## Congruent Triangles

While we can look at the dresses and see that they are similar, this doesn't constitute mathematical proof. So, how can you prove that two shapes are congruent or similar? Let's use a triangle, a common geometrical shape, to illustrate the relationships between similar and congruent shapes. First, we'll go over some ways to prove that two triangles are congruent. Remember, shapes that are congruent will be exactly the same size and shape.

There are several ways to prove that triangles are congruent. One way is to compare each of the three sides of a triangle, referred to as the **side-side-side (SSS)** proof. Two triangles are congruent if each side in the first triangle is congruent (equal) to a side in the second triangle.

Our second proof is referred to as **side-angle-side (SAS)**. Two triangles are congruent if two sides in one triangle and the angle in between them are congruent to the same sides and angle in the second triangle.

With the **angle angle side (AAS)** proof, we see that two triangles are congruent if two angles and one side (one that is not between the angles) in the first triangle are congruent to the same angles and side in the second triangle.

We can also look at a side between two angles. With the **angle side angle (ASA)**, we know that two triangles are congruent if two angles and the side between them are congruent to the same angles and included side in the second triangle.

If you can show that any of these are true, then you know that the triangles are congruent.

What about two triangles that have the same angle measurements? Are they congruent as well? They might be, but just knowing the angle measurements of the triangles is not enough to prove that the triangles are congruent. Similar triangles also have equal angle measurements, but the sides may be different lengths.

## Similar Triangles

Now that we know how to prove congruency, let's look at the ways to prove similarity between two shapes. The test for similar triangles is a bit simpler. Similar triangles have the same shape, but different sizes, so the only thing you need to prove to show that two triangles are similar is that the angles in the first triangle are equal to the angles in the second triangle. The side lengths can be different and the triangles will still be similar.

The side lengths in similar triangles will be proportional to each other, however, even when they are not the same length. Look at these two similar triangles and see if you can determine the length of the unknown side:

Because the triangles are similar, you know that the side lengths are proportional. This means that you can set up a ratio of side lengths to find the length of the unknown side.

Now, you can cross multiply to solve for the unknown side.

You can use a similar process to find the side lengths of any two similar triangles.

## Lesson Summary

In geometry, if two figures are **similar**, they are the same shape but different sizes. Triangles that are similar will have the same angle measurements, but may have sides that are different lengths. The side lengths of two similar triangles are directly proportional to one another, however, so you can construct a ratio of side lengths to find any that are not known. **Congruent** shapes are not only the same shape, but also the same size. Congruent triangles (and other congruent shapes), have the same angle measurements just as similar triangles do, but they also have the same side lengths. There are four ways that you can prove that two triangles are congruent:

**Side-side-side (SSS)**: Two triangles are congruent if each side in the first triangle is congruent (equal) to a side in the second triangle.
**Side-angle-side (SAS)**: Two triangles are congruent if two sides in one triangle and the angle in between them are congruent to the same sides and angle in the second triangle.
**Angle angle side (AAS)**: Two triangles are congruent if two angles and one side (that is not between the angles) in the first triangle are congruent to the same angles and side in the second triangle.
**Angle side angle (ASA)**: Two triangles are congruent if two angles and the side between them are congruent to the same angles and included side in the second triangle.