The interior angles of all triangles sum to 180°. If you know the measure of any two angles, you can easily find the third. That is the "secret" at the heart of the Angle-Angle Similarity Criterion, which says all pairs of triangles with two congruent interior angles are similar.

## Angles

Interior angles of triangles are formed by their sides. Their sides can be any length; in other words, the available measurements of triangle's sides are infinite.

Interior angles must adhere to different rules. The three angles must sum to 180°, so those three angles must fit into these three categories:

1. All acute, or
2. One obtuse and two acute, or
3. One right and two acute

That's it. No other possibilities exist for the interior angles, which means if you know the measure of any two angles, you know the measure of the third, by subtraction:

• 180° - TKA -TKB =TKC

The little symbol, TK , means "a measured angle."

[Scripted will not allow the measured angle symbol, which I have replaced with TK for editing purposes; please correct in final version]

The word criterion is the singular of criteria. It is the standard, the basis for judging something. You might set a criterion for purchasing a video game by requiring that it costs you no more than two weeks' allowance, for example.

The Angle-Angle Similarity Criterion tells us, "If two interior angles of two triangles are congruent, then the two triangles are similar."

## Similarity

Two polygons are similar if they meet two criteria:

1. Their interior angles are congruent (identical in measure)
2. Their sides are proportional; that is, the ratio of the measurements of one triangle is equal to the ratio of the measurements of the other triangle

When polygons are similar, one will appear to be a larger or smaller version of the other. This is called dilation, and in triangle is looks like this:

[insert drawing of two right 3-4-5 triangles, such as 3-4-5, 15-20-25; better, consider a video showing a small triangle growing and then sliding over to match the larger one]

Sometimes you can understand a concept better with a non-example. Here are two triangles of different size but also different proportions, so they cannot be similar:

[insert drawing small equilateral triangle and larger right triangle]

The angles are not equal and the sides are not proportional. These two triangles are not similar.

## AA Criterion

The Angle-Angle Similarity Criterion tells us that two known angles of two triangles can be congruent. What does this force us to conclude about the third angle? It also has to match, triangle for triangle. You cannot find two algebraic expressions that do this:

• A + B + C = 180° AND
• A + B + D = 180°

If, between the two triangles, you find ∠A and ∠B congruent, then ∠C must also be congruent for the three angles to sum to 180°.

Once you have three congruent angles, you have similar triangles. If you do not believe this, attempt to draw two triangles with congruent angles whose sides are not proportional to each other. You cannot do it.

## Proceed with Caution!

Even if two triangles are turned in different directions, are different sizes, or are mirror images of each other, they will still be similar if two of their interior angles are congruent.

Transformations (rotations, dilations, or reflections) do not affect similarity. Here are three pairs of similar triangles. For the first pair, one is rotated; the second pair are dilated. The third pair are reflections:

[insert drawing as described, possibly with equilateral triangles for the first pair, isosceles for the second, and right triangles for the third pair]

## Legitimate Shortcut

You probably have been cautioned against taking shortcuts in geometry. The Angle-Angle Similarity Criterion is a legitimate shortcut. Once you know two interior angles of two triangles are congruent, you should have no worries about relying on the AA Criterion to say the two triangles are similar

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Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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