Final answer:
An exterior angle of a triangle must always be greater in measure than either of its remote interior angles because it is equal to their sum. Since the sum of the interior angles of a triangle is 180 degrees, the exterior angle, being the sum of the two remote interior angles, will necessarily be larger than either of those angles.
Step-by-step explanation:
The reason it makes sense to claim that an exterior angle to a triangle must always be greater in measure than either of its two remote interior angles is because the exterior angle is equal to the sum of the two remote interior angles. This relationship is a consequence of the triangle's interior angles adding up to 180 degrees. By extending one side of the triangle, we create the exterior angle, which forms a linear pair with the adjacent interior angle, and the sum of these two angles must be 180 degrees. Therefore, the exterior angle is equal to 180 degrees minus the adjacent interior angle, which is the sum of the other two interior angles of the triangle.
For example, if we have a triangle with interior angles of 40 degrees, 60 degrees, and 80 degrees, and we extend the side opposite the 60-degree angle, we get an exterior angle. This exterior angle will be 120 degrees because it is equal to the sum of the 40-degree and 80-degree angles (which are the remote interior angles).
Because the sum of the two remote interior angles is always less than 180 degrees (since they are part of the triangle which totals 180 degrees), the exterior angle which equals their sum must necessarily be greater in measure than either of the individual remote interior angles.