Explanation

We are asked to find the sum of the angle measures . On to the statements! Separately first!

Statement (1) gives us the value of a. The angles a and b are both interior to a quadrilateral, and c and d are both supplementary to the other angles of the quadrilateral:



Therefore, . With the value of a, this doesn't give us b+d. One method to see this, a technique to analyze by cases in geometry, is to try to stretch or move elements of the figure, and see whether the stated data allow it. For example, the angle a is fixed here, so we can't move the top or left side of this quadrilateral freely. However, nothing stops us from grabbing the right side in our mind and tilting it left or right. As we do so, the complement of the angle not contributed by is passed back and forth, shared in different measure, by the angles and . That means that d can vary freely and thus so can . Insufficient.

Statement (2) is insufficient on similar grounds to Statement (1): the angle d is now constrained, but angle b can vary freely.

When we combine statements, both angles a and c are fixed. In our imaginations, that fixes the top and sides of the quadrilateral, allowing us to "grab and rotate" only the bottom side. As we adjust it (considering possible cases), that will change b and d, but maybe will be the same in these cases. Indeed, we now have , so , , , and hence , and .Apparently the difference of the angles of b and d is invariant now, but not the sum. If the angles we were looking for were both interior or both exterior angles, we would have sufficient here, but we want an interior angle and an exterior angle. For example, given , we can make the following cases



The statements together are insufficient.

The correct answer is (E).