Question

In the adjoining figure, ABC is an isosceles triangle in which AB = AC and PQ is parallel to BC. If ∠A = 40°, find ∠PQC.

Answer 1

Given:

AB = AC,

PQ || BC,

∠A = 40°.

Find ∠PQC.

Step-wise calculation:

1. Since AB = AC, the base angles are equal:

∠B = ∠C

= `(180^circ - 40^circ)/2`

= 70°

2. Q lies on AC and P lies on AB; PQ || BC.

The ray QP is parallel to BC and the ray QC points from Q toward C (i.e. along AC but toward C).

3. ∠PQC is the angle between the rays QP and QC.

Compare this with ∠C = ∠ACB, which is the angle between CA (from C to A) and CB (from C to B).

The ray QC is the opposite direction to CA (QC points toward C while CA at C points toward A).

So, the angle between QP || BC and QC is the supplement of ∠C. 

Therefore, ∠PQC

= 180° – ∠C 

= 180° – 70° 

= 110°