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For the cube shown above, what is the degree measure of PQR  [#permalink]

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Attachment: cube.gif [ 2.13 KiB | Viewed 58445 times ]
For the cube shown above, what is the degree measure of PQR?

A. 30
B. 45
C. 60
D. 75
E. 90

Originally posted by nocilis on 02 Feb 2005, 21:35.
Last edited by Bunuel on 30 Jun 2012, 06:40, edited 1 time in total.
Edited the question and added the OA.
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For the cube shown above, what is the degree measure of PQR  [#permalink]

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For the cube shown above, what is the degree measure of PQR?

A. 30
B. 45
C. 60
D. 75
E. 90 Note that triangle PQR is equilateral: it's made by the diagonals of the adjacent faces of the given cube (and as faces of a cube are squares its diagonals are equal). Thus angle BEG=60 degrees.

Attachment: cube.gif [ 2.28 KiB | Viewed 61620 times ]

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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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ankushbassi wrote:
Hi Bunuel, since the face of a cube forms the square and the diagonal of the square bisects the angles.The 2 diagonals shown in the pictures and each makes an angle of 45.Thus,why not the answer is 45+45=90?Can you please help me to understand this.

Hi Ankush,
The angle in question is as shown.Please refer the diagram below:
Attachment: g2.png [ 17.85 KiB | Viewed 53366 times ]

It would be an equilateral triangle made up by 3 face diagonals.

What you are referring are the two angles in two different planes.
which are angle POA and angle AOR each 45 degrees

They are on adjacent faces of the cube,which you cannot combine.
Attachment: G5.png [ 11.18 KiB | Viewed 53336 times ]

Hope this helps.
Press kudos if it was of help.

Kindly check the following post with the poll and revert with your replies
10-straight-lines-no-two-of-which-are-parallel-and-no-three-171525.html
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60 as well

because the figure in question is a cube sides pq,qr,pr are equal. Hence pqr is an equilateral triangle. So you get 60 for each included angle.
Had this been a cuboid, then it is a different story.
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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Hi Bunuel, since the face of a cube forms the square and the diagonal of the square bisects the angles.The 2 diagonals shown in the pictures and each makes an angle of 45.Thus,why not the answer is 45+45=90?Can you please help me to understand this.
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ankushbassi wrote:
Hi Bunuel, since the face of a cube forms the square and the diagonal of the square bisects the angles.The 2 diagonals shown in the pictures and each makes an angle of 45.Thus,why not the answer is 45+45=90?Can you please help me to understand this.

Because these two squares are not in the same plane. If the square were in the same plane (as shown below), then you would be correct: Does this make sense?

Attachment: squares.png [ 1.8 KiB | Viewed 54950 times ]

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For the cube shown above, what is the degree measure of PQR  [#permalink]

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Attachment:
cube.gif

For the cube shown above, what is the degree measure of PQR?

A. 30
B. 45
C. 60
D. 75
E. 90

Don't be in a hurry and say it is 90 degrees.
If you look closely , you will notice PQ and QR are two diagonals to the cube; therefore they will be equal in dimension. Now see If you join QR you will get another diagonal with.
NOW we get a triangle in which PQ=PR=RQ ; An equilateral triangle.

ALL angles of an equilateral triangle are 60 degrees.

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Schools: Fisher '19 (M$) GPA: 3.71 Re: For the cube shown above, what is the degree measure of PQR [#permalink] ### Show Tags Bunuel - What will be the value of each diagonal in the equilateral triangle that is formed here ? (triangle PQR) Math Expert V Joined: 02 Sep 2009 Posts: 65062 Re: For the cube shown above, what is the degree measure of PQR [#permalink] ### Show Tags Keats wrote: Bunuel - What will be the value of each diagonal in the equilateral triangle that is formed here ? (triangle PQR) Do you mean angles instead of diagonals? Triangle does not have diagonals. If you mean angles, then each angle in equilateral triangle is 60 degrees. _________________ Director  G Joined: 28 Nov 2014 Posts: 806 Concentration: Strategy Schools: Fisher '19 (M$)
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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Bunuel wrote:
Keats wrote:
Bunuel - What will be the value of each diagonal in the equilateral triangle that is formed here ? (triangle PQR)

Do you mean angles instead of diagonals? Triangle does not have diagonals. If you mean angles, then each angle in equilateral triangle is 60 degrees.

Sorry that I didn't put it up correctly. I wanted to ask that what will be the value of each side of the equilateral triangle (which happens to be the diagonal of the cube), Bunuel
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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Keats wrote:
Bunuel wrote:
Keats wrote:
Bunuel - What will be the value of each diagonal in the equilateral triangle that is formed here ? (triangle PQR)

Do you mean angles instead of diagonals? Triangle does not have diagonals. If you mean angles, then each angle in equilateral triangle is 60 degrees.

Sorry that I didn't put it up correctly. I wanted to ask that what will be the value of each side of the equilateral triangle (which happens to be the diagonal of the cube), Bunuel

If we assume that the length of a side of the cube is x, then the length of each diagonal = $$\sqrt{x^2 + x^2} = \sqrt{2}x$$
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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nocilis wrote:
Attachment:
The attachment cube.gif is no longer available
For the cube shown above, what is the degree measure of PQR?

A. 30
B. 45
C. 60
D. 75
E. 90

Draw a construction (Pink Line) and

Attachments

File comment: www.GMATinsight.com Answer 3.jpg [ 58.96 KiB | Viewed 24366 times ]

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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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interesting! I have get an actual wooden box to understand this concept
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ferrarih wrote:
Hi, I understood the answer and I agree, but I can't find out why my logic isn't right.

Since it's a cube, all angles are 90 degrees, and since PO and RO are the bisectrix of the angle 90, than PQR would have to be 90, which is not the answer. What am thinking wrong?

Thanks.

You are looking at it 2D imagine in 3D...
Also like everyone said all sides are equal hence angle in 60
Note its a triangle
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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All the three sides of triangle PQR are face diagonals of a cube.
If the side of the cube is ‘a’, then PQ = QR = PR = a x sqrt(2)
So, the triangle PQR is an equilateral triangle. Hence, all the angles are of 60.
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Re: For the cube shown above, what is the degree measure of PQR  [#permalink]

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nocilis wrote:
Attachment:
cube.gif
For the cube shown above, what is the degree measure of PQR?

A. 30
B. 45
C. 60
D. 75
E. 90

If we draw a line segment from P to R, we see that the length of PR = length of PQ = length of RQ, since each segment is the diagonal of a square face of the cube.

Thus, we have triangle PQR with all sides of equal lengths, and thus the degree measure of angle PQR is 60 degrees.

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