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In triangle PQR, the measure of angle P is 30 more than

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In triangle PQR, the measure of angle P is 30 more than  [#permalink]

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New post 26 Nov 2010, 13:11
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In triangle PQR, the measure of angle P is 30° more than twice the measure of angle Q. What is the measure of angle R?

(1) PQ = QR
(2) The measure of angle P is 78°.
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Re: DS-Geometry  [#permalink]

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New post 26 Nov 2010, 13:38
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rite2deepti wrote:
In triangle PQR, the measure of angle P is 30° more than twice the measure of angle Q.
What is the measure of angle R?
(1) PQ = QR
(2) The measure of angle P is 78°.

I am clueless about this question


Given: \(\angle{P}=2*\angle{Q}+30\). Question: \(\angle{R}=?\).

Now, as the sum of the angles in a triangle equals to 180 degrees then: \(\angle{P}+\angle{Q}+\angle{R}=180\) --> \(\angle{R}=180-(\angle{P}+\angle{Q})=180-(2*\angle{Q}+30+\angle{Q})=150-3*\angle{Q}\) or \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\). So, knowing the measure of \(\angle{P}\) ( or \(\angle{Q}\)) is sufficient to answer the question.

(1) PQ = QR --> triangle PQR is isosceles (base is PR) --> so \(\angle{R}=\angle{P}\). We also know that \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\) so we have two distinct linear equations with two unknowns and we can solve for them. Sufficient.
(2) The measure of angle P is 78 --> so \(\angle{P}=78\) then we can solve \(\angle{R}=150-3*\frac{\angle{P}-30}{2}\) for \(\angle{R}\). Sufficient.

Answer: D.

For more on this issues check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.
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Re: DS-Geometry  [#permalink]

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New post 27 Nov 2010, 13:11
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rite2deepti wrote:
In triangle PQR, the measure of angle P is 30° more than twice the measure of angle Q.
What is the measure of angle R?
(1) PQ = QR
(2) The measure of angle P is 78°.

I am clueless about this question


The best thing to do in geometry questions is to make a diagram:
Attachment:
Ques1.jpg
Ques1.jpg [ 7.12 KiB | Viewed 19320 times ]


I have got all angles in terms of angle Q.
angle R = 180 - Q - (30 + 2Q) = 150 - 3Q
(Since sum of all angles in a triangle is 180).
To find angle R, I only need angle Q.

Stmnt 1: If PQ = QR, angle P = angle R.
30 + 2Q = 150 - 3Q. you will get Q and hence the value of angle R.
Stemnt 2: If angle P is 78, 30 + 2Q = 78. You will get value of angle Q and hence value of angle R.
Both statements alone sufficient so answer is D.
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Re: In triangle PQR, the measure of angle P is 30 more than  [#permalink]

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New post 20 Sep 2016, 09:00
D is correct. Here's why:

(1) SUFFICIENT - We know from PQ=QR that the angles opposite these sides are equal. Therefore, we can write...

30+2Q+30+2Q+Q=180 --> Q =24

(2) P=78 - SUFFICIENT

30+2Q=78 --> Q=24

Solve for remaining side given 78 and 24
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Re: In triangle PQR, the measure of angle P is 30 more than  [#permalink]

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New post 19 Feb 2018, 14:00
Hi All,

This DS question is based completely on your knowledge of triangle rules.

We're told that, in triangle PQR, Angle P is 30 degrees MORE than TWICE the measure of Angle Q. We're asked for the measure of Angle R.

Fact 1: PQ = QR

This tells us the the triangle is ISOSCELES, and that Angle P = Angle R. By extension, Angle R is ALSO 30 degrees more than twice the measure of Angle Q.

Thus, our 3 angles are...
P = 30 + 2Q
Q
R = 30 + 2Q

Total = 180 degrees = 60 + 5Q. With this, we can solve for Q, which means we can solve for P and R (the angles end up as Q = 24, P = 78 and R = 78, but that math is unnecessary).
Fact 1 is SUFFICIENT.

Fact 2: Angle P = 78

With this, we can figure out Angle Q. Then, with those 2 angles, we can figure out the remaining angle (Angle R).
Fact 2 is SUFFICIENT

Final Answer:

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Re: In triangle PQR, the measure of angle P is 30 more than   [#permalink] 19 Feb 2018, 14:00
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