# 9.17: Corresponding Parts of Similar Figures

**At Grade**Created by: CK-12

**Practice**Corresponding Parts of Similar Figures

Remember the geometric skateboard park? Well, the students are working on all kinds of designs for the sidewalk that leads into the park.

Sam is designing some figures to be painted on the sidewalk for the skateboard park.

"I am going to use all primary colors," Sam told his friend Kara at lunch.

"Let me see," Kara said looking at his drawing.

Here are Sam's designs.

"Those aren't the same," Kara said looking at the drawing.

"I know that. These are similar figures. But I think they are cool."

"They are cool, but they aren't the same," Kara said again.

"That doesn't matter with similar figures. They have corresponding sides," Sam explained.

Kara is puzzled. She isn't sure how to identify corresponding sides of similar figures.

Do you know?

**In this Concept you will learn all about corresponding sides of similar figures, and you will know how to identify them too.**

### Guidance

We just finished identifying the corresponding parts of congruent figures, and we can also identify the corresponding parts of similar figures. We do this in the same way.

First, notice that all of the angle measures are the same. Whether figures are similar or congruent, the angle measures are the same in both.

The side lengths are different in similar figures. The side lengths are the same in congruent figures.

Triangle \begin{align*}ABC\end{align*} is similar to triangle \begin{align*}DEF\end{align*}. This means that while they are the same shape, they aren’t the same size. In fact, there is a relationship between the corresponding parts of the triangle.

**The side lengths are corresponding even though they aren’t congruent.**

\begin{align*}\overline{AB} \times \overline{DE} \\ \overline{BC} \times \overline{EF} \\ \overline{AC} \times \overline{DF}\end{align*}

**We use the symbol for similar ("~") to show the relationship between the corresponding sides of the two triangles.**

Use these similar figures to answer the following questions.

#### Example A

How many right angles are in the first two figures?

**Solution: 4 right angles**

#### Example B

What makes the two squares similar if they both have four right angles?

**Solution: The side lengths are different.**

#### Example C

In the triangle pair, are the two triangles similar or congruent? why?

**Solution: The triangles are similar because the side lengths are different. The angle measures are the same.**

Now back to Kara, Sam and the similar figures. Here is the original problem once again.

Sam is designing some figures to be painted on the sidewalk for the skateboard park.

"I am going to use all primary colors," Sam told his friend Kara at lunch.

"Let me see," Kara said looking at his drawing.

Here are Sam's designs.

"Those aren't the same," Kara said looking at the drawing.

"I know that. These are similar figures. But I think they are cool."

"They are cool, but they aren't the same," Kara said again.

"That doesn't matter with similar figures. They have corresponding sides," Sam explained.

Kara is puzzled. She isn't sure how to identify corresponding sides of similar figures.

When working with congruent figures, you had to match up the sides that were the same length. Similar figures aren't the same length, but based on the position of the figures, you can figure out which sides go together. These are the corresponding sides. Corresponding sides are in the same position on two different figures.

Here are the corresponding sides of these figures.

\begin{align*}\overline{HG}\end{align*} ~ \begin{align*}\overline{XW}\end{align*}

\begin{align*}\overline{HI}\end{align*} ~ \begin{align*}\overline{XY}\end{align*}

\begin{align*}\overline{IJ}\end{align*} ~ \begin{align*}\overline{XZ}\end{align*}

\begin{align*}\overline{GJ}\end{align*} ~ \begin{align*}\overline{WZ}\end{align*}

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Congruent
- having the same size and shape and measurement

- Similar
- having the same shape, but not the same size. Similar shapes are proportional to each other.

- Corresponding
- matching-corresponding sides between two triangles are sides that match up

### Guided Practice

Here is one for you to try on your own.

List all of the pairs of corresponding sides in the similar figures below.

**Answer**

Here are the pairs of corresponding sides.

\begin{align*}\overline{OP}\end{align*} and \begin{align*}\overline{RS}\end{align*}

\begin{align*}\overline{NO}\end{align*} and \begin{align*}\overline{QR}\end{align*}

\begin{align*}\overline{MP}\end{align*} and \begin{align*}\overline{TS}\end{align*}

\begin{align*}\overline{MN}\end{align*} and \begin{align*}\overline{TQ}\end{align*}

**This is our answer.**

### Video Review

Here are videos for review.

Khan Academy Congruent and Similar Triangles

James Sousa, Congruent and Similar Triangles

### Practice

Directions: Use the following figures to answer each question.

1. Are these two triangles similar or congruent?

2. How do you know?

3. Which side is congruent to \begin{align*}\overline{AB}\end{align*}?

4. Which side is congruent to \begin{align*}\overline{AC}\end{align*}?

5. Which side is congruent to \begin{align*}\overline{RS}\end{align*}?

6. Which angle is congruent to angle A?

7. Which angle is congruent to angle B?

8. Which angle is congruent to angle C?

9. Are the two figures similar or congruent?

10. Why?

11. Which side is congruent to \begin{align*}\overline{NO}\end{align*}?

12. Which side is congruent to \begin{align*}\overline{MN}\end{align*}?

13. Which side is congruent to \begin{align*}\overline{ST}\end{align*}?

14. Which side is congruent to \begin{align*}\overline{QT}\end{align*}?

15. Which side is congruent to \begin{align*}\overline{OP}\end{align*}?

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Congruent

Congruent figures are identical in size, shape and measure.Corresponding

The corresponding sides between two triangles are sides in the same relative position.Similar

Two figures are similar if they have the same shape, but not necessarily the same size.### Image Attributions

Here you'll learn to identify corresponding parts of similar figures.