In the given figure, AM ⊥ BC and AN is the bisector of ∠A. Then ∠MAN is

Question

In the given figure, AM

⊥

BC and AN is the bisector of

∠

A. Then

∠

MAN is

Options:

A . 3212∘

B . 1612∘

C . 16∘

D . 32∘

Answer: Option C
:
C
In given figure,

∠ B A C + ∠ A B C + ∠ A C B = 180^{\circ} (Angles of same triangle) \Rightarrow ∠ B A C + 65^{\circ} + 33^{\circ} = 180^{\circ} \Rightarrow ∠ B A C = 82^{\circ}

∠ C A N = \frac{∠ B A C}{2} = 41^{\circ} (Given that AN is an angle bisector) . Now, ∠ C A N + ∠ A N C + ∠ A C N = 180^{\circ} (Angles of same triangle) \Rightarrow 41^{\circ} + ∠ A N C + 33^{\circ} = 180^{\circ} \Rightarrow ∠ A N C = 106^{\circ}

∠ A N C + ∠ A N M = 180^{\circ} (Linear pair) \Rightarrow 106^{\circ} + ∠ A N M = 180^{\circ} \Rightarrow ∠ A N M = 74^{\circ}

∠ A N M + ∠ A M N + ∠ M A N = 180^{\circ} \Rightarrow 74^{\circ} + 90^{\circ} + ∠ M A N = 180^{\circ} \Rightarrow ∠ M A N = 16^{\circ} .

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