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# Is the measure of one of the interior angles of

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Is the measure of one of the interior angles of  [#permalink]

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Updated on: 18 Sep 2012, 00:48
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Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?

(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD

As the OA is not given, I have gone for answer C and this is how I reached. Please let me know whether I am right.

First, we know that the sum of 4 angles of a quadrilateral is 360 degrees.

Now, considering statement 1

Two angles are right angles. Not sufficient as it doesn't say anything about the other two angles.

Considering Statement 2

Not sufficient as it doesn't say anything about the other two angles.

Combining both statements

We can find the 4th angle as below

90+90+90+x = 360
270+x = 360
x = 90. where x is the 4th angle. So the answer is NO the angle won't be 60 degrees.

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Originally posted by enigma123 on 07 Feb 2012, 18:01.
Last edited by Bunuel on 18 Sep 2012, 00:48, edited 3 times in total.
Edited the OA
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07 Feb 2012, 18:07
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Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

Sum of inner angels of quadrilateral is 360 degrees. (Sum of inner angles of polygon=180*(n-2), where n is # of sides)

(1) Two of the interior angles of ABCD are right angles --> angles can be 90+90 + any combination of two angels totaling 180. Not sufficient.

(2) The degree measure of angle ABC is twice the degree measure of angle BCD --> <ABC=2<BCD. Not sufficient

(1)+(2) Angles can be 90+90+45+135 Or 90+90+60+120. Not sufficient.

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Re: Is the measure of one of the interior angles of  [#permalink]

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07 Feb 2012, 23:25
That question was a tricky sucker.
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Re: Is the measure of one of the interior angles of  [#permalink]

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06 Oct 2012, 06:30
Since two angles of the quad. are right angles, would other angles not be right angles as well? As sides emanating from right angles would only be straight, and hence the corresponding angles will also be right angles? Correct me if I am wrong. Thx.
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Re: Is the measure of one of the interior angles of  [#permalink]

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06 Oct 2012, 06:49
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chigiwigi wrote:
Since two angles of the quad. are right angles, would other angles not be right angles as well? As sides emanating from right angles would only be straight, and hence the corresponding angles will also be right angles? Correct me if I am wrong. Thx.

Consider the diagram below:
Attachment:

Trapezoid.png [ 1.62 KiB | Viewed 16807 times ]
As you can see we can have a quadrilateral with only two right angles.

Hope it helps.
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Re: Is the measure of one of the interior angles of  [#permalink]

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06 Oct 2012, 07:31
ok, makes sense. Thanks.
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Re: Is the measure of one of the interior angles of  [#permalink]

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18 Jan 2016, 22:13
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?

(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD

In the original condition, there are 4 variables(A,B,C,D) and 1 equation(A+B+C+D=180), which should match with the number of equations. So you need 3 more equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer. When 1) & 2)
Attachment:

GCDS enigma123 Is the measure of one of the interior (20160119).jpg [ 1.78 MiB | Viewed 11519 times ]

Just like the above, there are both yes and no, which is not sufficient. Therefore, the answer is E.

-> For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: Is the measure of one of the interior angles of  [#permalink]

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18 Feb 2017, 03:25
2
(C) is very tempting, but ignores a tricky possibility. The answer is actually (E).

(1) by itself tells us that the two remaining angles sum to 180, but we have no idea if they're 120/60 or some other combination, so it's insufficient.

(2) just tells us that one angle is twice another. We might have a 60 degree angle, but again we might not, so it's insufficient.

Combined, it's very tempting to say that the angles have to be 90/90/120/60. However, the angles also could be 90/90/45/135, since 90 is twice as much as 45, so statement (2) is still satisfied. Therefore, we may or may not have a 60 degree angle: choose (E).
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Re: Is the measure of one of the interior angles of  [#permalink]

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23 Jun 2018, 12:24
Bunuel wrote:
Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

Sum of inner angels of quadrilateral is 360 degrees. (Sum of inner angles of polygon=180*(n-2), where n is # of sides)

(1) Two of the interior angles of ABCD are right angles --> angles can be 90+90 + any combination of two angels totaling 180. Not sufficient.

(2) The degree measure of angle ABC is twice the degree measure of angle BCD --> <ABC=2<BCD. Not sufficient

(1)+(2) Angles can be 90+90+45+135 Or 90+90+60+120. Not sufficient.

How is 135 twice of 45?

I thought that was supposed to be 90.

Please explain this to me, thanks!

Posted from my mobile device
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Re: Is the measure of one of the interior angles of  [#permalink]

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24 Jun 2018, 00:28
MrJglass wrote:
Bunuel wrote:
Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

Sum of inner angels of quadrilateral is 360 degrees. (Sum of inner angles of polygon=180*(n-2), where n is # of sides)

(1) Two of the interior angles of ABCD are right angles --> angles can be 90+90 + any combination of two angels totaling 180. Not sufficient.

(2) The degree measure of angle ABC is twice the degree measure of angle BCD --> <ABC=2<BCD. Not sufficient

(1)+(2) Angles can be 90+90+45+135 Or 90+90+60+120. Not sufficient.

How is 135 twice of 45?

I thought that was supposed to be 90.

Please explain this to me, thanks!

Posted from my mobile device

In that case, 90 degree angle is twice 45 degree angle.
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Re: Is the measure of one of the interior angles of  [#permalink]

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24 Jun 2018, 12:38
Bunuel wrote:
MrJglass wrote:

How is 135 twice of 45?

I thought that was supposed to be 90.

Please explain this to me, thanks!

Posted from my mobile device

In that case, 90 degree angle is twice 45 degree angle.

Oh okay! Thanks for the clarification.
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Re: Is the measure of one of the interior angles of  [#permalink]

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28 Aug 2018, 16:04
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enigma123 wrote:
Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?

(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD

Target question: Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

Key concept: the 4 angles in a quadrilateral must add to 360 degrees

Statement 1: Two of the interior angles of ABCD are right angles.
Let's test some possible cases.
There are infinitely many quadrilaterals that satisfy statement 1. Here are two:
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The degree measure of angle ABC is twice the degree measure of angle BCD.
Let's test some possible cases.
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Re: Is the measure of one of the interior angles of   [#permalink] 28 Aug 2018, 16:04
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