Justin wants to make a kite, so he created the following pattern.

Justin needs to be sure that each triangle is congruent. He knows that there is a way to do this, but he can’t remember how to figure this out.

In this concept, you will learn to identify and apply theorems to test triangle congruence.

### Triangle Theorems

If you can identify the congruent sides and angles of polygons then you can use this to determine congruence. If the corresponding sides and the corresponding angles of any two polygons are congruent, then you know that the two polygons are also congruent.

You can also work with triangles, because after all a triangle is also a polygon. Triangles are unique because there are a few rules that can be used to help identify whether or not two triangles are congruent. If you learn these rules, then you won’t have to compare every angle and every side to determine whether or not two triangles are congruent.

The first rule represents the side-side-side, or **SSS**, relationship. It says that if each side of one triangle is congruent to a corresponding side of a second triangle, then the two triangles are congruent. It is not necessary to check the angles. In a triangle the longest side is always opposite the largest angle. The shortest side is always opposite the smallest angle. If the corresponding sides of two triangles are congruent then the corresponding angles opposite these sides are also congruent.

The second rule says that if one angle and the sides adjacent to it in one triangle are congruent to an angle and its adjacent sides in the second triangle, the triangles will be congruent. You call this the side-angle-side **(SAS)** rule. In other words, if two sides of a triangle and the angle they make are congruent to two sides and the angle they make of a second triangle, then the triangles are congruent. Remember that the angle must be located between the two sides.

The third rule says that if two angles and the side between them in one triangle are congruent to two angles and the side between them in the second triangle, the two triangles are congruent. This is the angle-side-angle **(ASA)** rule. You can determine congruence just by knowing two angles and one side.

Let’s look at an example.

Are the triangles below congruent? Explain your reasoning.

First, start by looking at the triangles and looking for the given information. Three side measurements must be known for SSS. A side measurement, an angle measure and a side measurement is necessary for SAS. An angle measure, a side measurement and an angle measure must be known for ASA.

For the above diagram, the sides which have a measurement of 4 cm are corresponding sides of the triangles. Likewise, the sides which have a measurement of 7.5 cm are also corresponding sides of the triangles. This is not information to call the triangles congruent.

Next, decide which rule can be used to check for congruence? The SSS relationship cannot be used because only the lengths of two sides are known. The SAS relationship can be used. Remember, to use SAS, the angle must be between the two sides. In the first triangle, the angle between the two sides is a right angle which has a measure of 90°. The second triangle also has a right angle between the 7.5 and 4 centimeter sides. Using the SAS rule, the triangles can be compared: 7.5 centimeters (side), 90° (angle), 4 centimeters (side).

The answer is the triangles are congruent by applying the SAS rule.

Let’s look at another example.

Are these two triangles congruent? Explain your reasoning.

First, start by looking at the triangles and looking for the given information. Three side measurements must be known for SSS. A side measurement, an angle measure and a side measurement is necessary for SAS. An angle measure, a side measurement and an angle measure must be known for ASA.

Next, decide which rule can you use to check for congruence? No measures are given for any of the angles in either triangle so neither the SAS nor the ASA rule can be used to show congruence. However, the measurements of the sides are given. The SSS rule can be used to compare the triangles to determine congruence. Remember the sides that are equal in length must be corresponding sides of the triangles. In the first triangle, the side lengths are 4 in (side), 5 in (side), and 6 in (side). These are the same measurements of the corresponding sides of the second triangle.

The answer is the triangles are congruent because of the SSS rule.

### Examples

#### Example 1

Earlier, you were given a problem about the pattern that Justin drew for his kite.

First, let’s think about which theorem he can use to prove that the two triangles are congruent. These triangles share a common side

, this side is congruent in both triangles. The same tic marks show that the other two sides are congruent as well.Next, since three sides of one triangle are congruent to three corresponding sides of the other triangle, then the triangles are congruent using the SSS Rule.

The answer is the triangles are congruent using the SSS Rule.

Justin has been successful in proving congruency.

#### Example 2

Use this illustration to answer the following question.

Which theorem could you use to prove that these triangles are congruent? Explain your thinking.

First, notice that no angle measures or side measurements have been given. However, same tic marks indicate same side lengths. Each side of triangle \begin{align*}ABC \end{align*}

The answer is the triangles are congruent because of the SSS rule.

Name the theorem that you could use to prove triangle congruence based on each description.

#### Example 3

You have been given two side lengths and the included angle measure in each of two triangles.

The answer is ASA.

#### Example 4

You have been given two angle measures and one side length in each of two triangles

The answer is SAS.

#### Example 5

You have been given three side lengths and no angle measures in each of two triangles.

The answer is SSS.

### Review

Use the given information to figure out each congruence statement.

1.

2.

3.

4.

5.

6.

7. If line segment

has a length of 8, which other segment also has a length of 8?8. If angle

has a measure of 55°, which other angle has a measure of 55°?9. If angle

has a measure of 45°, which other angle has a measure congruent to that?10. If these two triangles are congruent, are the side lengths and angle measures the same?

11. Would these two triangles look identical?

12. Which two letters would represent the vertex of each triangle?

Name the theorem that would best prove triangle congruence based on each description.

13. Three side lengths, 6 inches, 5 inches and 4 inches.

14. One angle measure and two side lengths.

15. Two angle measures and one side length.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.11.