The information for the triangles below looks to be “AAS”. How could you use “ASA” to verify that the triangles are congruent?

### ASA and AAS Triangle Congruence

If two triangles are **congruent** it means that all corresponding angle pairs and all corresponding sides are congruent. However, in order to be sure that two triangles are congruent, you do not necessarily need to know that all angle pairs and side pairs are congruent. Consider the triangles below.

In these triangles, you can see that \begin{align*}\angle A \cong \angle D\end{align*}, \begin{align*}\angle B \cong \angle E\end{align*}, and \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}. The information you know about the congruent corresponding parts of these triangles is an **a**ngle, a **s**ide, and then another **a**ngle. This is commonly referred to as “angle-side-angle” or “ASA”.

**The** **ASA criterion** **for triangle congruence states that if two triangles have two pairs of congruent angles and the common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.**

In the examples, you will use rigid transformations to show why the above ASA triangles must be congruent overall, even though you don't know the lengths of all the sides and the measures of all the angles.

“Angle-angle-side” or “AAS” is another criterion for triangle congruence that directly follows from ASA.

**The AAS criterion** **for triangle congruence states that if two triangles have two pairs of congruent angles** **and a non-****common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.**

#### Performing a Rigid Transformation

Perform a rigid transformation to bring point \begin{align*}D\end{align*} to point \begin{align*}A\end{align*}.

#### Drawing a Vector

Draw a vector from point \begin{align*}D\end{align*} to point \begin{align*}A\end{align*}. Translate \begin{align*}\Delta DEF\end{align*} along the vector to create \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*}.

#### Rotating Triangles

1. Rotate \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*} to map \begin{align*}\overline{D^\prime E^\prime}\end{align*} to \begin{align*}\overline{AB}\end{align*}.

2. Measure \begin{align*}\angle B D^\prime E^\prime\end{align*}. In this case, \begin{align*}m \angle BD^\prime E^\prime=84^\circ\end{align*}.

3. Rotate \begin{align*}\Delta D^\prime E^\prime F^\prime\end{align*} clockwise that number of degrees about point \begin{align*}D^\prime\end{align*} to create \begin{align*}\Delta D^{\prime\prime} E^{\prime\prime} F^{\prime\prime}\end{align*}. Note that because \begin{align*}\overline{DE} \cong \overline{AB}\end{align*} and rigid transformations preserve distance, \begin{align*}\overline{D^{\prime\prime}E^{\prime\prime}}\end{align*} matches up perfectly with \begin{align*}\overline{AB}\end{align*}.

#### Performing Reflections

1. Reflect \begin{align*}\Delta D^{\prime\prime}E^{\prime\prime}F^{\prime\prime}\end{align*} to map it to \begin{align*}\Delta ABC\end{align*}. Can you be confident that the triangles are congruent?

2. Reflect \begin{align*}\Delta D^{\prime\prime}E^{\prime\prime}F^{\prime\prime}\end{align*} across \begin{align*}\overline{D^{\prime\prime}E^{\prime\prime}}\end{align*} (which is the same as \begin{align*}\overline{AB}\end{align*}).

Because \begin{align*}\angle F^{\prime\prime}D^{\prime\prime}E^{\prime\prime} \cong \angle CAB\end{align*} and \begin{align*}\angle F^{\prime\prime}E^{\prime\prime}D^{\prime\prime} \cong \angle CBA\end{align*}, the triangles must match up exactly (in particular, \begin{align*}F^{\prime\prime\prime}\end{align*} must map to \begin{align*}C\end{align*}), and the triangles are congruent.

This means that even though you didn't know all the side lengths and angle measures, because you knew two pairs of angles and the included sides were congruent, the triangles had to be congruent overall. At this point you can use the ASA criterion for showing triangles are congruent without having to go through all of these transformations each time (but make sure you can explain why ASA works in terms of the rigid transformations!).

**Examples**

**Example 1**

Earlier, you were asked how could you use "ASA" to verify that the triangles are congruent.

Because the three angles of a triangle always have a sum of \begin{align*}180^\circ \end{align*}, \begin{align*}m \angle B=56^\circ\end{align*} and \begin{align*}m \angle E=56^\circ\end{align*}. Therefore, the triangles are congruent by ASA due to the fact that \begin{align*}\angle A \cong \angle D\end{align*}, \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}, \begin{align*}\angle B \cong \angle E\end{align*}.

This example shows how if ASA is a criterion for triangle congruence, then AAS must also be a criterion for triangle congruence.

#### Example 2

Are the following triangles congruent? Explain.

The triangles are congruent by ASA.

#### Example 3

Are the following triangles congruent? Explain.

The triangles are not necessarily congruent. The information for \begin{align*}\Delta ABC\end{align*} is AAS while the information for \begin{align*}\Delta EFG\end{align*} is ASA. There is not enough information about corresponding sides that are congruent.

#### Example 4

Are the following triangles congruent? Explain.

What additional information would you need in order to be able to state that the triangles below are congruent by AAS?

You would need to know that \begin{align*}\angle G \cong \angle C\end{align*}.

### Review

1. What does ASA stand for? How is it used?

2. What does AAS stand for? How is it used?

3. Draw an example of two triangles that must be congruent due to ASA.

4. Draw an example of two triangles that must be congruent due to AAS.

For each pair of triangles below, state if they are congruent by ASA, congruent by AAS, or if there is not enough information to determine whether or not they are congruent.

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10. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by AAS? Assume that points \begin{align*}B\end{align*}, \begin{align*}C\end{align*}, and \begin{align*}E\end{align*} are collinear.

11. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by ASA? Assume that points \begin{align*}B\end{align*}, \begin{align*}C\end{align*}, and \begin{align*}E\end{align*} are collinear.

12. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by AAS?

13. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by ASA?

14. If you can show that two triangles are congruent with AAS, can you also show that they are congruent with ASA?

15. Show how the ASA criterion for triangle congruence works: use rigid transformations to help explain why the triangles below are congruent.

### Review (Answers)

To see the Review answers, open this PDF file, and look for section 3.2.