Step-by-step EXPLANATION:

Step 1:

Suppose that  ANGLE  ''A'' =  2x.

Looking at the fact that  AN is bisector of angle  A you may write that

angle "BAN" = "NAC"  i.e. = x

Step 2:

In the triangle ( ΔABC) the total of all the angles must be 180 degrees, as per common law rules.

So the sum of all angles of triangle must be  ∠B  + ∠C + ∠A = 180°

Step 3: We have following missing values:-

65 + 30 + angle "A" = 180°

=180-( 65+30)

=180-(95)

=85 degrees.

You found value for angle "A"

Step 4:

Now you must do angle "A"/2 i.e.  x = 85 degrees /2

Step 5:

in triangle ABN another triangle ANC is the exterior angle so use this law of total of exterior angles to find out missing value.

Where; x + 65  =  angle "ANC"

85/2 + 56 = angle "ANC "

∠ANC = 215/2

Step 6:

After finding the angle LOOK at MNA =  180 degrees

Next you do this

180- (215/2) = 143/2

Step 7:

in the triangle AMN, this is the situation that; 143/2 + 90 +  angle MAN = 180°

thus; angle MAN = 90 - 143/2

Making answer  angle MAN = 35/2

final answer: 17.5 degrees