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# In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2577
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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06 Feb 2019, 18:59
00:00

Difficulty:

45% (medium)

Question Stats:

50% (01:00) correct 50% (01:03) wrong based on 20 sessions

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In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the sum of all possible values of ∠P?

A. 70
B. 80
C. 100
D. 130
E. 150

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Joined: 09 Mar 2018
Posts: 1000
Location: India
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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Updated on: 06 Feb 2019, 19:49
1
EgmatQuantExpert wrote:
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the sum of all possible values of ∠P?

A. 70
B. 80
C. 100
D. 130
E. 150

IMO E

Now PQR is an isosceles triangle, < Q = 80

3 Cases
1) when <P = < R = 50
2) when <Q = <P = 80
3) when <Q = <R = 80, <P = 20

Sum = 150
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If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.

Originally posted by KanishkM on 06 Feb 2019, 19:13.
Last edited by KanishkM on 06 Feb 2019, 19:49, edited 1 time in total.
Director
Joined: 06 Jan 2015
Posts: 587
Location: India
Concentration: Operations, Finance
GPA: 3.35
WE: Information Technology (Computer Software)
Re: In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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06 Feb 2019, 19:44
1
KanishkM wrote:
EgmatQuantExpert wrote:
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the sum of all possible values of ∠P?

A. 70
B. 80
C. 100
D. 130
E. 150

IMO E

Now PQR is an isosceles triangle, < Q = 80

3 Cases
1) when <P = < R = 50
2) when <Q = <P = 80
3) when <Q = <R = 160, <P = 20

Sum = 150

Hey KanishkM,

3) when <Q = <R = 160, <P = 20 How is it possible? While Sum of angles of triangle is 180
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Joined: 09 Mar 2018
Posts: 1000
Location: India
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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06 Feb 2019, 19:46
NandishSS wrote:
KanishkM wrote:

IMO E

Now PQR is an isosceles triangle, < Q = 80

3 Cases
1) when <P = < R = 50
2) when <Q = <P = 80
3) when <Q = <R = 160, <P = 20

Sum = 150

Hey KanishkM,

3) when <Q = <R = 160, <P = 20 How is it possible? While Sum of angles of triangle is 180

hahahah NandishSS

I think i did some big time typo , let me correct it now
_________________

If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.

Intern
Joined: 05 Jan 2017
Posts: 4
Re: In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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07 Feb 2019, 11:13
This is an isosceles triangle, so two sides are equal and two angles must be of the same measure.

There are three cases:

1) angle Q is formed by two equal sides, so the other two angles (P and R) must be the same.

$$80 (angle Q) + angle P + angle R = 180$$ (sum of all angles in a triangle)

$$angle P + angle R = 100$$

As mentioned before, angle P and angle R must be equal, hence P = 50 and R = 50.

2) angle Q is formed by the base and one of the sides, and the angle (let's say R) is formed by the base and the other side. It means, that Q=R, so $$P = 180-80-80 = 20$$

3) angle Q is formed by the base and one of the sides, and the angle (this time it's P) is formed by the base and the other side. It means, that Q=P, so P = 80

Sum of all possible values of angle $$P = 50 + 20 + 80 = 150$$.

Manager
Joined: 22 Sep 2018
Posts: 244
Re: In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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07 Feb 2019, 21:43
EgmatQuantExpert wrote:
In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the sum of all possible values of ∠P?

A. 70
B. 80
C. 100
D. 130
E. 150

P can be equal to 80, if side PR is the same length as QR.

P can be 20 if side PQ is the same as PR

P can be 50 if side PQ and RQ are the same.

total is 150
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2577
Re: In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...  [#permalink]

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07 Feb 2019, 23:09

Solution

Given:
• PQR is an isosceles triangle
• ∠Q = 80 degrees

To find:
• Sum of all possible values of ∠P

Approach and Working:
We know that in an isosceles triangle two angles are equal
Case 1: If ∠Q = ∠P, then ∠P = 80 degrees
Case 2: If ∠Q = ∠R, then ∠P = 180 – 80 – 80 = 20 degrees
Case 3: If ∠P = ∠R, then ∠P = $$\frac{(180 – 80)}{2} = 50$$ degrees

Therefore, sum of all possible values of ∠P = 80 + 50 + 20 = 150

Hence the correct answer is Option E.

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Re: In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the ...   [#permalink] 07 Feb 2019, 23:09
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