*Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:*

Two lines that intersect produce 2 pairs of identical angles. Call these angles

*x*and*y*(that is, the degree measures of those angles). We have one pair of angles, each measuring*x*°, while each of the other two angles measures*y*°. Moreover, we know that*x*+*y*= 180.We are asked whether

*x*or*y*is greater than 120. For that to be the case, the other angle would have to be less than 60, to make the sum 180.Statement (1): SUFFICIENT. The product of the four angles is less than the given number. Writing each angle as

*x*or*y*, we have this:*x*2

*y*2 < 2103454

Take the square root of all sides to simplify.

*xy*< 253252

Before expanding & evaluating the right side, let’s pause and consider the “break” point: 120°. What if

*x*actually*were*120? Then*y*would be 60. Figure out the product of*x*and*y*, specifically its prime factorization:(120)(60) = 2(60)(60) = 2(2×2×3×5)(2×2×3×5) = 253252. This is exactly the prime factorization above.

Alternatively, you could compute 253252 = 23322252 = 2332102 = (8)(9)(100) = 7,200. This is the product of 120 and 60.

Now, what does it mean that

*xy*is LESS than this number? It means that one of the two variables is GREATER than 120, while the other is LESS than 60. If two variables add up to a constant, then their product is maximum when the two variables are equal. In this case, we’d have a maximum for*xy*when*x*=*y*= 90. As the two numbers become more unequal, the product decreases. You can see this phenomenon in the extreme – if*x*= 179 and*y*= 1, then*xy*= 179, much smaller than 7,200. Making*y*larger, you increase the product toward 7,200. If the product is less than 7,200, then either*x*or*y*is greater than 120.For proof, write

*x*and*y*in this way:*x*= 90 +

*z*

*y*= 90 –

*z*

By writing the angles this way, we know that they add to 180. Assume

*z*is positive (if it’s not, just flip the names*x*and*y*). The product*xy*then looks like the difference of squares:*xy*= (90 +

*z*)(90 –

*z*) = 902 –

*z*2

The bigger

*z*is – that is, the more*unequal**x*and*y*are – the smaller the product, because you’re subtracting off a bigger number from 902.Statement (2): NOT SUFFICIENT. The product of the four angles is less than the given number. Still writing each angle as

*x*or*y*, we have this:*x*2

*y*2 < 21454

Take the square root of all sides to simplify.

*xy*< 2752

Compute the right side by regrouping: 2752 = 252252 = 25102 = (32)(100) = 3,200. By trial and error, you can discover that 3,200 = (160)(20). But you don’t need to do this. Since (120)(60) = 7,200, we know that

*xy*could be less than 7,200 (giving us angles greater than 120) OR*xy*could be greater than 7,200, giving us more nearly equal angles (e.g., perfectly equal angles of 90°), with none over 120°.**The correct answer is A.**