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Third Angle Theorem

Third angles are the same if the other two sets are congruent.

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Third Angle Theorem

What if you were given and and you were told that and ? What conclusion could you draw about and ? After completing this Concept, you'll be able to make such a conclusion.

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CK-12 Foundation: Chapter4TheThirdAngleTheoremA

Guidance

Find and .

The sum of the angles in each triangle is by the Triangle Sum Theorem. So, for and . For , and is also .

Notice that we were given that and and we found out that . This can be generalized into the Third Angle Theorem.

Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent.

In other words, for triangles and , if and , then .

Notice that this theorem does not state that the triangles are congruent. That is because if two sets of angles are congruent, the sides could be different lengths. See the picture below.

Example A

Determine the measure of the missing angles.

From the markings, we know that and . Therefore, the Third Angle Theorem tells us that . So,

Example B

The Third Angle Theorem states that if two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. What additional information would you need to know in order to be able to determine that the triangles are congruent?

In order for the triangles to be congruent, you need some information about the sides. If you know two pairs of angles are congruent and at least one pair of corresponding sides are congruent, then the triangles will be congruent.

Example C

Determine the measure of all the angles in the triangle:

First we can see that . This means that also because they are alternate interior angles. was given. This means by the Triangle Sum Theorem that . This means that also because they are alternate interior angles. Finally, by the Triangle Sum Theorem.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter4TheThirdAngleTheoremB

Concept Problem Revisited

For two given triangles and , you were told that and .

By the Third Angle Theorem, .

Guided Practice

Determine the measure of all the angles in the each triangle.

1.

2.

3.

Answers:

1. , and by the Triangle Sum Theorem .

, and by the Triangle Sum Theorem, .

2. . By the Triangle Sum Theorem .

3. , and by the Triangle Sum Theorem, . because they are alternate interior angles and the lines are parallel. because they are alternate interior angles and the lines are parallel. because they are vertical angles.

Explore More

Determine the measures of the unknown angles.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.5. 

Vocabulary

Third Angle Theorem

Third Angle Theorem

If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles is also congruent.

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