Bunuel wrote:

If four lines intersect as shown in the figure above, what is the value of (x + y)?

(A) 140

(B) 130

(C) 120

(D) 110

(E) 100

We can see there are several triangles in this diagram, but let’s only concentrate on two of them. The first one (the smaller one) is the one with the two 30-degree angles. The second one (the larger one) is the one that is composed of the triangle with the x-degree angle and the quadrilateral with the y-degree angle.

In the larger triangle we’ve described above, we see that it has the x-degree and the y-degree angle, and the remaining angle (unlabeled) is actually an exterior angle of the smaller triangle we’ve described above. Recall that an exterior angle of a triangle is equal to the two remote interior angles of the triangle. Thus, this unlabeled angle of the larger triangle is equal to the sum of the two 30-degree angles of the smaller triangle, which is 60 degrees. Thus, we have:

60 + x + y = 180

x + y = 120

Alternate Solution:

Let’s look at the smaller triangle, the one that has x as one of its interior angles. Another interior angle of this triangle is 30, and the remote exterior angle is 70. Since an exterior angle is the sum of the two remote interior angles, x + 30 = 70, and thus x = 40.

Next, let’s look at the triangle that has y and 30 as two of its interior angles. The remaining interior angle is found to be 70, and since the interior angles add up to 180, we must have y + 30 + 70 = 180. Then, y = 80, and thus x + y = 40 + 80 = 120.

Answer: C

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