GMAT Changed on April 16th - Read about the latest changes here

 It is currently 23 Apr 2018, 08:32

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Line m and line n intersect, forming 4 angles. Does any of these angle

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 44636
Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

28 Apr 2015, 05:18
Expert's post
9
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

38% (01:58) correct 62% (02:19) wrong based on 144 sessions

### HideShow timer Statistics

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.
[Reveal] Spoiler: OA

_________________
Manager
Joined: 26 May 2013
Posts: 96
Re: Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

28 Apr 2015, 09:14
1
KUDOS
1
This post was
BOOKMARKED
St (1) is sufficient.

I viewed this problem as a min/max problem. The angles formed by line m and n, can be considered x, y. Since they intersect, you will have 2x and 2y. The problem is essentially asking if x^2(y^2) contains an angle combination greater than 120.

To solve, I set angle X to 120 and angle Y to 60. This means that x*y = 120*60 = 2^5*3^2*5^2. Therefore (xy)^2 equals St(1). If (xy)^2 is less than that number, no angle is greater than 120 degrees.

St (2) is insufficient.
Retired Moderator
Joined: 06 Jul 2014
Posts: 1266
Location: Ukraine
Concentration: Entrepreneurship, Technology
GMAT 1: 660 Q48 V33
GMAT 2: 740 Q50 V40
Re: Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

28 Apr 2015, 12:04
2
This post was
BOOKMARKED
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.

We know that intersecting lines create angles which supplement each other. so we will have $$2$$ $$x$$ and $$2$$ $$y$$ angles and we know that $$x+y = 180°$$
Product of all this angles will be equal to $$x^2*y^2$$
From the rule "The maximum area of rectangles will be received from square" we can infer that maximum product of this angles will be given than $$x = y$$
Let's start from $$x = 120$$ and $$y = 60$$
$$x^2*y^2 = 2^{10}*3^4*5^4$$

1) from this statement we know that $$x^2*y^2 < 2^{10}*3^4*5^4$$ So it is possible only if $$x$$ bigger than $$120$$ because if x less than $$120$$ then $$x^2*y^2$$ will be bigger than $$2^{10}*3^4*5^4$$
and if x = 120 than $$x^2*y^2$$ will be equal to $$2^{10}*3^4*5^4$$

2) from this statement we know that $$x^2*y^2 > 2^{14}*5^4$$
so $$x^2*y^2$$ can be $$2^{10}*3^4*5^4$$ when $$x = 120$$
or $$x^2*y^2$$ can be on one $$2$$ bigger $$2^{11}*3^4*5^4$$ when $$x$$ less than $$120$$ and this number will be still bigger than $$2^{14}*5^4$$
Insufficient

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 44636
Re: Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

04 May 2015, 04:41
2
KUDOS
Expert's post
1
This post was
BOOKMARKED
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°.

_________________
Intern
Joined: 30 Nov 2016
Posts: 15
Re: Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

12 Jan 2017, 07:32
2
KUDOS
Bunuel wrote:
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°.

you made a little typo, $$x^2y^2 > 2^{14}*5^4$$ NOT $$x^2y^2 < 2^{14}*5^4$$
Non-Human User
Joined: 09 Sep 2013
Posts: 6655
Re: Line m and line n intersect, forming 4 angles. Does any of these angle [#permalink]

### Show Tags

16 Mar 2018, 20:51
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: Line m and line n intersect, forming 4 angles. Does any of these angle   [#permalink] 16 Mar 2018, 20:51
Display posts from previous: Sort by