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# AA Similarity

## Two triangles are similar if two pairs of angles are congruent.

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AA Triangle Similarity

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

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

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

Example B

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

Solution: From Example A, you know that BE\begin{align*}\angle B^\prime \cong \angle E\end{align*}AF\begin{align*}\angle A^\prime \cong \angle F\end{align*} and AB¯¯¯¯¯¯¯FE¯¯¯¯¯\begin{align*}\overline{A^\prime B^\prime} \cong \overline{FE}\end{align*}. This means that ΔABCΔFED\begin{align*}\Delta A^\prime B^\prime C^\prime \cong \Delta FED\end{align*} by ASA\begin{align*}ASA \cong\end{align*}. Therefore, there must exist a sequence of rigid transformations that will carry ΔFED\begin{align*}\Delta FED\end{align*} to ΔABC\begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}.

ΔABCΔFED\begin{align*}\Delta ABC \sim \Delta FED\end{align*} because a series of rigid transformations will carry ΔFED\begin{align*}\Delta FED\end{align*} to ΔABC\begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}, and then a dilation will carry ΔABC\begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} to ΔABC\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. Congruent figures will always have corresponding angles and sides that are congruent as well.

similarity transformation is one or more rigid transformations followed by a dilation.

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. Similar figures will always have corresponding angles congruent and corresponding sides proportional.

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

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

1. 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*}.
2. 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*}.
3. 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.

### Vocabulary Language: English

AA Similarity Postulate

AA Similarity Postulate

If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
Dilation

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.
Triangle Sum Theorem

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.
Rigid Transformation

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.