Question

ABCD is a rhombus, ΔPAB is equilateral, ∠D = 68°. Find the values of x and y.

Answer 1

Step-by-step

Step 1: Utilize Properties of Rhombus and Equilateral Triangle

It is known that in a rhombus, all sides are equal in length. 

Therefore, AB = BC = CD = DA.

It is also known that in an equilateral triangle, all sides are equal and all angles are 60°.

Thus, in ΔPAB, PA = PB = AB and ∠PAB = ∠PBA = ∠APB = 60°.

Step 2: Determine Angles in the Rhombus

In a rhombus, opposite angles are equal.

Given ∠D = 68°, it follows that ∠ABC = ∠D = 68°.

Adjacent angles in a rhombus are supplementary.

Therefore, ∠A = 180° – ∠B = 180° – 68° = 112°.

Step 3: Identify Isosceles Triangle PBC

From Step 1, it is established that AB = PB and AB = BC. Consequently, PB = BC.

This implies that ΔPBC is an isosceles triangle and the base angles opposite the equal sides are equal, so ∠BCP = ∠BPC = x.

Step 4: Calculate Angle PBC

The angle ∠PBC is formed by the sum of ∠PBA and ∠ABC.

∠PBC = ∠PBA + ∠ABC = 60° + 68° = 128°.

Step 5: Solve for x in Triangle PBC

The sum of angles in ΔPBC is 180°.

Therefore, ∠BPC + ∠BCP + ∠PBC = 180°.

Substituting the known values, x + x + 128° = 180°.

This simplifies to 2x = 180° – 128° = 52°.

Solving for x, x = `52^circ/2` = 26°.

Step 6: Solve for y

It is given that ∠APB = 60°.

The angle ∠APB is composed of ∠APD and ∠DPB.

Also, ∠APB = ∠APD + ∠DPB.

From the diagram, it is observed that ∠APB = ∠APD + ∠DPB = y + x.

Therefore, 60° = y + x.

Substituting the value of x found in Step 5, 60° = y + 26°.

Solving for y, y = 60° – 26° = 34°.