## Table of Contents

- AAS Theorem
- AAS Proof
- AAS (Angle-Angle-Side) Property
- Examples of AAS (Angle-Angle-Side)
- Lesson Summary

Angle-angle-side (AAS) congruence is used to prove two triangles are congruent. The proof of AAS congruency is simple, and examples are included.
Updated: 12/16/2021

- AAS Theorem
- AAS Proof
- AAS (Angle-Angle-Side) Property
- Examples of AAS (Angle-Angle-Side)
- Lesson Summary

The **angle-angle-side congruency**, or **AAS**, is a theorem that allows the determination of whether two triangles are congruent. Two triangles are **congruent** if they have three sides of the same length and three angles of the same measure. The theorem states that if two triangles have two equal angles and one side adjacent to only one of the angles, then the two triangles are congruent.

The two triangles above share an equal side adjacent, and the two angles are equal. The equal side is only adjacent to one of the angles that are equal between the two triangles. This is why it is known as angle-angle-side: there are two angles next to each other and then one side next to only one of the angles. It reads as angle, angle, side. This is different from **angle-side-angle**, or ASA, where the sides that are equal are between the two angles. In order to use AAS, all that is necessary is identifying two equal angles in a triangle, then finding a third side adjacent to only one of the angles in each of the triangles such that the two sides are equal. This is enough to prove the two triangles are congruent.

The **proof** of the AAS congruence theorem is simple. To start, consider two triangles ABC and DEF such that there are two equal angles and one equal side adjacent to only one of the angles.

As above, the angle A equals and angle E, and the angle B equals angle F. Notice that because the sum of the interior angles of a triangle equal 180 degrees, it is true that A + B + C = 180, so C = 180 - A - B = 180 - (A + B), and D + E + F = 180, so D = 180 - E - F = 180 - (E + F). Furthermore, A + B = E + F. Thus, it is true that angle C has the same measure as angle D. This means that there are two equal angles with an equal side between them.

Therefore, by the ASA congruency theorem, the triangle ABC is congruent to the triangle DEF, because A = E, C = D, and |AC| = |DE|.

There are two necessary properties for two triangles to be congruent by angle-angle-side. First, the two triangles must have two equal angles. Then, there must be a side adjacent to only one of the angles in each triangle such that the sides are equal. AAS is not used when the equal side is between the two angles. In that case, use the angle-side-angle, or ASA, theorem. If there are three equal sides, then use the side-side-side, or SSS, theorem. If there are two sides that are equal and one angle between the two sides that are equal, then use the side-angle-side, or SAS, theorem. It is important to note that if there are two equal sides and an angle adjacent to only one of those sides, known as side-side-angle or SSA, then the two triangles may not be congruent. Furthermore, if there are only two equal angles and no other information is given, it is impossible to determine whether the two triangles are congruent.

Consider two triangles arranged like so:

It is given that BC is parallel to EF, angle C is equal in measure to angle F, and |AD| = |BE|. Then, it is true that B = E, because corresponding angles of parallel lines are congruent. Then, because |AD| = |BE|, it is true that |AD| + |BD| = |BE| + |BD| by the addition property of equality. Furthermore, |AD| + |BD| = |AB|, |BE| + |BD| = |DE| by the diagram. Use the substitution property to find that |AB| = |DE|. So, it is true that C = F, B = E, and |AB| = |DE|. In other words, there are two equal angles and a side adjacent to only one of the angles such that the two sides are equal. This satisfies the AAS theorem. Therefore, triangle ABC is congruent to triangle DEF.

Another example uses two triangles arranged in a bow-tie manner.

It is given that AC is parallel to DE, and B is the midpoint of AE. First, notice that |AB| = |BE| because B is the midpoint of AE. Then, angle A is equal in measure to angle E because the alternate interior angles of two parallel lines are congruent. Furthermore, angle C is equal in measure to angle D for the same reason. Thus, there are two equal angles and an equal side. This satisfies the AAS theorem. Therefore, triangle ABC is congruent to triangle BDE.

For an example of how it is used when the angle measures and the side is given, consider the example from earlier. It is given that there is an angle of measure 53.1 degrees, an angle of measure 90 degrees, and a side adjacent only to the 90-degree angle of length 4. Although the triangles are positioned differently, it must be true that by AAS the two triangles are congruent, because they share two equal angles and a side of equal length.

The **angle-angle-side** (**AAS**)theorem is used to determine when two triangles are congruent. By the theorem, two triangles are **congruent** if they have two angles of equal measure and one side adjacent to only one of the angles that are equal in length. It differs from the **angle-side-angle** theorem because angle-side-angle requires the sides that are equal to be between the two equal angles. Other theorems include the side-side-side theorem, which states that two triangles are congruent if they have three equal sides, and the side-angle-side theorem, which states that two triangles are congruent if they have two equal sides with an equal angle between them. If there are two or three equal angles, then congruency cannot be determined. Furthermore, side-side-angle, where two sides are equal and there is an equal angle adjacent to only one of the sides, does not determine that the two triangles are equal. The **proof** of the angle-angle-side theorem requires finding that the third angle is equal between the two triangles. Then, it follows from angle-side-angle theorem that the two triangles are congruent.

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Frequently Asked Questions

The theorem states that if two triangles have two equal angles and an equal side on each triangle adjacent to only one of the equal angles, then the two triangles are congruent.

The AAS, or angle-angle-side, congruency rule states that if two triangles have two equal angles and a side adjacent to only one of the angles that are equal, then the two triangles are congruent.

The rule states that if two triangles share two equal angles and a side that is equal in each triangle, but is only adjacent to one of the two equal angles, then the triangles are congruent.

Consider two triangles ABC and DEF lied side by side such that AD = BE , and angle C equals angle F, and BC is parallel to EF. Then the two triangles are congruent.

Or, consider two triangles ABC and BDE arranged in a bowtie such that AB is parallel to DE and B is the midpoint of AE. Then the two triangles are congruent.

If two angles and a side are given, then it is possible to use the AAS rule if the two angles are equal and the length of the sides are equal.

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