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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?

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 ΔABC with a scale factor of FEAB to create ΔABC. What do you know about the sides and angles of ΔABC?

Below, ΔABC is dilated about point P with a scale factor of FEAB to create ΔABC.

Corresponding angles are congruent after a dilation is performed, so BB. Therefore, BE as well. Similarly, AA, and therefore AF. Because the scale factor was FEAB, AB=FEABAB=FE. So, AB¯¯¯¯¯¯¯¯¯¯FE¯¯¯¯¯¯¯¯.

2. Use your work from #1 to prove that ΔABCΔFED.

From #1, you know that BEAF and AB¯¯¯¯¯¯¯¯¯¯FE¯¯¯¯¯¯¯¯. This means that ΔABCΔFED by ASA. Therefore, there must exist a sequence of rigid transformations that will carry ΔFED to ΔABC.

ΔABCΔFED because a series of rigid transformations will carry ΔFED to ΔABC, and then a dilation will carry ΔABC to ΔABC.

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 AA.

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 180. Therefore, three pairs of congruent angles is more information than you need.

Example 2

Prove that ΔCBAΔGDA.

Note that CB¯¯¯¯¯¯¯¯  GD¯¯¯¯¯¯¯¯. This means that AGDACB because they are corresponding angles. A  is shared by both triangles. Two pairs of angles are congruent, so ΔCBAΔGDA by AA.

Example 3

Prove that ΔCBAΔCEF.

Note that AB¯¯¯¯¯¯¯¯  FE¯¯¯¯¯¯¯¯. This means that BACEFC because they are corresponding angles. C is shared by both triangles. Two pairs of angles are congruent, so ΔCBAΔCEF by AA.

Example 4

Prove that ΔCBAΔGHF.

In #1 you found that AGDACB. In #2 you found that BACEFC. Two pairs of angles are congruent, so ΔCBAΔGHF by AA.

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. 

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Vocabulary

AA Similarity Postulate

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

Dilation

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

Triangle Sum Theorem

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

Rigid Transformation

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

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