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

_________________

Hey KanishkM,

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

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 \(P = 50 + 20 + 80 = 150\).

Answer E.

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.

Answer: E


_________________