4.6: Triangle Congruence using AAS
Learning Objectives
- Understand and apply the AAS Congruence Theorem.
AAS Congruence
Another way you can prove congruence between two triangles is by using two angles and the non-included side.
Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.
In the AAS Theorem, you use two _______________ and a _____________________________ side to prove congruence.
This is a theorem because it can be proven. First, we will do an example to see why this theorem is true, then we will prove it formally. Like the ASA Postulate, the AAS Theorem uses two angles and a side to prove triangle congruence. However, the order of the letters (and the angles and sides they stand for) is different.
The AAS Theorem is equivalent to the ASA Postulate because when you know the measure of two angles in a triangle, you also know the measure of the third angle. The pair of congruent sides in the triangles will determine the size of the two triangles. We will explore this further in the last section of this lesson.
Notice that when you look at the congruent triangles in a clockwise direction (beginning at \begin{align*}\angle C\end{align*} and \begin{align*}\angle Z\end{align*}), the congruent parts spell A-A-S, but when you look at them in a counter-clockwise direction, they spell S-A-A.
The AAS Theorem is similar to the _____________________________________________ because they both use two angles and a side to prove congruence.
Example 1
What information would you need to prove that these two triangles were congruent using the AAS Theorem?
A. the measures of sides \begin{align*}\overline{TW}\end{align*} and \begin{align*}\overline{XZ}\end{align*}
B. the measures of sides \begin{align*}\overline{VW}\end{align*} and \begin{align*}\overline{YZ}\end{align*}
C. the measures of \begin{align*}\angle VTW\end{align*} and \begin{align*}\angle YXZ \end{align*}
D. the measures of angles \begin{align*}\angle TWV\end{align*} and \begin{align*}\angle XZY \end{align*}
If you are to use the AAS Theorem to prove congruence, you need to know that pairs of two angles are congruent and the pair of sides adjacent to one of the given angles are congruent.
You already have one side and its adjacent angle, but you still need another angle. It needs to be the angle not touching the known side, rather than adjacent to it. Therefore, you need to find the measures of \begin{align*}\angle TWV \end{align*} and \begin{align*}\angle XZY\end{align*} to prove congruence.
Then you would have:
\begin{align*}\angle TWV & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;} (A)\\ \angle WVT & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;} (A)\\ \overline{VT} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;} (S)\end{align*}
These spell A-A-S.
The correct answer is D.
When you use AAS (or any triangle congruence postulate) to show that two triangles are congruent, you need to make sure that the corresponding pairs of angles and sides actually align.
When using triangle congruence postulates, it is important for ____________________________ angles and sides to match up.
For instance, look at the diagram below:
Even though two pairs of angles and one pair of sides are congruent in the triangles, these triangles are NOT congruent. Why?
Notice that the marked side in \begin{align*}\Delta TVW\end{align*} is \begin{align*}\overline{TV}\end{align*}, which is between the unmarked angle and the angle with two arcs.
However in \begin{align*}\Delta KML\end{align*}, the marked side is between the unmarked angle and the angle with one arc.
Since the corresponding parts DO NOT match up, you CANNOT use AAS to say these triangles are congruent.
If you want to prove that two triangles are ________________________________, you must be careful to make sure that _______________________________ parts of the triangles match up!
Reading Check:
1. In the space below, sketch two congruent triangles and mark the parts that are congruent by the AAS Theorem.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. In the space below, sketch two congruent triangles and mark the parts that are congruent by the ASA Postulate.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
AAS and ASA Congruence
The AAS Triangle Congruence Theorem is logically the exact same as the ASA Triangle Congruence Postulate. Look at the following diagrams to see why:
You can see the following in the figure on the previous page:
- \begin{align*}\angle C \cong \angle Z\end{align*} because both angles have two arcs
- \begin{align*}\angle B \cong \angle Y\end{align*} because both angles have one arc
The congruent parts in the figure spell A-A-S, so based on these markings you see, the triangles are congruent because of the AAS Theorem.
ALSO,
Since \begin{align*}\angle C \cong \angle Z\end{align*} and \begin{align*}\angle B \cong \angle Y\end{align*}, we can conclude from the Third Angle Theorem that \begin{align*}\angle A \cong \angle X\end{align*}. This is because the sum of the measures of the three angles in each triangle is \begin{align*}180^\circ\end{align*} and if we know the measures of two of the angles, then the measure of the third angle is already determined. We therefore know that all three angles in both triangles are congruent.
We know that \begin{align*}\angle A \cong \angle X\end{align*} because all three angles in a triangle add up to _______________.
Marking \begin{align*}\angle A \cong \angle X\end{align*}, the diagram becomes this:
Now we can see that:
- \begin{align*}\angle A \cong \angle X \ (A)\end{align*}
- \begin{align*}\overline{AB} \cong \overline{XY} \ (S)\end{align*}
- and \begin{align*}\angle B \cong \angle Y \ (A)\end{align*}
which shows that \begin{align*}\Delta ABC \cong \Delta XYZ\end{align*} is also true by the ASA Postulate.
Reading Check:
1. True/False: When you use the AAS Congruence Theorem OR the ASA Congruence Postulate, you can prove that all angles in both triangles are congruent with the Third Angle Theorem.
2. True/False: The AAS Theorem and the ASA Postulate are logically completely different.
Graphic Organizer for Lessons 3-5
Type of Congruency | Letters stand for... | Postulate or Theorem? | Draw a picture | Describe the corresponding congruent parts |
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SSS | ||||
SAS | ||||
ASA | ||||
AAS | ||||
HL |
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