Section 2.5: Similar Figures

Property

Two triangles are similar if either one of the following conditions is true:

1. Their corresponding angles are equal.
2. Their corresponding sides are proportional.

Examples

• One triangle has angles $30^\circ$ and $70^\circ$. Another has angles $70^\circ$ and $80^\circ$. The third angle in the first is $80^\circ$ and in the second is $30^\circ$. Since all three corresponding angles are equal, the triangles are similar.
• A triangle has sides of length 5, 12, and 13. A second triangle has sides 10, 24, and 26. The ratios of corresponding sides are all equal to 2 ($\frac{10}{5} = \frac{24}{12} = \frac{26}{13} = 2$), so they are similar.
• A large triangle is formed by a 20-foot flagpole and its 15-foot shadow. A person who is 6 feet tall stands nearby. Their shadow forms a smaller, similar triangle. The length of the person's shadow, $s$, can be found by the proportion $\frac{6}{s} = \frac{20}{15}$, so $s=4.5$ feet.

Explanation

Triangles have a special shortcut for similarity. You only need to prove one of the two conditions. If their angles match, their sides must be proportional. If their sides are proportional, their angles must match. This makes them easier to work with.

Property

The symbol $\sim$ means "is similar to" and is used to write similarity statements between figures. When writing similarity statements like $\triangle ABC \sim \triangle DEF$, the order of vertices must match the correspondence between the figures.

Examples

Property

A proportion is a statement that two ratios are equal. We can clear the fractions from a proportion by cross-multiplying.

If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Examples

• To solve $\frac{x}{10} = \frac{3}{5}$, we cross-multiply to get $5x = 10(3)$. This gives $5x=30$, so $x=6$.