Bunuel wrote:

Line m and line n intersect, forming 4 angles. Does any of these angles measure greater than 120°?

(1) The product of the measures, in degrees, of the four angles is less than 2^10*3^4*5^4.

(2) The product of the measures, in degrees, of the four angles is greater than 2^14*5^4.

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION: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:

\(x^2y^2 < 2^{10}3^45^4\)

Take the square root of all sides to simplify.

\(xy < 2^53^25^2\)

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:

\((120)(60) = 2(60)(60) = 2(2×2×3×5)(2×2×3×5) = 2^53^25^2\). This is exactly the prime factorization above.

Alternatively, you could compute \(2^53^25^2 = 2^33^22^25^2 = 2^33^210^2 = (8)(9)(100) = 7,200\). 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) = 90^2 – 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 90^2.

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:

\(x^2y^2 < 2^{14}*5^4\)

Take the square root of all sides to simplify.

\(xy < 2^7*5^2\)

Compute the right side by regrouping: \(2^75^2 = 2^52^25^2 = 2^510^2 = (32)(100) = 3,200\). 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.you made a little typo, \(x^2y^2 > 2^{14}*5^4\) NOT \(x^2y^2 < 2^{14}*5^4\)