In a `Delta ABC, CD` is the angle bisector of interior `angle C`. Which meets AB at D . M is a point on CD such that `AD=AM.` IF `angle B=59^(@)` find `angle MAC`.

Answer

To solve the problem step by step, we will analyze the triangle ABC with the given conditions:

### Step 1: Understand the Triangle and Given Information
- We have triangle ABC.
- CD is the angle bisector of angle C, which means that angle ACD = angle BCD.
- We know that angle B = 59°.
- M is a point on CD such that AD = AM.

### Step 2: Define Angles
Since CD is the angle bisector, we can denote:
- Let angle ACD = x
- Therefore, angle BCD = x as well (because CD bisects angle C).

### Step 3: Use the Exterior Angle Theorem
In triangle BDC, the exterior angle ADC can be expressed as:
- ADC = angle B + angle BCD
- Thus, ADC = 59° + x.

### Step 4: Relate Angles ADM and AMD
Since AD = AM, triangles ADM and AMD are isosceles triangles. Therefore:
- angle ADM = angle AMD.

### Step 5: Establish the Relationship Between Angles
From the previous steps, we know:
- angle ADC = angle ADM (because they are the same angle).
- Therefore, angle ADM = 59° + x.

### Step 6: Analyze Triangle AMC
In triangle AMC, we can apply the exterior angle theorem again:
- angle AMD = angle MAC + angle ACM.
- We know angle ACM = x (since it is equal to angle ACD).

### Step 7: Substitute Values
Now we can substitute the values:
- angle AMD = 59° + x,
- angle ACM = x.

So we have:
- 59° + x = angle MAC + x.

### Step 8: Solve for angle MAC
By simplifying the equation:
- 59° + x - x = angle MAC,
- Therefore, angle MAC = 59°.

### Final Answer
Thus, angle MAC = 59°.

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