Similar Triangles

AA, SAS, and SSS similarity theorems

Similar Triangles

Definition

Two triangles are similar if:

All corresponding angles are congruent
All corresponding sides are proportional

Symbol: $\triangle ABC \sim \triangle DEF$

Similarity Postulates

AA (Angle-Angle)

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Note: If two angles match, the third must also match (angle sum = 180°)

SAS (Side-Angle-Side)

If two sides of one triangle are proportional to two sides of another triangle AND the included angles are congruent, the triangles are similar.

SSS (Side-Side-Side)

If all three pairs of corresponding sides are proportional, the triangles are similar.

Scale Factor

The ratio of corresponding sides: $k = \frac{\text{side in triangle 1}}{\text{corresponding side in triangle 2}}$

Properties

If $\triangle ABC \sim \triangle DEF$ with scale factor $k$ :

Perimeters are in ratio $k:1$
Areas are in ratio $k^2:1$

Applications

Finding unknown lengths
Indirect measurement
Proving geometric relationships

📚 Practice Problems

1Problem 1easy

❓ Question:

In $\triangle ABC$ , $\angle A = 50°$ and $\angle B = 60°$ . In $\triangle DEF$ , $\angle D = 50°$ and $\angle E = 60°$ . Are the triangles similar?

💡 Show Solution

Two angles of $\triangle ABC$ are congruent to two angles of $\triangle DEF$ :

$\angle A \cong \angle D$ (both $50°$ )
$\angle B \cong \angle E$ (both $60°$ )

By AA (Angle-Angle) similarity, the triangles are similar.

Answer: Yes, $\triangle ABC \sim \triangle DEF$ by AA

2Problem 2medium

❓ Question:

If $\triangle ABC \sim \triangle XYZ$ with sides $AB = 6$ , $BC = 8$ , $AC = 10$ and $XY = 9$ , find $YZ$ and $XZ$ .

💡 Show Solution

Find the scale factor using corresponding sides: $k = \frac{XY}{AB} = \frac{9}{6} = \frac{3}{2}$

For YZ (corresponds to BC): $\frac{YZ}{BC} = \frac{3}{2}$ $\frac{YZ}{8} = \frac{3}{2}$ $YZ = 12$

For XZ (corresponds to AC): $\frac{XZ}{AC} = \frac{3}{2}$ $\frac{XZ}{10} = \frac{3}{2}$ $XZ = 15$

Answer: $YZ = 12$ , $XZ = 15$

3Problem 3hard

❓ Question:

A tree casts a shadow 24 feet long at the same time a 6-foot person casts a 4-foot shadow. How tall is the tree?

💡 Show Solution

The sun creates similar triangles (same angle).

Set up proportion: $\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{person height}}{\text{person shadow}}$

$\frac{h}{24} = \frac{6}{4}$

Solve: $4h = 144$ $h = 36$

Answer: The tree is 36 feet tall

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