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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.
Two figures are called similar if there is some combination of translations, rotations, dilations, and reflections which makes them coincide with each other.
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.
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.
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 $\cl"blue"{∠B'}$.
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 $\cl"blue"{∠B'}$).
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 $\cl"blue"\ov{A'B'}$.
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:
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.
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 $\cl"red"{▵ABC}$.
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 $\cl"blue"{▵A'B'C'}$.
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:
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 $$3/2 = 1.5$$.
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.