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

Let ∠BAN = ∠NAC = x [∵ AN bisects ∠A]

∴ ∠ANM = x + 33° [Exterior angle property]

In ΔAMB

∠BAM = 90° − 65° = 25° [Exterior angle property]

∴ ∠MAN = ∠BAN − ∠BAM = (x − 25)°

Now in ΔMAN,

(x - 25)° + (x + 33)° + 90° = 180° [Angle sum property]

⇒ 2x + 8° = 90°

⇒ 2x = 82°

⇒ x = 41°

∴ MAN = x − 25°

= 41° − 25°

= 16°