC$ is similar to triangle $DEF$") if and only if:
- Corresponding angles are congruent: $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$
- Corresponding sides are proportional: $\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}$
The common ratio is the scale factor.
AA — Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
You only need two pairs — the third pair is automatic (Triangle Angle-Sum).
This is the most commonly used similarity shortcut.
SSS~ — Side-Side-Side Similarity
If the ratios of all three pairs of corresponding sides are equal, the triangles are similar.
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$
SAS~ — Side-Angle-Side Similarity
If two pairs of corresponding sides are proportional AND the included angles are congruent, the triangles are similar.
$$\frac{AB}{DE} = \frac{BC}{EF} \quad \text{and} \quad \angle B \cong \angle E$$
Example 1 — AA Similarity
$\triangle ABC$ has $\angle A = 50°$ and $\angle B = 70°$. $\triangle DEF$ has $\angle D = 50°$ and $\angle F = 60°$.
$\angle C = 60°$ and $\angle E = 60°$.
$\angle A = \angle D$ and $\angle B = \angle E$ → AA → $\triangle ABC \sim \triangle DEF$.
Example 2 — Finding Missing Sides
$\triangle ABC \sim \triangle DEF$. $AB = 6$, $BC = 8$, $AC = 10$; $DE = 9$. Find $EF$ and $DF$.