To prove that the measure of the exterior angle of a triangle is equal to the sum of the measurements of the two remote interior angles, we can use the following steps:
1. Draw a triangle ABC and extend one of its sides, say BC, to form an exterior angle DBC.
2. Draw a parallel line to BC through A and label the point where it intersects BD as E.
3. By the alternate interior angles theorem, we have angle ABE = angle ABC and angle AED = angle ACB.
4. we have angle EDB = angle BCD by the corresponding angles theorem.
5. By adding the equal angles, we get angle ABE + angle AED + angle EDB = angle ABC + angle ACB + angle BCD.
6. we get angle AED + angle EDB = angle DBC by simplifying.
7. Therefore, the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.