Question

In Fig. 16.21, the bisectors of ∠A and ∠B meet at a point P. If ∠C = 100° and ∠D = 50°, find the measure of ∠APB.

Answer 1

$$\begin{gathered} \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{D}=360° \\ \Rightarrow \angle \mathrm{A}+\angle \mathrm{B}+100°+50°=360° \\ \Rightarrow \angle \mathrm{A}+\angle \mathrm{B}=210°...\left(\mathrm{i}\right) \\ \mathrm{In}△\mathrm{APB},\mathrm{we}\mathrm{have}: \\ \frac{1}{2}\angle \mathrm{A}+\frac{1}{2}\angle \mathrm{B}+\angle \mathrm{APB}=180° \\ \Rightarrow \angle \mathrm{APB}=180°-\frac{1}{2}\left(\angle \mathrm{A}+\angle \mathrm{B}\right) \\ \mathrm{From}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ \Rightarrow \angle \mathrm{APB}=180°-\left(\frac{1}{2}\times 210°\right) \\ \therefore \angle \mathrm{APB}=75° \end{gathered}$$