How can you use the SSS criterion for triangle congruence to show that the triangles below are congruent?

### SSS Triangles

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 all three pairs of sides are congruent. This is commonly referred to as “side-side-side” or “SSS”.

**The SSS** **criterion** **for triangle congruence states that if two triangles have** **three** **pairs of congruent** **sides,** **then the triangles are congruent.**

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

**Solve the following problems**

Perform a rigid transformation to bring point \begin{align*}E\end{align*} to point \begin{align*}B\end{align*}.

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

Rotate \begin{align*}\triangle 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*}.

Measure \begin{align*}\angle{ABD^\prime}\end{align*}. In this case, \begin{align*}m\angle{ABD^\prime}=26^\circ\end{align*}.

Rotate \begin{align*}\triangle D^\prime E^\prime F^\prime\end{align*} clockwise that number of degrees to create \begin{align*}\triangle 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*}.

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

Reflect \begin{align*}\triangle 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*}).

In this case, it looks like the triangles match up exactly and are therefore congruent, but how can you always be confident that \begin{align*}F^{\prime \prime}\end{align*} will map to \begin{align*}C\end{align*}? Consider the previous step, with the two triangles below:

You know that wherever \begin{align*}F^{\prime \prime}\end{align*} ends up after it is reflected, it has to stay 5 units away from \begin{align*}E^{\prime \prime}\end{align*} and 9 units away from \begin{align*}D^{\prime \prime}\end{align*}. Create a circle centered at \begin{align*}E^{\prime \prime}\end{align*} with radius 5 units to find all the points besides \begin{align*}F^{\prime \prime}\end{align*} that are 5 units away from \begin{align*}E^{\prime \prime}\end{align*}. Also create a circle centered at \begin{align*}D^{\prime \prime}\end{align*} with radius 9 units to find all the points besides \begin{align*}F^{\prime \prime}\end{align*} that are 9 units away from \begin{align*}D^{\prime \prime}\end{align*}. Notice that there are only two points in the whole plane that are both 5 units away from \begin{align*}E^{\prime \prime}\end{align*} and 9 units away from \begin{align*}D^{\prime \prime}\end{align*}: point \begin{align*}F^{\prime \prime}\end{align*} and point \begin{align*}C\end{align*}. Since reflections preserve distance, when \begin{align*}F^{\prime \prime}\end{align*} is reflected, it must end up at point \begin{align*}C\end{align*}. Therefore, a reflection will always map \begin{align*}\triangle D^{\prime\prime}E^{\prime\prime} F^{\prime\prime}\end{align*} to \begin{align*}\triangle ABC\end{align*} at this step.

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

**Examples**

**Example 1**

Earlier, you were asked how can you use the SSS criterion for triangle congruence to show that the triangles below are congruent.

Because these are right triangles, you can use the Pythagorean Theorem to find the third side of each triangle. The third side of each triangle will be \begin{align*}\sqrt{15^2-12^2}=9\end{align*}. Now you know that all three pairs of sides are congruent, so the triangles are congruent by SSS.

In general, anytime you have the hypotenuses congruent and one pair of legs congruent for two right triangles, the triangles are congruent. This is often referred to as “HL” for “hypotenuse-leg”. Remember, it only works for right triangles because you can only use the Pythagorean Theorem for right triangles.

#### Example 2

Are the following triangles congruent? Explain.

Yes, the triangles are congruent by SSS.

#### Example 3

Are the following triangles congruent? Explain.

There is not enough information to determine if the triangles are congruent. You need to know how the unmarked side compares to the other sides, or if there are right angles.

#### 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 HL?

You would need to know that the triangles are right triangles in order to use HL.

### Review

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

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

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

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

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

5.

6.

7.

8.

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 HL?

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

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

13. Point \begin{align*}A\end{align*} is the center of the circle below. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by SSS?

14. If you can show that two triangles are congruent by HL, can you also show that they are congruent by SAS?

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

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 3.4.