Consider a triangle. Let $\angle A$ be an exterior angle of the triangle, and let $\angle B$ and $\angle C$ be the remote interior angles of $\angle A.$

The measure of the exterior angle $A$ 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 $B$ and $C$ is greater than the measure of $\angle B.$ Similarly, the sum of the measures of $\angle B$ and $\angle C$ is greater than the measure of $\angle C.$ Substituting $m\angle A$ for $m\angle B+m\angle C$ in each inequality, the required inequalities are obtained.