Question

P is a point on the arc APB of a circle witg centre O. Show that the sum of angles PAB and PBA is constant.

Answer 1

Let ∠OBP = ∠OPB = α, and ∠OAP = ∠OPA = β
∠OAB = ∠OBA = γ
Note 2α + 2β + 2γ = 180°
which implies α + β + γ = 90°
Therefore,
∠PAB + ∠PBA
= γ + β + γ + α = 90° + γ
which is independent of α, β

Hence the sum of ∠PAB and ∠PBA is constant