Question

In ABC, PQ || BC. If PA = PC, ∠B = 74° and ∠PCB = 56°, find angles x and y.

Answer 1

To solve for the angles x and y in triangle ABC with the given conditions, we can follow these steps:

Given:

PQ || BC (PQ is parallel to BC)

PA = PC (PA and PC are equal)

∠B = 74°

∠PCB = 56°

Step 1: Use the Property of Parallel Lines (Alternate Interior Angles)

Since PQ || BC, we know that corresponding angles formed by a transversal with two parallel lines are equal.

Therefore, ∠PQA = ∠PCB because PQ || BC.

Hence, ∠PQA = 56° (since ∠PCB = 56°).

Step 2: Triangle PBC

Now, consider triangle PBC, where PA = PC.

This makes triangle PCA an isosceles triangle, so we have:

∠PCA = ∠PCB = 56°.

Step 3: Sum of Angles in Triangle ABC

In triangle ABC, the sum of the angles is always 180°.

So, we can find ∠A angle at vertex A by using the fact that ∠A + ∠B + ∠C = 180°.

We already know ∠B = 74° and ∠C is related to ∠PCB = 56°.

Step 4: Finding Angles x and y

1. Since PA = PC, the triangle PCA is isosceles,

So, x = ∠PCA = 56°, the angle formed by the line segments PA and PC.

2. Now, since ∠B = 74° and we are asked to find y:

The remaining angle y

= 180° – ∠B – ∠C 

= 180° – 74° – 56°

= 81°

Thus, we have:

x = 25°

y = 81°