Theorems About Triangles
Rule

Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles, or remote interior angles.
Triangle with an exterior angle marked
Based on the diagram above, the following relation holds true.

Proof

Using Properties of Angles

Consider a triangle with vertices and and one of the exterior angles corresponding to

Triangle with an exterior angle marked
The diagram shows that and form a linear pair, so the sum of their measures is Additionally, by the Triangle Angle Sum Theorem, the sum of the angle measures of is
Now can be isolated in Equation (I).
Next, the expression of can be substituted into Equation (II).
Solve for
It has been proven that the measure of is equal to the sum of the measures of and Therefore, it can be said that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Proof

Using Transformations

Consider where and are the midpoints of and respectively. Let be one exterior angle of

Triangle with an exterior angle marked
Now, can be rotated about Since a rotation is a rigid motion, the image of after the rotation is congruent to Corresponding parts of congruent figures are congruent, so the measures of the angles and the lengths of the sides remain unchanged.
Triangle interior angles included side rotation
Since a rotation is equivalent to a reflection, is parallel to and is parallel to Therefore, is a parallelogram and is congruent to Now the parallelogram will be rotated about
Triangle exterior angle parallelogram rotation
By the Parallelogram Opposite Angles Theorem, is congruent to Congruent angles have the same measure by the definition.
Since is equal to the sum of and and because of the Transitive Property of Equality, is equal to the sum of and
This completes the proof.
Exercises
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