Question

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

Answer 1

In the given ΔABC, AM⊥BC, ANis the bisector of ∠A, ∠B = 65 and ∠C = 33

We need to find ∠MAN

Now, using the angle sum property of the triangle

In ΔAMC, we get,

∠MAC + ∠AMC + ∠ACM = 180°

             ∠MAC + 90° + 33° = 180°

                    ∠MAC + 123° = 180

                                ∠MAC= 180° - 123°

∠MAC = 57°…….(1) 

Similarly,

In ΔABM, we get,

∠ABM + ∠AMB+ ∠BAM = 180°

           ∠BAM + 90°+ 65° = 180°

                     ∠AM + 155° = 180°

                             ∠BAM = 180° - 155°

∠BAM …..(2) 

So, adding (1) and (2)

∠BAM + ∠MAC = 25° + 57°

∠BAM + ∠MAC = 82°

Now, since AN is the bisector of  ∠A

∠BAN = ∠NAC

Thus, 

∠BAN + ∠NAC = 82°

             2∠BAN =82°

           `   ∠BAN = (82°)/2`

                         = 41

Now,

∠MAN = ∠BAN - ∠BAM

 = 41 - 25

= 16

Therefore, ∠MAN = 16.