Why don't you have to verify that all three pairs of corresponding angles are congruent in order to show that two triangles are similar?

### AA Triangle Similarity

If two triangles are **similar** it means that all corresponding angle pairs are congruent and all corresponding sides are proportional. However, in order to be sure that two triangles are similar, you do not necessarily need to have information about all sides and all angles.

**The AA** **criterion for triangle similarity** **states that if two triangles have two pairs of congruent angles****, then the triangles are similar****.**

In the examples, you will use similarity transformations and criteria for triangle congruence to show why AA is a criterion for triangle similarity.

Let's take a look at some problems regarding AA triangle similarity.

1. Consider the triangles below. Dilate \begin{align*}\Delta ABC\end{align*} with a scale factor of \begin{align*}\frac{FE}{AB}\end{align*} to create \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}. What do you know about the sides and angles of \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}?

Below, \begin{align*}\Delta ABC\end{align*} is dilated about point \begin{align*}P\end{align*} with a scale factor of \begin{align*}\frac{FE}{AB}\end{align*} to create \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}.

Corresponding angles are congruent after a dilation is performed, so \begin{align*}\angle B \cong \angle B^\prime\end{align*}. Therefore, \begin{align*}\angle B^\prime \cong \angle E\end{align*} as well. Similarly, \begin{align*}\angle A \cong \angle A^\prime\end{align*}, and therefore \begin{align*}\angle A^\prime \cong \angle F\end{align*}. Because the scale factor was \begin{align*}\frac{FE}{AB}\end{align*}, \begin{align*}A^\prime B^\prime=\frac{FE}{AB} \cdot AB=FE\end{align*}. So, \begin{align*}\overline{A^\prime B^\prime} \cong \overline{FE}\end{align*}.

2. Use your work from #1 to prove that \begin{align*}\Delta ABC \sim \Delta FED\end{align*}.

From #1, you know that \begin{align*}\angle B^\prime \cong \angle E\end{align*}, \begin{align*}\angle A^\prime \cong \angle F\end{align*} and \begin{align*}\overline{A^\prime B^\prime} \cong \overline{FE}\end{align*}. This means that \begin{align*}\Delta A^\prime B^\prime C^\prime \cong \Delta FED\end{align*} by \begin{align*}ASA \cong\end{align*}. Therefore, there must exist a sequence of rigid transformations that will carry \begin{align*}\Delta FED\end{align*} to \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}.

\begin{align*}\Delta ABC \sim \Delta FED\end{align*} because a series of rigid transformations will carry \begin{align*}\Delta FED\end{align*} to \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}, and then a dilation will carry \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} to \begin{align*}\Delta ABC\end{align*}.

**All that was known about the original two triangles in #1 was two pairs of congruent angles. Therefore, you have proved that AA is a criterion for triangle similarity.**

Now, let's take a look at a problem about determining whether two triangles are similar.

Are the triangles below similar? Explain.

One pair of angles is marked as being congruent. You also have another pair of congruent angles due to the vertical angles in the center of the picture. Therefore, the triangles are similar by \begin{align*}AA \sim\end{align*}.

### Examples

#### Example 1

Earlier, you were asked why don't you have to verify that all three pairs of corresponding angles are congruent in order to show that two triangles are similar.

Why don't you have to verify that all three pairs of corresponding angles are congruent in order to show that two triangles are similar?

If two pairs of angles are congruent, then three pairs of angles must be congruent due to the fact that the sum of the measures of the interior angles of a triangle is \begin{align*}180^\circ\end{align*}. Therefore, three pairs of congruent angles is more information than you need.

#### Example 2

Prove that \begin{align*}\Delta CBA \sim \Delta GDA\end{align*}.

Note that \begin{align*}\overline{CB} \ \| \ \overline{GD}\end{align*}. This means that \begin{align*}\angle AGD \cong \angle ACB\end{align*} because they are corresponding angles. \begin{align*}\angle A\end{align*} is shared by both triangles. Two pairs of angles are congruent, so \begin{align*}\Delta CBA \sim \Delta GDA\end{align*} by \begin{align*}AA \sim\end{align*}.

#### Example 3

Prove that \begin{align*}\Delta CBA \sim \Delta CEF\end{align*}.

Note that \begin{align*}\overline{AB} \ \| \ \overline{FE}\end{align*}. This means that \begin{align*}\angle BAC \cong \angle EFC\end{align*} because they are corresponding angles. \begin{align*}\angle C\end{align*} is shared by both triangles. Two pairs of angles are congruent, so \begin{align*}\Delta CBA \sim \Delta CEF\end{align*} by \begin{align*}AA \sim\end{align*}.

#### Example 4

Prove that \begin{align*}\Delta CBA \sim \Delta GHF\end{align*}.

In #1 you found that \begin{align*}\angle AGD \cong \angle ACB\end{align*}. In #2 you found that \begin{align*}\angle BAC \cong \angle EFC\end{align*}. Two pairs of angles are congruent, so \begin{align*}\Delta CBA \sim \Delta GHF\end{align*} by \begin{align*}AA \sim\end{align*}.

### Review

1. What does AA stand for? How is it used?

2. Draw an example of two triangles that must be similar due to AA.

For each pair of triangles below, state if they are congruent, similar or not enough information. If they are similar or congruent, write a similarity or congruence statement. Explain your answer.

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14. Can you use the AA criteria to show that other shapes besides triangles are similar?

15. Use similarity transformations to explain in your own words why two triangles with two pairs of congruent angles must be similar. *Hint: Look at Examples A and B for help.*

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.3.