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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

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. This is called the Third Angle Theorem.

If and , then .

What if you were given and and you were told that and ? What conclusion could you draw about and ?

 

 

Examples

Example 1

Determine the measure of all the angles in each triangle.

 

. By the Triangle Sum Theorem .

Example 2

Determine the measure of all the angles in each triangle.

, 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.

Example 3

Determine the measure of the missing angles.

From the Third Angle Theorem, we know . From the Triangle Sum Theorem, we know that the sum of the interior angles in each triangle is .

Example 4

Explain why the Third Angle Theorem works.

The Third Angle Theorem is really like an extension of the Triangle Sum Theorem. Once you know two angles in a triangle, you automatically know the third because of the Triangle Sum Theorem. This means that if you have two triangles with two pairs of angles congruent between them, when you use the Triangle Sum Theorem on each triangle to come up with the third angle you will get the same answer both times. Therefore, the third pair of angles must also be congruent.

Example 5

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.

Review

Determine the measures of the unknown angles.

You may assume that .

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.5. 

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Vocabulary

Triangle Sum Theorem

The Triangle Sum Theorem states that the measure of the three interior angles of any triangle will add up to 180^\circ.

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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