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?

#### Watch This

https://www.youtube.com/watch?v=vqwXmupMpsA James Sousa: Similar Triangles Using Angle-Angle

#### Guidance

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.

**Example A**

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*}?

**Solution:** 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*}.

**Example B**

Use your work from Example A to prove that \begin{align*}\Delta ABC \sim \Delta FED\end{align*}.

**Solution:** From Example A, 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 Example A was two pairs of congruent angles. Therefore, you have proved that AA is a criterion for triangle similarity.**

**Example C**

Are the triangles below similar? Explain.

**Solution:** 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*}.

**Concept Problem Revisited**

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.

#### Vocabulary

** Rigid transformations** are transformations that preserve distance and angles. The rigid transformations are reflections, rotations, and translations.

Two figures are ** congruent** if a sequence of rigid transformations will carry one figure to the other.

**will always have corresponding angles and sides that are congruent as well.**

*Congruent figures*
A ** similarity transformation** is one or more rigid transformations followed by a dilation.

A ** dilation** is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point \begin{align*}P\end{align*}, multiplying the distance from the center point by a common scale factor, \begin{align*}k\end{align*}.

Two figures are ** similar** if a similarity transformation will carry one figure to the other.

**will always have corresponding angles congruent and corresponding sides proportional.**

*Similar figures*
** AA, or Angle-Angle**, is a criterion for triangle similarity. The AA criterion for triangle similarity states that if two triangles have two pairs of congruent angles, then the triangles are similar.

#### Guided Practice

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

**Answers:**

- 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*}.
- 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*}.
- 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*}.

#### Practice

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.*