Is measure of one of the interior angles of quadrilateral AB

[Guest]

Source : Gmatprep 2

Is 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.

Ans. E

Please explain

[Saurav]

Is 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.

Sum of angles of any quadrilateral = 360. If two angles are 90 deg each, then sum of other two triangles = 180. Not Sufficient

2) The degree measure of angle ABC is twice the degree measure of angle BCD.

This gives

ang ABC + ang BCD + ang CDA + ang DAB = 360

2X + X + ang CDA + ang DAB = 360 (3 unknowns) Not Sufficient


Using 1 and 2

ang ABC + ang BCD + ang CDA + ang DAB = 360
2X + X + ang CDA + ang DAB = 360

Case 1
ang ABC = 90 then ang BCD = 45, ang CDA / ang DAB = 90, 135 - No angle is 60 deg

Case 2
ang BCD = 90 then ang ABC = 180, ang CDA / ang DAB = cant determine - Not possible

Case 3
ang ABC = 2X then ang BCD = X, ang CDA / ang DAB = 90, 90
ang ABC + ang BCD = 3X = 180
ang ABC = 120 then ang BCD = 60, ang CDA / ang DAB = 90, 90 - One angle is 60 deg

Hence using both 1 and 2, one angle can be either 60 deg (case 3) or none angle is 60 deg (case 1)

hence not sufficient using both.

[RonPurewal]

saurav's solution is good once again.

i'm partial to explanations with more words than algebraic expressions, though, so here's an equivalent version that has more words in it.

statement (1)
we know that the angles are 90°, 90°, x, y, and that all of them sum to 360°.
by subtraction, x + y must be 180°.
it's possible that one of them could be 60° (if the other one is 120°), but it's also possible for neither of them to be 60° (if they have any measures other than 60° and 120°).
insufficient.

statement (2)
you could pick any 1:2 ratio, from 0.0000001° and 0.0000002° all the way up to 89.99999° and 179.99998°, and just select the remaining 2 angles so that the sum of all four of them is 360°.
this includes possibilities in which one of the angles is 60°, as well as possibilities in which none of the angles has that measure.
insufficient.

(together)
the trap here is to assume that the angles are 90°, 90°, x°, and 2x°. that's one possibility, but not the only one. in this case, x = 60 and 2x = 120.
however, it's possible that the 90° angle is the "2x" in this problem. this would mean that a third angle was 45° (so that 90° and 45° provide the required 2:1 ratio), and, by subtraction, the last angle is 135°.
therefore, 60° could be either present or absent.
still insufficient.

answer (e).

[sd]

Ron, I understand why the annswer is E. But for questions like this which are not accompanied by any figures, is the order of vertices set in stone. I mean the question states that the quadrilateral is ABCD. Should we assume the vertices only in that order, or can we visualize a quadrilateral ADBC.

Please see this image to see what I mean -

http://www.postimage.org/image.php?v=gx_4810

If we take statement 2, then we get different pictures depending on the order of the vertices. I totally agree that either way, the answer is E for this question. But I want to know if both figure 1 and figure 2 are valid visualizations for questions such as these.

[Ben Ku]

I mean the question states that the quadrilateral is ABCD. Should we assume the vertices only in that order, or can we visualize a quadrilateral ADBC.

A polygon is ALWAYS named in the order of the vertices, either going clockwise or counterclockwise. This way, there is no ambiguity about what our polygon looks like. So quadrilateral ABCD is NOT the same as quadrilateral ADBC.

Hope that helps.

[djfunny]

statement (1)
we know that the angles are 90°, 90°, x, y, and that all of them sum to 360°.
by subtraction, x + y must be 180°.
it's possible that one of them could be 60° (if the other one is 120°), but it's also possible for neither of them to be 60° (if they have any measures other than 60° and 120°).
insufficient.

if two interior angles of a quadrilateral are 90° then all the four angles are 90°. Is it not? Therefore, the right answer should be A.

[agha79]

statement (1)
we know that the angles are 90°, 90°, x, y, and that all of them sum to 360°.
by subtraction, x + y must be 180°.
it's possible that one of them could be 60° (if the other one is 120°), but it's also possible for neither of them to be 60° (if they have any measures other than 60° and 120°).
insufficient.

if two interior angles of a quadrilateral are 90° then all the four angles are 90°. Is it not? Therefore, the right answer should be A.

A quadrilateral is any four sided polygon. It could be a rhombus, trapezoid, parallelogram, square or a rectangle. The statement 1 would be sufficient if the question mentions that the quadrilateral is either a square or a rectangle. Then we would have got the remaining two angles 90 degrees, choosing A as answer.

[djfunny]

A quadrilateral is any four sided polygon. It could be a rhombus, trapezoid, parallelogram, square or a rectangle. The statement 1 would be sufficient if the question mentions that the quadrilateral is either a square or a rectangle. Then we would have got the remaining two angles 90 degrees, choosing A as answer.

If a quadrilateral has two 90 degree angles, which is the case here, then it's either a square or a rectangle. Would you agree?

[agha79]

If a quadrilateral has two 90 degree angles, which is the case here, then it's either a square or a rectangle. Would you agree?

i maybe wrong but i took the statement 1 this way that it tells that the two of the interior angles are 90 degrees but it doesn’t tell that all lines segments, making a quadrilateral, are also parallel. When two angles are 90 degrees in a qudlilateral then one pair of line segments is parallel to each other. However, the other pair of line segments can be or can not be parallel. Thus, the interior angles of the other two line segments are unable to determine with statement 1.

[djfunny]

i maybe wrong but i took the statement 1 this way that it tells that the two of the interior angles are 90 degrees but it doesn’t tell that all lines segments, making a quadrilateral, are also parallel. When two angles are 90 degrees in a qudlilateral then one pair of line segments is parallel to each other. However, the other pair of line segments can be or can not be parallel. Thus, the interior angles of the other two line segments are unable to determine with statement 1.

it doesn’t have to tell you that all lines segments, making a quadrilateral, are parallel, because this follows from the statement 1.
Can you draw\imagine a quadrilateral with two right angles which is not a rectangle?

[RonPurewal]

it doesn’t have to tell you that all lines segments, making a quadrilateral, are parallel, because this follows from the statement 1.
Can you draw\imagine a quadrilateral with two right angles which is not a rectangle?

you should check your reasoning against the correct answer before posting -- if your statement here were true, then the correct answer to the problem would be (a), not (e). therefore, your statement must be wrong.

as for a quadrilateral that has two right angles, but isn't a rectangle -- sure, you can have a kite (like this one: http://www.tutornext.com/system/files/u21/4a.JPG) with right angles where "B" and "D" are located on the diagram.
(or you could have several other, irregularly shaped quadrilaterals with two right angles -- but the kite is the easiest one to visualize.)

[GAURAV.1119]

correct me if i am wrong..

the answer is C.

consider a trapezoid ABCD with angle DAB = 90, and angle CDA = 90 (statement -1 )

Now the other two angles are 60 and 120 (since sum of interior angles = 360) these angles are angle ABC and angle BCD.. double of each other (statement - 2)

hence with both statements we can say that at least one interior angle is 60 degrees.

[RonPurewal]

correct me if i am wrong..

the answer is C.

consider a trapezoid ABCD with angle DAB = 90, and angle CDA = 90 (statement -1 )

Now the other two angles are 60 and 120 (since sum of interior angles = 360) these angles are angle ABC and angle BCD.. double of each other (statement - 2)

hence with both statements we can say that at least one interior angle is 60 degrees.

no, this is incorrect -- and the correct answer (e) has been amply explained earlier in the thead. please do not post without reading the preceding discussion.

here is a direct link to the post (on this same thread) on which the answer has already been explained:
post20150.html#p20150