Helga is making a quilt for her grandmother. She plans to use a pattern that has medium sized squares and small squares. She wants the pattern to look uniform, so she plans to stop halfway through to make sure that the medium sized squares and the small squares are still similar. She sets up proportions that compare the medium sized squares to the small squares. If one side of the medium square is 3 inches and the proportional corresponding sides have a quotient of 3, what is the side length of the smaller square?

In this concept, you will learn about similarity.

### Similarity of Figures

Some figures look identical except they are different sizes. When you have figures that are proportional to each other, you call these figures **similar** figures. Similar figures have the same angle measures but different side lengths.

Squares are similar shapes because they always have four \begin{align*}90^\circ\end{align*} angles and four equal sides, even if the lengths of their sides differ. Other shapes can be similar too, if their angles are equal.

Let’s look at a pair of similar shapes.

Notice that the figures look the same, but one is smaller than the other. Since they are not the same size, they are not congruent. However, they have the same angles, so they are similar.

Unlike congruent figures, similar figures are not exactly the same. In similar figures, the angles are congruent, even if the sides are not.

Let's look at an example of similarity between figures.

Is the pair of figures below similar?

Each triangle has a \begin{align*}50^{\circ}\end{align*} angle. All three angles must be congruent, however, so let’s solve for the missing angle in each angle. Remember, the sum of the three angles is always \begin{align*}180^{\circ}\end{align*} for a triangle.

\begin{align*}&\text{Triangle 1} && \text{Triangle 2}\\ &50 + 60 + \text{angle} \ 3 = 180 && 50 + 80 + \text{angle} \ 3 = 180\\ &110 + \text{angle} \ 3 = 180 && 130 + \text{angle} \ 3 = 180\\ &\text{angle} \ 3 = 180 - 110 && \text{angle} \ 3 = 180 - 130 \\ &\text{angle} \ 3 = 70^{\circ} && \text{angle} \ 3 = 50^{\circ}\end{align*}

The angles in the first triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}60^{\circ}\end{align*}, and \begin{align*}70^{\circ}\end{align*}. The angles in the second triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}50^{\circ}\end{align*}, and \begin{align*}80^{\circ}\end{align*}. These triangles are not similar because their angle measures are different.

There is another way to prove similarity.

List the corresponding angles in the figures below and set up proportions for the side lengths .

Angles \begin{align*}G\end{align*} and \begin{align*}W\end{align*} are both right angles, so they correspond to each other. Imagine you can turn the figures to line up the right angles. You might even trace the small figure so that you can place it on top of the larger one.

Angles \begin{align*}H\end{align*} and \begin{align*}X\end{align*} correspond to each other. So do angles \begin{align*}I\end{align*} and \begin{align*}Y\end{align*} and angles \begin{align*}J\end{align*} and \begin{align*}Z\end{align*}. Now let's name these two quadrilaterals: \begin{align*}GHIJ\end{align*} is similar to \begin{align*}WXYZ\end{align*}.

The sides in similar figures are proportional. Proportions have the same ratio. Look at \begin{align*}GHIJ\end{align*} and \begin{align*}WXYZ\end{align*} again. Write each pair of sides as a proportion.

\begin{align*}\frac{GH}{WX},\frac{HI}{XY},\frac{IJ}{YZ},\frac{GJ}{WZ}\end{align*}

The sides from one figure are on the top, and the corresponding sides of the other figure are on the bottom. If the proportions each have the same quotient, the figure is similar.

Here is an example.

Is the pair of figures below similar?

First, let’s write out the pairs of proportional corresponding sides:

\begin{align*}\frac{6}{3}, \frac{6}{3}, \frac{4}{1}\end{align*}

The proportions show side lengths from the large triangle on the top and its corresponding side in the small triangle on the bottom. The pairs of sides must have the same proportion in order for the triangles to be similar.

Next, test whether the three proportions are the same by dividing each. If the quotient is the same, the pairs of sides must exist in the same proportion to each other.

\begin{align*}\frac{6}{3} = 2\\ \frac{6}{3} = 2\\ \frac{4}{1} = 4\end{align*}

Then, evaluate the results.

Only two pairs of sides have the same proportion. The third pair of sides does not exist in the same proportion as the other two, so these triangles cannot be similar.

The answer is that the figures are not similar.

### Examples

#### Example 1

Earlier, you were given a problem about Helga and her quilt.

She wants the pattern to be uniform and only includes similar squares. She sets up proportions that compare the medium sized squares to the small squares. If one side of the medium square is 3 inches and the proportional corresponding sides have a quotient of 3, what is the side length of the smaller squares?

First, determine which side length is the numerator.

3

Next, set up a proportion to match the given information.

\begin{align*}\frac{3}{x} = 3\end{align*}

Then, solve for the missing side.

\begin{align*}x = 1\end{align*}

The answer is that the side length of the smaller square is 1 inch.

#### Example 2

Is the pair of figures below similar?

First, note the given information.

The measures of three of the angles.

Next, check the relationship between corresponding angles.

You know the measure of three angles in each figure. In fact, they are all corresponding angles. As you know, the four angles in a quadrilateral must have a sum of \begin{align*}360^{\circ}\end{align*}. Because the three known angles are the same for both figures, you don’t even need to solve for the fourth to know that it will be the same in both figures.

Then, determine if the pair is similar.

These two figures are similar because their angle measures are all congruent.

The answer is that the pair is similar.

#### Example 3

Is the pair below similar?

First, look at the given information.

All of the angles are given.

Next, compare the angles to see if the corresponding angles are equal.

Yes.

Then, determine if the pair is similar.

These two figures are similar because their angle measures are all congruent.

The answer is that the pair is similar.

#### Example 4

The two figures below are similar. What is the measure of angle \begin{align*}Z\end{align*}?

First, look at the given information.

Three angles are provided for the first figure and one angle for the second figure.

Next, identify the angle that corresponds to angle \begin{align*}Z\end{align*}.

Angle \begin{align*}J\end{align*} is the corresponding angle.

Then, determine the value of angle \begin{align*}Z\end{align*}.

Angle \begin{align*}Z = 125^o\end{align*}.

The answer is that the measure of angle \begin{align*}Z\end{align*} is \begin{align*}125^o\end{align*}.

#### Example 5

List all of the pairs of corresponding sides in the figures below as proportions.

First, line up the figures by their angles.

Trace one figure and rotate it until it matches the other.

Next, choose a side to begin.

\begin{align*}OP\end{align*} and \begin{align*}RS\end{align*} are the shortest sides in each figure. They are proportional, so you can write

\begin{align*}\frac{OP}{RS}\end{align*}

Then, list the rest of the pairs.

\begin{align*}\frac{NO}{QR}, \frac{MP}{TS}, \frac{MN}{TQ}\end{align*}

The answer is \begin{align*}\frac{OP}{RS},\frac{NO}{QR}, \frac{MP}{TS}, \frac{MN}{TQ}\end{align*}.

### Review

Tell whether the pairs of figures below are congruent, similar, or neither.

- True or false. If triangles \begin{align*}DEF\end{align*} and \begin{align*}GHI\end{align*} are similar, then the side lengths are different but the angle measures are the same.
- True or false. Similar figures have exactly the same size and shape.
- Triangles \begin{align*}LMN\end{align*} and \begin{align*}HIJ\end{align*} are similar. If this is true, then the side lengths are the same, true or false.
- What is a proportion?
- True or false. To figure out if two figures are similar, then their side lengths form a proportion.
- Define similar figures

- Are the two figures similar or congruent?
- Define congruent figures.
- True or false. A proportion is formed by a pair of equal ratios.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.13.