When two triangles have two pairs of sides and their included angles congruent, the triangles are congruent. What if the angles aren't included angles?

### SAS 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 G\cong \angle D\end{align*}, \begin{align*}\overline{IG}\cong\overline{FD}\end{align*}, and \begin{align*}\overline{GH}\cong\overline{DE}\end{align*}. The information you know about the congruent corresponding parts of these triangles is a **s**ide, an **a**ngle, and then another **s**ide. This is commonly referred to as “side-angle-side” or “SAS”.

**The** **SAS criterion** **for triangle congruence states that if two triangles have two pairs of congruent** **sides and the included angle in one triangle is congruent to the included angle in the other triangle, then** **the triangles are congruent.**

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

Let's take a look at some example problems.

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

2. Draw a vector from point \begin{align*}G\end{align*} to point \begin{align*}D\end{align*}. Translate \begin{align*}\triangle GHI\end{align*} along the vector to create \begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*}.

3. Rotate \begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*} to map to \begin{align*}\overline{G^\prime I^\prime}\end{align*} to \begin{align*}\overline{DF}\end{align*}.

4. Measure \begin{align*}\angle I^\prime DF\end{align*}. In this case, \begin{align*}m\angle I^\prime DF=148^\circ\end{align*}.

5. Rotate \begin{align*}\triangle G^\prime H^\prime I^\prime\end{align*} counterclockwise that number of degrees about point \begin{align*}G^\prime\end{align*} to create \begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*}. Note that because \begin{align*}\overline{GI}\cong\overline{DF}\end{align*} and rigid transformations preserve distance, \begin{align*}\overline{G^{\prime \prime}I^{\prime \prime}}\end{align*} matches up perfectly with \begin{align*}\overline{DF}\end{align*}.

6. Reflect \begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*} to map it to \begin{align*}\triangle DEF\end{align*}. Can you be confident that the triangles are congruent?

7. Reflect \begin{align*}\triangle G^{\prime\prime} H^{\prime\prime} I^{\prime\prime}\end{align*} across \begin{align*}\overline {G^{\prime\prime}I^{\prime\prime}}\end{align*} (which is the same as \begin{align*}\overline{DF}\end{align*}).

Because \begin{align*}\angle EDF\cong \angle{H^{\prime\prime} G^{\prime\prime} I^{\prime\prime}}\end{align*} and \begin{align*}\overline {G^{\prime\prime} H^{\prime\prime}}\cong \overline{DE}\end{align*}, the triangles must match up exactly (in particular, \begin{align*}H^{\prime \prime \prime}\end{align*} must map to \begin{align*}E\end{align*}), and the triangles are congruent.

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

**Examples**

**Example 1**

Earlier, you were asked what if the angles are not included angles.

Even though these triangles have two pairs of sides and one pair of angles that are congruent, the triangles are clearly not congruent. SSA is NOT a criterion for triangle congruence. In order to use two pairs of sides and one pair of angles to show that triangles are congruent, the pair of angles must be included between the pairs of congruent sides.

#### Example 2

Are the following triangles congruent? Explain.

The triangles are congruent by SAS.

#### Example 3

The triangles are not necessarily congruent. The given angle is not the included angle in both triangles.

#### Example 4

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

You would need to know that \begin{align*}\overline{AB}\cong\overline{AD}\end{align*}.

### Review

1. What does SAS stand for? What does it have to do with congruent triangles?

2. What does SSA stand for? What does it have to do with congruent triangles?

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

4. Draw an example of two triangles that **are not congruent** because all you know is SSA.

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

5.

6.

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9.

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

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

13. Do you think you always need at least three pairs of congruent sides/angles to show that two triangles are congruent? Explain.

14. If the two pairs of legs are congruent on two right triangles, are the triangles congruent? Explain. Draw a picture to support your reasoning.

15. Show how the SAS 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 3.3.