In the diagram, JKLM is a square, and P is the midpoint of KL. Is JQM

I think C

(1) Angle KPQ = 90° => There are many points which satisfy this conditions, i mean it lies on the lines which is the perpendicular with JM and pass on the midpont of JM => Insufficient
(2) Angle JQP = 150° => Also, there are many points satisfy this conditions, i mean it lies on the circular => Insufficient

Combine (1) (2) => Q is the intersect between the lines in (1) and the circLE in (2)

=> answer: C

attached image for my solution to this problem .

minor correction , Corrected the angle names..

(1) INSUFFICIENT- only with KPQ= 90 and no information about length of PQ. if PQ increases then JQM increases. so we need additional information.
(2) INSUFFICIENT- PQJ= 150. but no information whether JK || PQ or not. So with P being the mid point and KPQ \(\neq{90}\), PQJ can be 150. We need additional information.

Considering both

KPQ= 90 & PQJ= 150

=>KJP= 30=>QJM=60. same way it can be shown that JMQ= 60. ultimately JQM=60.
Hence JQM= equilateral.

Ans- C.

Original Statement says P is midpoint of KL, so we can say some mirror image of P on JM must be a midpoint too.
To prove it an equilateral triangle, we need to find atleast one angle or 3 sides.

1) Ang KPQ=90 => PQ, if extended, will meet the JM at the mirrored midpoint, bisecting the base of the triangle in 2. This is a property of isosceles & equilateral triangles. --> NOT SUFFICIENT
2) Ang JQP=150 => No information regarding angles of triangle or sides --> NOT SUFFICIENT

1) & 2)
When PQ is extended, it forms 180degs (say PQS) => Ang JQS=AngPQS-AngJQP=30; Similarly, AngMQS=30 => Ang JQM=60

Hence C

Fairly straightforward.

KPQJ is a quadrilateral hence the sum of interior angles must equal 360. Likewise, the triangle's interior angles must equal 180.

A says KPQ is 90. Clearly this is insufficient by itself as it means nothing
B says JPQ is 150 which means half of angles Q must be 30 as it forms a straight line with P; hence, Q is 60. Just because Q is 60 does not mean the remaining 2 interior angles of the triangle are 60. Insufficient.

Combining both:
We know JPQ is 150 and KPQ is 90. We also know all interior angles of a square connecting 2 sides is 90. Hence, angles J and K are 90 each. Regardless, K + P + Q + J = 360. We know from above K = 90, P = 90, PQJ = 150; hence, J outside of triangle is 360 - 330 = 30. Since whole of J is 90, 90-30 = 60. 2 angles = 60 --> third angle is 60.

In this question on squares, it’s very easy to get into the trap of assuming that KP and PQ are perpendicular to each other, because they LOOK LIKE. Remember that it is not mentioned in the question and therefore cannot be assumed to be true.

A point to note on Geometry questions on GMAT Quant is to not trust the figure to be to scale.

If JQM is an equilateral triangle, angle JQM = angle QJM = angle QMJ = 60 degrees.

From statement I alone, angle KPQ = 90 degrees. Only now can we say that KP is perpendicular to PQ. However, this is all we can say. We do not have any information about angles PQJ or PQM.

If PQJ = 150 degrees and PQM = 150 degrees, then angle JQM = 60 degrees; also, angle QJK = angle QML = 30 degrees. Therefore, angle QJM = angle QMJ = 60 degrees. In this case, triangle JQM is an equilateral triangle.

If PQJ = 140 degrees and PQM = 140 degrees, then angle JQM = 80 degrees; also, angle QJK = angle QML = 40 degrees. Therefore, angle QJM = angle QMJ = 50 degrees. In this case, triangle JQM is NOT an equilateral triangle.

Statement I alone is insufficient. Answer options A and D can be eliminated, possible answer options are B, C or E.

From statement II alone, we know that angle JQP = 150 degree. But, we do not know if angle KPQ is a right angle. Therefore, we would not be able to find out the exact value of the other angles in diagram.
Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we see that angle KPQ = 90 degree and angle PQJ = 150 degree. This is a case we already took while evaluating statement I, and in doing so, we understood that this case will make triangle JQM an equilateral triangle. The combination of statements is sufficient.

The correct answer option is C.

Hope this helps!

This is a not drawn to scale on steroids.

Statement I - Not suff. because it you still don't know the how far down PQ goes. which would affect the lengths of JQ and MQ. It could, and it couldn't not be equal.

Statement II - Not suff because even if you know that's 150 degrees you are without that 90 degree certainty from statement I and thus the two sides could be equal - but could not be also.

Putting them together you know if PQ is 90 degrees and JQP is 150 then the other two angles must be 90 and 30 since it makes a trap. and it has to add to 360 degrees. Well it says it's a square and you know KJM is 90 and tht KJQ is 90 so QJM must be 60 degrees. But if PQ is 90 degrees then you know each side is equal so LMX is also 60 degress. That leaves 60 degrees for JQM and you can for sure say this bad boy is an equilateral triangle.

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