In the figure shown, the measure of angle PRS is how many degrees grea

In the figure shown, the measure of angle PRS is how many degrees greater than the measure of angle PQR.


Question: \(\angle PRS - \angle PQR=?\)


(1) The measure of angle QPR is 30°.

\(\angle QPR=30°\);
In triangle QPR three angles sum \(= 180° = \angle QPR+\angle PQR +\angle PRQ\);
\(180°=30°+\angle PQR+(180°-\angle PRS)\);
\(30° = \angle PRS-\angle PQR\). SUFFICIENT.


(2) The sum of the measures of angles PQR and PRQ is 150°.

Basically the same information is given \(\angle PQR + \angle PRQ=150°\);
\(\angle PQR + 180° - \angle PRS=150°\);
\(30° = \angle PRS - \angle PQR\). SUFFICIENT.


Answer: D.


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This is a hard problem to solve without a pen and a paper. Hope Img makes it clear.

grafical solution is always easier.
Answer is D.

1) Angle PRS = QPR + PQR
PRS = 30 + PQR
PRS - PQR = 30 SUFFICIENT
2) PQR + PRQ = 150
PQR + 180-PRS = 150
PRS - PQR = 30 SUFFICIENT

ANS. D
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Hi, there! I'm happy to help with this one.

What's tricky about this problem is that there are three different triangles in the diagram, and we have to apply the "180 degree Triangle Theorem" in each one.

First of all, from the diagram, we know in triangle PRS, that (angle SPR) + (angle PRS) + 90 = 180, or in other words, (angle SPR) + (angle PRS) = 90

In triangle PQS, we know that (angle SPQ) + (angle SQR) + 90 = 180, or in other words, (angle SPQ) + (angle PQR) = 90.

Set those two equal:

(angle SPR) + (angle PRS) = (angle SPQ) + (angle PQR)

(angle SPQ) - (angle SPR) = (angle PRS) - (angle PQR)

The question is asking: "the measure of angle PRS is how many degrees greater than the measure of angle PQR?" In other words, they are asking for (angle PRS) - (angle PQR), and our equation above tells us that: if we know (angle SPQ) - (angle SPR), then we know (angle PRS) - (angle PQR).

Statement #1: (angle QPR) = 30 degrees

We know that (angle SPQ) = (angle SPR) + (angle QPR) (big angle equals the sum of the two little angles the comprise it)

Therefore (angle SPQ) = (angle SPR) + 30 --> (angle SPQ) - (angle SPR) = 30 ---> (angle PRS) - (angle PQR) = 30

Statement #1 is sufficient by itself. .

Statement #2: (angle PQR) + (angle PRQ) = 150 degrees

Well, in triangle PQR, we know that: (angle PQR) + (angle PRQ) + (angle QPR) = 180 degrees

If (angle PQR) + (angle PRQ) = 150 degrees, then 150 + (angle QPR) = 180 degrees ---> (angle QPR) = 30, and we have the same information we had in statement #1, so statement #2 is also sufficient by itself.

Answer Choice D. Does that make sense?

Here's another GMAT DS question on the 180 degree Triangle Theorem, just for practice.

https://gmat.magoosh.com/questions/1009

The question at that link should be followed by a video explanation of the answer.

Please let me know if you have any more questions.

Mike

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Hi, sorry for the stupid question but it is something I can't grasp.

The text when it refers to angles PRS or PGR to which angles is it visually referring to?

I literally can't understand on the chart which are the angles the problem is asking about.

Thanks for your help.

An angle can be identified like this: ∠PQR. The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs.

So in the figure above the red angle would be ∠PQR or ∠RQP (so long as the vertex is the middle letter, the order is not important).

Hope it's clear.
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angle PRS = QPR ( 30) + PQR is this exterior angle property?

Yes it is. Angle PRS is an exterior angle of angle PRQ: <PRS = 180 - <PRQ = 180 - (180 - <QPR - <PQR) = <QPR + <PQR
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Hi I used the following method. Turned out to be the quickest for me.

Theorem: An exterior angle of a triangle is equal to the sum of the opposite interior angles.

So, as per the theorem, for Triangle PRQ we have ...
PRS = RPQ + PQR

which gives,

RPQ= PRS - PQR .

Now if we know the measure of RPQ. We are sorted. :D

1) RPQ = 30. SUFFICIENT.

2) same as 1. SUFFICIENT

So D.

Solution:

We need to determine by how many degrees the measure of angle PRS exceeds the measure of angle PQR.

Statement One Only:

The measure of angle QPR is 30°.

Let the measure of RPS be x. Then, the two interior angles of the triangle PQS are 90 and 30 + x. Thus, the third interior angle of the same triangle is 180 - (90 + 30 + x) = 180 - 120 - x = 60 - x. So, in terms of x, the angle PQS measures 60 - x.

Now, look at the triangle PRS. Two of the interior angles of this triangle are 90 and x; thus, the third interior angle (which is angle PRS) is 180 - (90 + x) = 90 - x.

Using the two expressions, we can determine that the angle PRS exceeds the angle PQS by 90 - x - (60 - x) = 90 - x - 60 + x = 30 degrees.

Statement one is sufficient to answer the question.

Statement Two Only:

The sum of the measures of angles PQR and PRQ is 150°.

This means angle QPR is 180 - 150 = 30°. Using the same procedure that we used in statement one, we can determine the difference between the measures of angles PRS and PQR using QPR = 30.

Statement two is sufficient to answer the question.

Answer: D

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Well, is it safe to assume that QRS is a straight line in this question? I marked E because I did not assume that QRS is a straight line, and thus angle QRP + angle PRS = 180 was not something that I could consider in my solution.

OFFICIAL GUIDE:

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.

Hope it helps.
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Video solution from Quant Reasoning starts at 15:25
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

Here is my solution

Redrawing the figure is a good strategy to apply in this question; it makes it easy to identify that angle PRS is the exterior angle wrt triangle PQR.



In a triangle, the exterior angle is equal to the sum of the non-adjacent opposite interior angles. As shown in the diagram, angle PRS (highlighted in red) is the exterior angle and therefore equal to the sum of angles PQR and QPR (highlighted in green).
Angle PRS = Angle PQR + Angle QPR.

The above equation can be reorganized to give us exactly what we want. Since we are trying to find the value of Angle PRS – Angle PQR, we can rewrite the above equation as

Angle PRS – Angle PQR = Angle QPR.
Any information about angle QPR is sufficient information.

From statement I alone, angle QPR is 30 degrees. Sufficient.
Answer options B, C and E can be eliminated. Possible answer options are A or D.

From statement II alone, sum of angles PQR and PRQ is 150 degrees. Since the sum of the three interior angles of a triangle is 180 degrees, this means that angle QPR is 30 degrees.
Statement II alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

Hope that helps!
Aravind B T

Hi guys,

After solving the question using a lot of x-s and y-s, I gave some more thought to this problem and noticed very easy thing here. We can solve this question in 20 seconds without even using any calculation.

So the logics is that angles of triangle must sum to 180 right? So assume we only had the smaller triangle first. If one angle is right (90 degrees), than the others must sum to 90. ok, if now we move the upper point (move R all the way to Q), than the angle of P increases, by 30 degrees in this case. Accordingly, the angle of the other corner must reduce by the same degree right? so that the sum of the two is 90 degrees again.

So the other angle must reduce by the same amount as the leftmost angle increases.