# Similarity

You have already learned about rigid motions and dilations. Two figures that are related by a combination of rigid motions and dilations are called similar. Roughly speaking, this means they are the same shape as each other (but perhaps not the same size). In this lesson, you will study similarity, and learn about ways to determine that two triangles are similar to each other.

## Definition and properties of similarity

Two figures are called similar if there is some combination of translations, rotations, dilations, and reflections which makes them coincide with each other.

 Using the sliders to the left, what combination of translations, rotations, dilations, and reflections do you need to make the red sailboat coincide with the black sailboat? (Start by using translations to make the point \$\cl"red"A\$ coincide with the corresponding point on the black sailboat. Then use the other controls to get the rest of the red sailboat to coincide.)
 Are the red sailboat and the black sailboat similar?
 Using the sliders to the left, what combination of translations, rotations, dilations, and reflections do you need to make the red sailboat coincide with the black sailboat?
 Are the red sailboat and the black sailboat similar?

A combination of rigid motions and dilations is called a similarity transformation. A triangle \$\cl"red"{▵ABC}\$ is drawn to the left, with sliders that allow you to transform it using a similarity transformation made up of several dilations and rigid motions.

 As you slide the sliders, do the sides of \$\cl"red"{▵ABC}\$ remain straight?
 Do the measures of \$\cl"red"{▵ABC}\$’s angles change or stay the same?
 Do the lengths of its sides change or stay the same?

Because any rigid motion or dilation takes straight line segments to straight line segments and preserves angle measures, any similarity transformation does as well. Because any rigid motion preserves lengths, and any dilation multiplies every length by the same number, any similarity transformation multiplies every length by the same number.

## Triangle similarity — AA

Two triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ are drawn to the left. There is one slider that lets you change the measures of \$\cl"red"{∠A}\$ and \$\cl"blue"{∠A'}\$, and another slider that lets you change the measures of \$\cl"red"{∠B}\$ and \$\cl"blue"{∠B'}\$.

 When the measure of \$\cl"red"{∠A}\$ is 40˚ and the measure of \$\cl"red"{∠B}\$ is 80˚, what is the measure of \$\cl"red"{∠C}\$?
 When the measure of \$\cl"blue"{∠A'}\$ is 40˚ and the measure of \$\cl"blue"{∠B'}\$ is 80˚, what is the measure of \$\cl"blue"{∠C'}\$?
 However you slide the sliders, \$\cl"red"{∠A}\$ and \$\cl"blue"{∠A'}\$ will always have equal measure, as will \$\cl"red"{∠B}\$ and \$\cl"blue"{∠B'}\$. Will \$\cl"red"{∠C}\$ and \$\cl"blue"{∠C'}\$ always have equal measure?

As you can see:

If you have any two triangles (such as \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$) with two pairs of related angles that are equal in measure (for example, \$\cl"red"{∠A}\$ and \$\cl"blue"{∠A'}\$ are equal in measure, as are \$\cl"red"{∠B}\$ and \$\cl"blue"{∠B'}\$), then the third pair of related angles (in this case \$\cl"red"{∠C}\$ and \$\cl"blue"{∠C'}\$) will also be equal in measure.

In aa-to-aaa-qn, you saw that, in any two triangles with two pairs of related equal-measured angles, the third pair of related angles were also equal-measured. Now we’ll explore what else can be said about triangles with equal-measured related angles.

The two triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ drawn to the left both have a 45˚ angle (\$\cl"red"{∠A}\$ and \$\cl"blue"{∠A'}\$) and a 60˚ angle (\$\cl"red"{∠B}\$ and \$\cl"blue"{∠B'}\$).

 You can slide the slider to dilate \$\cl"red"{▵ABC}\$. How much do you need to dilate \$\cl"red"{▵ABC}\$ by to make the side \$\cl"red"\ov{AB}\$ have the same length as the side \$\cl"blue"\ov{A'B'}\$?
 Once the sides \$\cl"red"\ov{AB}\$ and \$\cl"blue"\ov{A'B'}\$ are equally long, the triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ have two pairs of corresponding equal-measured angles, and the corresponding sides between those angles have equal length. That is, the triangles have corresponding measurements equal for an Angle, then a Side, and then another Angle. Which congruence rule applies to two triangles like this: SAS, ASA, or SSS?
 If the slider is set so that the sides \$\cl"red"\ov{AB}\$ and \$\cl"blue"\ov{A'B'}\$ are equally long, are \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ congruent to each other?

Click . Now the original triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ are drawn, along with the dilated copy \$\cl"green"{▵A''B''C''}\$ of \$\cl"red"{▵ABC}\$ in which \$\cl"green"\ov{A''B''}\$ has the same length as \$\cl"blue"\ov{A'B'}\$.

 What rigid motion (combination of translations, rotations, and reflections) do you need to apply to the green triangle \$▵A''B''C''\$ to make it coincide with the blue triangle \$▵A'B'C'\$?
 You have found a dilation which takes the red triangle \$▵ABC\$ to the green triangle \$▵A''B''C''\$, and a combination of translations, rotations, and reflections which takes the green triangle \$▵A''B''C''\$ to the blue triangle \$▵A'B'C'\$. Is there a similarity transformation (combination of dilations, translations, rotations, and reflections) which takes the red triangle to the blue triangle?
 Are the triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ similar?

If you have two triangles with two pairs of corresponding equal-measured angles, as in \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$, then you can always make the triangles coincide with a similarity transformation in this way:

• Using a dilation, make the sides between the two angles equally long.
• Now, by the ASA criterion, these two triangles are congruent. Use rigid motions to make them coincide.

So the two triangles are similar. This is called the Angle-Angle or AA rule for similarity of triangles. Because of what you learned in similarity-transformation-qn, that also means that the length of every side of one of those triangles is the same multiple of the length of the corresponding side of the other triangle.

 Three triangles are drawn to the left, along with some of their measurements. Which triangle must be similar to the red triangle (\$\cl"red"{▵ABC}\$) by the AA rule: the blue triangle (\$\cl"blue"{▵A'B'C'}\$) or the green triangle (\$\cl"green"{▵A''B''C''}\$)?

## Triangle similarity — proportional sides

Two triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ are drawn to the left. Notice that every side of \$\cl"blue"{▵A'B'C'}\$ is twice as long as the corresponding side of \$\cl"red"{▵ABC}\$.

 You can slide the slider to dilate \$\cl"red"{▵ABC}\$. How much do you need to dilate \$\cl"red"{▵ABC}\$ by to make the side \$\cl"red"\ov{AB}\$ have the same length as the side \$\cl"blue"\ov{A'B'}\$?
When \$\cl"red"\ov{AB}\$ has the same length as \$\cl"blue"\ov{A'B'}\$, what are the lengths of \$\cl"red"\ov{BC}\$ and \$\cl"red"\ov{AC}\$?
 length of \$\cl"red"\ov{BC}\$: length of \$\cl"red"\ov{AC}\$:
 Are these lengths equal to the lengths of the sides \$\cl"blue"\ov{B'C'}\$ and \$\cl"blue"\ov{A'C'}\$ of \$\cl"blue"{▵A'B'C'}\$?
 When two triangles have all corresponding Side lengths equal, which congruence rule applies to those triangles: SAS, ASA, or SSS?
 If the slider is set so that the sides \$\cl"red"\ov{AB}\$ and \$\cl"blue"\ov{A'B'}\$ are equally long, are \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ congruent to each other?

Click . Now the original triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ are drawn, along with the dilated copy \$\cl"green"{▵A''B''C''}\$ of \$\cl"red"{▵ABC}\$ that has the same corresponding side lengths as \$\cl"blue"{▵A'B'C'}\$.

 What rigid motion do you need to apply to the green triangle \$▵A''B''C''\$ to make it coincide with the blue triangle \$▵A'B'C'\$?
 You have found a dilation which takes the red triangle \$▵ABC\$ to the green triangle \$▵A''B''C''\$, and a combination of translations, rotations, and reflections which takes the green triangle \$▵A''B''C''\$ to the blue triangle \$▵A'B'C'\$. Is there a similarity transformation (combination of dilations, translations, rotations, and reflections) which takes the red triangle to the blue triangle?
 Are the triangles \$\cl"red"{▵ABC}\$ and \$\cl"blue"{▵A'B'C'}\$ similar?

If you can multiply the side lengths of one triangle by a single number \$r\$ to get the side lengths of another triangle (the way each side of \$\cl"blue"{▵A'B'C'}\$ was 2 times as long as the corresponding side of \$\cl"red"{▵ABC}\$), then you can always make them coincide with a similarity transformation in this way:

• Using a dilation, make their corresponding sides equally long.
• Now, by the SSS criterion, these two triangles are congruent. Use rigid motions to make them coincide.

So the two triangles are similar. This is called the proportional sides rule for similarity of triangles. Because of what you learned in similarity-transformation-qn, that also means that every angle of each triangle has a corresponding equal-measured angle in the other triangle.

Three triangles are drawn to the left, along with some of their measurements. We would like to use the proportional sides rule to determine whether the blue triangle (\$\cl"blue"{▵A'B'C'}\$) or the green triangle (\$\cl"green"{▵A''B''C''}\$) is similar to the red triangle (\$\cl"red"{▵ABC}\$).

Notice that the shortest side of the red triangle has a length of 2, while the shortest side of the blue triangle has a length of 3. So, if you can multiply the side lengths of the red triangle by a single number to get the side lengths of the blue triangle, that number must be \$\$3/2 = 1.5\$\$.

 If you multiply the side lengths of the red triangle by 1.5, do you get the side lengths of the blue triangle?
 If you can multiply the side lengths of the red triangle by a single number to get the side lengths of the green triangle, what must that number be?
 If you multiply the side lengths of the red triangle by that number, do you get the side lengths of the green triangle?
 Which triangle must be similar to the red triangle (\$\cl"red"{▵ABC}\$) by the proportional sides rule: the blue triangle (\$\cl"blue"{▵A'B'C'}\$) or the green triangle (\$\cl"green"{▵A''B''C''}\$)?

## Non-similar figures

It’s possible to decide that two triangles are similar just by looking at their angles (by using the AA rule) or just by looking at their sides (by using the proportional sides rule). Now we’ll study whether or not there are rules like this which allow you to determine that two quadrilaterals (four-sided polygons) are similar.

 Two quadrilaterals \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ are drawn to the left, with a slider that dilates \$\cl"red"{ABCD}\$. Is there any way you can slide the slider so that \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ are congruent?
 Are \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ similar?
 Notice that all of the angles of both quadrilaterals are right angles, so they are all equal. Can there be a rule for quadrilaterals that tells you they are similar only by looking at their angles (like the AA rule for triangles)?
 Two quadrilaterals \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ are drawn to the left, with a slider that dilates \$\cl"red"{ABCD}\$. Is there any way you can slide the slider so that \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ are congruent?
 Are \$\cl"red"{ABCD}\$ and \$\cl"blue"{A'B'C'D'}\$ similar?
 Notice that both quadrilaterals have four equal sides, which means that however you dilate \$\cl"red"{ABCD}\$ the two quadrilaterals will have proportional sides. Can there be a rule for quadrilaterals that tells you they are similar only by looking at their sides (like the proportional sides rule for triangles)?