Question

ABCD is a rhombus. ΔPAD is equilateral, ∠B = 70°. ∴ The measure of angle x is ______.

Options

• 20°

• 25°

• 30°

• 35°

Answer 1

ABCD is a rhombus. ΔPAD is equilateral, ∠B = 70°. ∴ The measure of angle x is 25°.

Explanation:

Given the rhombus ABCD and the equilateral triangle ΔPAD constructed externally on side AD and with ∠B = 70°:

Step 1: Find the angles of the rhombus

In a rhombus, opposite angles are equal.

Therefore, ∠D = ∠B = 70°.

Step 2: Determine the angles of ΔCDP

Since ΔPAD is equilateral, its angle ∠ADP = 60°. Because ΔPAD is external to the rhombus, the total angle at vertex D is the sum of the rhombus angle and the triangle angle:

∠CDP = ∠ADC + ∠ADP

= 70° + 60° 

= 130°

In a rhombus, all sides are equal CD = AD.

In an equilateral triangle, all sides are equal AD = DP. 

This means CD = DP, so ΔCDP is an isosceles triangle.

Step 3: Solve for x

In isosceles triangle ΔCDP, the angles opposite the equal sides are equal ∠PCD = ∠CPD = x. 

The sum of angles in any triangle is 180°.

2x + 130° = 180°

2x = 50°

x = 25°

The measure of angle x is 25°.