Question

PQRS is a square. ΔOPQ is equilateral. The measure of angle a is ______.

Options

• 80°

• 70°

• 75°

• 60°

Answer 1

PQRS is a square. ΔOPQ is equilateral. The measure of angle a is 75°.

Explanation:

Step 1: Use the properties of a square and an equilateral triangle

Given that PQRS is a square, all its sides are equal in length PQ = PS. All interior angles of a square are 90°, so ∠SPQ = 90°.

Given that ΔOPQ is an equilateral triangle, all its sides are equal PQ = PO. All interior angles of an equilateral triangle are 60°, so ∠OPQ = 60°.

Step 2: Identify the isosceles triangle

Since PQ = PS from the square and PQ = PO from the equilateral triangle, it follows that PS = PO. This means that ΔSPO is an isosceles triangle.

Step 3: Calculate the angles of the isosceles triangle

First, find the measure of the angle at the vertex P in ΔSPO.

∠SPO = ∠SPQ – ∠OPQ

= 90° – 60°

= 30°

In an isosceles triangle, the base angles are equal. The base angles are ∠PSO and ∠POS.

Let’s assume the angle ‘a’ in the problem is ∠PSO. The sum of angles in a triangle is 180°.

∠SPO + ∠PSO + ∠POS = 180°

30° + a + a = 180°

30° + 2a = 180°

Step 4: Solve for ‘a’

Subtract 30° from both sides of the equation:

2a = 180° – 30°

2a = 150°

Divide by 2 to find the value of a:

a = `150^circ/2`

a = 75°