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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.

A. Statement (1) ALONEis sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONEis sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHERare sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

My approach to the problem :

Each statement individually cannot help solve the question. Using combined information from two statements:

It isn't clear from combined information whether angle BCD is right angle. So two cases arises here.

Case 1: If angle BCD is not the a right angle. In this case Angle ABC is 60 degrees since ABC + BCD = 180 degress as sum of other two angles of quadrilateral is 180 degress.

Case 2: If angle BCD is the right angle. In this case angle ABC is 45 degrees and other angles of quadrilateral are 90 and 135 degrees.

So we cannot clearly say that angle ABC is 60 with both information combined and so my answer to this is E but the official answer is otherwise.

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.

A. Statement (1) ALONEis sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONEis sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHERare sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

My approach to the problem :

Each statement individually cannot help solve the question. Using combined information from two statements:

It isn't clear from combined information whether angle BCD is right angle. So two cases arises here.

Case 1: If angle BCD is not the a right angle. In this case Angle ABC is 60 degrees since ABC + BCD = 180 degress as sum of other two angles of quadrilateral is 180 degress.

Case 2: If angle BCD is the right angle. In this case angle ABC is 45 degrees and other angles of quadrilateral are 90 and 135 degrees.

So we cannot clearly say that angle ABC is 60 with both information combined and so my answer to this is E but the official answer is otherwise.

Please correct if my logic is incorrect.

Merging similar topics. please refer to the solution above.

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.

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 11354 times ]

As you can see we can have a quadrilateral with only two right angles.

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

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28 Jul 2014, 08:52

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

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09 Oct 2015, 10:50

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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 6036 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

(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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