BTGmoderatorDC 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.

OA A

Source: Manhattan Prep

When Line m and line n intersect, they form 4 angles; two of them are equal opposite angles and the other two are equal opposite angles.

Say the 4 angles are x, x, y and y. Thus, x + y = 180.

Say x = 120 , thus, y = 60.

The product of 4 angles x*x*y*y = x^2*y^2 = 120^2*60^2 = 2^10*3^4*5^4

Let's take each statement one by one.

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

Since x^2*y^2 < 2^10*3^4*5^4, where x = 120 and y = 60, x ≠ 120 and y ≠ 60. Now the question is whether x > 120.

We see that x + y = 180. The product of x*y would be maximum when x = y = 60 and the product would decrease in value when the gap between increases. Thus, if x = 179º and yº = 1º, the product xy is least. Thus, we can conclude that x > 120º. Sufficient.

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

=> x^2*y^2 > 2^14*5^4. Since at x = 120º and y= 60º, we have x^2*y^2 = 2^10*3^4*5^4 and 2^10*3^4*5^4 > 2^14*5^4, x can be less than 120º, equal to 120º or greater than 120º. No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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