Consider a triangle. Let ∠A be an exterior angle of the triangle, and let ∠B and ∠C be the remote interior angles of ∠A.
The measure of the exterior angle
A is equal to the sum of the measures of its remote interior angles by the .
m∠A=m∠B+m∠C
In a triangle, every interior angle has measure. Then, the sum of the measures of the
B and
C is greater than the measure of
∠B. Similarly, the sum of the measures of
∠B and
∠C is greater than the measure of
∠C.
m∠B+m∠C>m∠Bm∠B+m∠C>m∠C
Substituting
m∠A for
m∠B+m∠C in each inequality, the required inequalities are obtained.