Triangle Similarity Theorems

When two triangles are similar ($\sim$), they have the exact same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are proportional.

Instead of checking all three angles and all three sides to prove two triangles are similar, you can use three shortcuts: the AA, SAS, and SSS similarity theorems.

The Three Similarity Theorems

1. Angle-Angle (AA) Similarity

If two angles of one triangle are congruent (equal in measure) to two angles of another triangle, then the two triangles are similar.

• Why it works: Since all angles in a triangle add up to $180^\circ$, if two pairs of angles are equal, the third pair must automatically be equal.

2. Side-Angle-Side (SAS) Similarity

If two sides of one triangle are proportional to two sides of another triangle, and their included angles (the angles between the proportional sides) are equal, then the triangles are similar.

• Example: If $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D$, then $\triangle ABC \sim \triangle DEF$.

3. Side-Side-Side (SSS) Similarity

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

• Example: If $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$, then $\triangle ABC \sim \triangle DEF$.

Proving Similarity with Parallel Lines

A common geometry scenario involves a line drawn parallel to one side of a triangle, creating a smaller triangle inside a larger one.

Problem: Prove $\triangle ABC \sim \triangle ADE$ where $DE \parallel BC$ (with $D$ on side $AB$ and $E$ on side $AC$).

Proof: Because $DE$ is parallel to $BC$, the corresponding angles are equal:

1. $\angle ADE = \angle ABC$
2. $\angle AED = \angle ACB$

Since two pairs of angles are equal, by the AA Similarity Theorem, $\triangle ABC \sim \triangle ADE$. (Note: They also share $\angle A$, which is another way to establish AA).

Practical Application: Indirect Measurement

Similar triangles are incredibly useful for finding distances that are hard to measure directly, like the height of a tall tree or building.

Example Problem: A $6$-foot person casts a $4$-foot shadow. At the same time, a tree casts a $20$-foot shadow. Find the tree's height.

Solution: The sun's rays hit the ground at the same angle, creating two similar right triangles (AA Similarity). We can set up a proportion comparing the height to the shadow length:

$\frac{\text{Person's Height}}{\text{Person's Shadow}} = \frac{\text{Tree's Height}}{\text{Tree's Shadow}}$

$\frac{6}{4} = \frac{x}{20}$

To solve for $x$, cross-multiply:

$4x = 6 \times 20$ $4x = 120$ $x = 30$

By using similar triangles, we easily find that the tree is $30$ feet tall.