Rule

Exterior Angle Inequality Theorem

In a triangle, the measure of an exterior angle is greater than the measure of either of its remote interior angles.
An angle with an exterior angle marked

Based on the above diagram, the following inequalities hold true.

Proof

Exterior Angle Inequality Theorem

Consider a triangle. Let be an exterior angle of the triangle, and let and be the remote interior angles of

An angle with an exterior angle marked
The measure of the exterior angle is equal to the sum of the measures of its remote interior angles by the Triangle Exterior Angle Theorem.
In a triangle, every interior angle has positive measure. Then, the sum of the measures of the interior angles and is greater than the measure of Similarly, the sum of the measures of and is greater than the measure of
Substituting for in each inequality, the required inequalities are obtained.
Exercises
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