Question

In the given figure, AM ⊥ BC and AN is the bisector of ∠A. If ∠ABC = 70° and ∠ACB = 20°, then ∠MAN = ?

Answer 1

In $∆ABC$, we have:
$$\begin{gathered} \angle A+\angle B+\angle C=180°\left[\text{Sum of the angles of a triangle}\right] \\ \Rightarrow \angle A+70°+20°=180° \\ \Rightarrow \angle A=90° \\ \Rightarrow \frac{1}{2}\angle A=45° \\ \Rightarrow \angle BAN=45° \end{gathered}$$
In $∆ABM$, we have:
$$\begin{gathered} \angle ABM+\angle AMB+\angle BAM=180°\left[\text{Sum of the angles of a triangle}\right] \\ \Rightarrow 70°+90°+\angle BAM=180° \\ \Rightarrow \angle BAM=20° \\ \therefore \angle MAN=\angle BAN-\angle BAM \\ \Rightarrow \angle MAN=45°-20° \\ \Rightarrow \angle MAN=25° \end{gathered}$$