Question

In Fig. 9.37, AM ⊥ BC and AN is the bisector of ∠A. If ∠B =65° and ∠C =33°, find ∠MAN.

Answer 1

In Fig. 9.37, AM ⊥ BC and AN is the bisector of ∠A. If ∠B =65° and ∠C =33°,  ∠MAN = 16°

Given,

AM ⊥ BC

AN is the bisector of ∠A

∠B =65°

∠C =33°

Consider the following attached figure while going through the following steps.

In Δ AMC

∠ MAC + ∠ AMC + ∠ ACM = 180°

∠ MAC + 90° + 33° = 180°

∠ MAC = 180° - 90° - 33°

∴ ∠ MAC = 57° ..........(1)

In Δ ABM

∠ ABM + ∠ AMB + ∠ BAM = 180°

∠ BAM + 90° + 65° = 180°

∠ BAM = 180° - 90° - 65°

∴ ∠ BAM = 25° ..........(2)

Adding (1) and (2), we get,

∠ BAM + ∠ MAC = 25° + 57°

∠ BAM + ∠ MAC = 82°

∠ BAN + ∠ NAC (given - AN is the bisector of ∠A)

Therefore,

∠ BAN + ∠ NAC = 82°

2 ∠ BAN = 82°

∠ BAN = 82°/2

∴ ∠ BAN = 41°

Now, consider

∠ MAN = ∠ BAN  - ∠ BAM

= 41° - 25°

∴ ∠ MAN = 16°