Question

The diagram shows triangle PQR.

PSR is a straight line.
size of angle PQS : size of angle SQR = 3:1
S
Work out the size of angle PSQ

Answer 1

Angle PSQ is 61°

Explanation:

In the triangle ΔPQR, we have;

m∠QPS = 65°

m∠PRQ = 43°

Line PSR is a straight line

m∠PQS : m∠SQR = 3:1

Therefore;

∠PQR = 180° - (∠QPS + ∠PRQ), by the angle sum theorem of a triangle

∴ ∠PQR = 180° - (65° + 43°) = 72°

∠PQR = 72°

∠PQR = m∠PQS + m∠SQR by angle addition postulate

m∠PQS : m∠SQR = 3:1

∴ m∠PQS = 3 × m∠SQR

∴ ∠PQR = m∠PQS + m∠SQR = 3 × m∠SQR + m∠SQR = 4 × m∠SQR

∠PQR = 4 × m∠SQR

∴ 4 × m∠SQR = ∠PQR = 72°

m∠SQR = 72°/4 = 18°

m∠PQS = 3 × m∠SQR = 3 × 18° = 54°

m∠PQS = 54°

∠PSQ = 180° - (∠QPS + ∠PQS) by the angle sum theorem

∴ ∠PSQ = 180° - (65° + 54°) = 61°

Angle ∠PSQ = 61°.