Question

In ΔPQR, PQ = QR and ΔPSR is equilateral. If ∠Q = 26°, find y.

Answer 1

Given: In triangle (ΔPQR):

PQ = QR so it is isosceles with PQ = QR.

ΔPSR is equilateral.

∠Q = 26°.

We need to find the value of y as labeled in the given figure.

Step 1: Analyze triangle (ΔPQR )

Since PQ = QR, triangle PQR is isosceles with PQ = QR.

Thus, angles opposite these sides are equal:

∠P = ∠R = y

Given: ∠Q = 26°

The sum of all angles in a triangle is 180°:

∠P + ∠Q + ∠R = 180°

y + 26° + y = 180° 

2y = 180° – 26° 

2y = 154° 

y = 77°

Step 2: Check impact of equilateral triangle (ΔPSR)

Since ΔPSR is equilateral, all its angles are 60° and all sides PS = SR = PR.

This fact establishes PR = PS = SR.

This constraints the angles in the bigger triangle PQR, but since the problem only asks for y, the isosceles property provides the answer directly.

So, the value of y is 77°.