Question

In ∆ABC, if ∠A − ∠B = 33° and ∠B − ∠C = 18°, find the measure of each angle of the triangle.

Answer 1

$$\begin{gathered} \angle A-\angle B=33°------\left(1\right) \\ \mathrm{and} \\ \angle B-\angle C=18°------\left(2\right) \\ \mathrm{From}\left(1\right)-\left(2\right),\mathrm{we}get: \\ \left(\angle A-\angle B\right)-\left(\angle B-\angle C\right)=33°-18° \\ \Rightarrow \angle A-\angle B-\angle B+\angle C=15° \\ \Rightarrow \angle A-2\angle B+\angle C=15°-----\left(3\right) \\ \mathrm{By}\mathrm{the}\mathrm{angle}\mathrm{sum}\mathrm{property},\mathrm{we}\mathrm{know}\mathrm{that} \\ \angle A+\angle B+\angle C=180°------\left(4\right) \\ \mathrm{From}\left(4\right)-\left(3\right),\mathrm{we}\text{get}: \\ \left(\angle A+\angle B+\angle C\right)-\left(\angle A-2\angle B+\angle C\right)=180°-15° \\ \Rightarrow \angle A+\angle B+\angle C-\angle A+2\angle B-\angle C=165° \\ \Rightarrow 3\angle B=165° \\ \Rightarrow \angle B=55° \\ Putting\angle B=55°\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get}: \\ \angle A-55°=33° \\ \Rightarrow \angle A=88° \\ \text{Putting}\angle B=55°\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get}: \\ 55°-\angle C=18° \\ \Rightarrow \angle C=55°-18°=37° \\ \mathrm{Therefore},\angle A=88°,\angle B=55°\mathrm{and}\angle C=37° \end{gathered}$$