Question

in a triangle ABC , CD is the bisector of interior angle C which meets AB at D M is the point onCD such that AD = AM if angle B=59 find angle MAC​

Answer 1

Given :

Triangle ABC, in which CD is an angel bisector of angle C.

∠ B = 59°   and      AM = AD

To Find :

Value of angle ∠ MAC

Solution :

In the attached figure below,

Let the value of the angle ∠ MAC = β°

and the value of the angle ∠ C = 2α°

∴ ∠ ACD = ∠ DCB = α      ( internal angle bisector)

Now, in triangle ADM,

angle∠ MDA is the exterior angle of triangle BCD.

which means the value of the angle ∠ MDA will be equal to the sum of the angle ∠DCM and ∠CBD.

As we know that the exterior angle of a triangle is equal to the sum of the two opposite interior angle of the triangle.

∴ ∠ MDA = ∠ DCM + ∠ CBD

  ∠ MDA = α + 59°

and ∵ AM = AD

∴  ∠ MDA = ∠ AMD = α + 59°

now, in triangle ADM and AMC,

angle ∠ AMD is the exterior angle of the triangle AMC,

∴ ∠ AMD = ∠ MAC + ∠ ACM

∴ ∠ AMD = β + α

∴ α + 59° = β + α

∴ β = 59°

So,  angle ∠ MAC = 59°