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

1. 70
2. 80
3. 100
4. 130
5. 150

Correct Answer: (E)

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• PQR is an Isosceles Triangle
• ∠Q = 80 Degrees

Find out:

• The sum of all possible values of ∠P

As per the principles of a triangle, the sum of all angles of a triangle = 180°.

In the case of the isosceles triangle, two angles opposite to the identical sides are congruent with each other.

Therefore, since ∠Q = 80°, there exist three cases.

1. When ∠P = ∠R = 50°
2. When ∠Q = ∠P = 80°
3. When ∠Q = ∠R = 80°, then ∠P = 20°

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

Approach Solution 2:

The problem statement suggests that:

Given:

• PQR is an Isosceles Triangle
• ∠Q = 80 Degrees

Find out:

• The sum of all possible values of ∠P

Since in an isosceles triangle, two sides are equal, then two angles must be identical.

As per the condition of the question, it leads to three cases:

1) Since angle Q is developed by two equal sides, the remaining two angles (P and R) ought to be the same.

Angle Q + Angle P + Angle R=180 (since, the sum of all three angles in a triangle=180)
Therefore, we can say, 80 + angle P + angle R = 180
=> angle P + angle R = 100

As mentioned earlier, angle P and angle R ought to be equal, therefore P = 50 and R = 50.

2) Angle Q is developed by the base and one of the sides, and the Angle (let's assume R) is developed by the base and the other side.
Therefore, we can say, Q = R,
Hence, P = 180−80−80 = 20

3) Angle Q is developed by the base and one of the sides, and the Angle (Let’s say P) is developed by the base and the other side.
Therefore, we can say, Q = P, then P = 80

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

Approach Solution 3:

The problem statement implies that:

Given:

• PQR is an Isosceles Triangle
• ∠Q = 80 Degrees

Find out:

• The sum of all possible values of ∠P

As per the principle of an isosceles triangle, we know that two angles are equal,

Therefore, we can say,

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 = (180–80)/2 =50

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

“In an isosceles triangle PQR, if ∠Q = 80 degrees, then what is the''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book "GMAT Official Guide 2021". GMAT Problem Solving questions test the candidates’ abilities and knowledge to solve numerical problems. GMAT Quant practice papers enable the candidates to analyse several sorts of questions that will help them to improve their mathematical learning.

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