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4.11: Third Angle Theorem

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Third angles are equal if the other two sets are each congruent.

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 AD and BE, then CF.

f-d_471d734fd862d213f14b32d06f34ffc18e11324d937d1ca29dec099c+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.1

What if you were given ΔFGH and ΔXYZ and you were told that FX and GY? What conclusion could you draw about H and Z?

Example 4.11.1

Determine the measure of all the angles in each triangle.

f-d_56d05c34153466a65ae6254dbc9e4f76e75ca66606faebf3975c19e6+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.2

Solution

mC=mA=mY=mZ=35. By the Triangle Sum Theorem mB=mX=110.

Example 4.11.2

Determine the measure of all the angles in each triangle.

f-d_282193c299a005b0e5b8aafc431192df59ad3de99ac2df1e6cf46476+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.3

Solution

mA=28, mABE=90 and by the Triangle Sum Theorem, mE=62. mD=mE=62 because they are alternate interior angles and the lines are parallel. mC=mA=28 because they are alternate interior angles and the lines are parallel. mDBC=mABE=90 because they are vertical angles.

Example 4.11.3

Determine the measure of the missing angles.

f-d_28fc2a1fee2abf230cfe9034dc130e4ec09fe93ba86e16490597b79a+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.4

Solution

From the Third Angle Theorem, we know CF. From the Triangle Sum Theorem, we know that the sum of the interior angles in each triangle is 180.

mA+mB+mC=180mD+mB+mC=18042+83+mC=180mC=55=mF

Example 4.11.4

Explain why the Third Angle Theorem works.

Solution

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 4.11.5

Determine the measure of all the angles in the triangle:

f-d_da675016fc17472beb403a2cca18997a80c652648e6d9d4ee078cf90+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.5

Solution

First we can see that mDCA=15. This means that mBAC=15 also because they are alternate interior angles. mABC=153 was given. This means by the Triangle Sum Theorem that mBCA=12. This means that mCAD=12 also because they are alternate interior angles. Finally, mADC=153 by the Triangle Sum Theorem.

Review

Determine the measures of the unknown angles.

f-d_cf693ed363f17f9595fb7a76897e06181b3298dbd21456d9bc520b3b+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.6
  1. Y
  2. x
  3. N
  4. L

f-d_c359e41cca6a5bff9b26bc1974d60c87ea6a7e8da0d84f28c68e39af+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.7
  1. E
  2. F
  3. H
f-d_99224d57d978d8467f1dca05760e7afd5022d151a11e6b3bb0b93e2a+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.8

You may assume that BCHI.

  1. ACB
  2. HIJ
  3. HJI
  4. IHJ
f-d_d81c43676fa4d30a28ca49e8a9aa7fdf9258e9f44f8e2d72f66c7941+IMAGE_TINY+IMAGE_TINY.png
Figure 4.11.9
  1. RQS
  2. SRQ
  3. TSU
  4. TUS

Review (Answers)

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

Vocabulary

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

Additional Resources

Video: The Third Angle Theorem Principles - Basic

Activities: Third Angle Theorem Discussion Questions

Study Aids: Triangle Congruence Study Guide

Practice: Third Angle Theorem

Real World: Third Angle Theorem

  1. RQS
  2. SRQ
  3. TSU
  4. TUS

4.11: Third Angle Theorem is shared under a CC BY-NC license and was authored, remixed, and/or curated by LibreTexts.

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