Lesson 35: Similar and Congruent Polygons

New Concept

This course bridges arithmetic with algebra and geometry. We will use familiar math skills to describe the world around us in new ways.

What’s next

To begin, we will define similarity and congruence. Then, we'll apply these rules with worked examples to find unknown lengths and explore scale factors.

Property

Similar polygons have corresponding angles which are the same measure and corresponding sides which are proportional in length.

Examples

A rectangle with sides 4 and 6 is similar to one with sides 8 and 12, since $\frac{4}{8} = \frac{6}{12} = \frac{1}{2}$.
If quadrilateral ABCD $\sim$ EFGH, then $\angle A \cong \angle E$ and the side ratio $\frac{AB}{EF}$ is constant for all corresponding sides.

Explanation

Think of it like a photo and its smaller print. They show the same image, just in different sizes! All angles match perfectly, but side lengths are scaled by the same amount. It’s the same shape, just zoomed in or out.

Property

Congruent polygons have corresponding angles which are the same measure and corresponding sides which are the same length.

Examples

A triangle with sides 3, 4, 5 is congruent to another with sides 3, 4, 5, so $\triangle ABC \cong \triangle DEF$.
If square JKLM $\cong$ PQRS and side JK is 10 cm, then side PQ must also be 10 cm.

Explanation

These are identical twins! Not only do they have the same shape, but they also have the exact same size. You could place one directly on top of the other, and it would be a perfect match. No scaling needed, just maybe a flip or a turn!

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).