The angles of a triangle are in the ratio 3 : 4 : 5. If l be the incentre of `DeltaABC` then find the measure of `angleADC` where AD, is the bisector of `angleA`.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Identify the Angles of the Triangle:**
The angles of the triangle are given in the ratio 3:4:5. Let the angles be represented as:
- Angle A = 3x
- Angle B = 4x
- Angle C = 5x

2. **Set Up the Equation:**
The sum of the angles in a triangle is always 180 degrees. Therefore, we can set up the equation:
\[ 3x + 4x + 5x = 180 \]

3. **Combine Like Terms:**
Combine the terms on the left side:
\[ 12x = 180 \]

4. **Solve for x:**
Divide both sides by 12 to find the value of x:
\[ x = \frac{180}{12} = 15 \]

5. **Calculate the Angles:**
Now substitute the value of x back into the expressions for the angles:
- Angle A = 3x = 3(15) = 45 degrees
- Angle B = 4x = 4(15) = 60 degrees
- Angle C = 5x = 5(15) = 75 degrees

6. **Identify the Incenter and Angle Bisector:**
Let D be the point where the angle bisector of angle A intersects side BC. Since AD is the angle bisector, angle DAC = angle DAB = 22.5 degrees (since angle A = 45 degrees).

7. **Use the Triangle ADC:**
In triangle ADC, we can use the fact that the sum of the angles in a triangle is 180 degrees:
\[ \angle ACD + \angle DAC + \angle ADC = 180 \]
Here, angle ACD = angle C = 75 degrees and angle DAC = 22.5 degrees.

8. **Set Up the Equation:**
Substitute the known values into the equation:
\[ 75 + 22.5 + \angle ADC = 180 \]

9. **Solve for Angle ADC:**
Combine the known angles:
\[ 97.5 + \angle ADC = 180 \]
Now, isolate angle ADC:
\[ \angle ADC = 180 - 97.5 = 82.5 \text{ degrees} \]

### Final Answer:
The measure of angle ADC is **82.5 degrees**.