Question

In ∆ABC, if ∠A − ∠B = 42° and ∠B − ∠C = 21°, ∠B = ?
(a) 95°
(b) 53°
(c) 32°
(d) 63°

Answer 1

(b) 53°

Let
$$\begin{gathered} \angle A-\angle B=42°...\left(i\right)\text{and} \\ \angle B-\angle C=21°...\left(ii\right) \\ \mathrm{Adding}\left(i\right)\mathrm{and}\left(ii\right),\text{we get:} \\ \angle A-\angle C=63° \\ \angle B=\angle A-42°\left[\text{Using}\left(i\right)\right] \\ \angle C=\angle A-63°\left[\text{Using}\left(iii\right)\right] \end{gathered}$$

$$\begin{gathered} \therefore \angle A+\angle B+\angle C=180°\left[\text{Sum of the angles of a triangle}\right] \\ \Rightarrow \angle A+\angle A-42°+\angle A-63°=180° \\ \Rightarrow 3\angle A-105°=180° \\ \Rightarrow 3\angle A=285° \\ \Rightarrow \angle A=95° \\ \therefore \angle B=\left(95-42\right)° \\ \Rightarrow \angle B=53° \end{gathered}$$