Lesson 9-4: Similarity and Corresponding Parts

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

If two polygons are similar, their corresponding sides are proportional. You can set up an equation to solve for a missing side length by equating the ratios of corresponding sides:

For example, if $\Delta ABC \sim \Delta XYZ$, then:

Property

Angle-Angle (AA) Similarity Criterion:

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

Property

If two triangles have proportional corresponding side lengths, then the triangles are similar.

Examples

A triangle with sides 3, 5, 7 is similar to one with sides 6, 10, 14 because $\frac{3}{6} = \frac{5}{10} = \frac{7}{14} = \frac{1}{2}$.
Given $\triangle ABC$ with sides 5, 12, 13 and $\triangle XYZ$ with sides 10, 24, 26, then $\triangle ABC \sim \triangle XYZ$.

Explanation

If you can match up the sides of two triangles and find that they all share the same growth factor, you've proven they're similar! It's like checking if a recipe was just doubled; if all ingredients are scaled perfectly, the final dish will have the same taste (shape).