Question

In the Fig., PQR is an isosceles triangle with PQ = PR and ∠PQR = 35°. Find ∠QSR.

Answer 1

Given:

Triangle PQR is isosceles with PQ = PR.

∠PQR = 35°

Need to find ∠QSR.

Since PQR is isosceles with PQ = PR, the base is QR.

In an isosceles triangle, the angles opposite equal sides are equal.

Given ∠PQR = 35°, so ∠PRQ = 35° (since PQ = PR).

Sum of angles in triangle PQR = 180°

∠PQR + ∠PRQ + ∠QPR = 180°

35° + 35° + ∠QPR = 180°

∠QPR = 180° - 70° = 110°

Point S lies on line segment PR (inside the circle), not on the circumference.

Therefore, we cannot use “same chord, same arc” properties with point S.

∠QSR is an exterior angle of triangle PQS.

An exterior angle of a triangle equals the opposite interior angle.

∠QSR = ∠QPS.

But ∠QPS is part of ∠QPR, and ∠QPR = 110°.

∠QSR = 110°.