The angles of a triangle are in the ratio 4 : 5 : 6 , find their measures in degrees .

Answer

To find the measures of the angles of a triangle that are in the ratio 4:5:6, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles in Terms of a Variable**:
Since the angles are in the ratio 4:5:6, we can express the angles as:
- Angle A = 4x
- Angle B = 5x
- Angle C = 6x

2. **Use the Triangle Angle Sum Property**:
We know that the sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ 4x + 5x + 6x = 180 \]

3. **Combine Like Terms**:
Combine the terms on the left side of the equation:
\[ (4x + 5x + 6x) = 15x \]
So, the equation becomes:
\[ 15x = 180 \]

4. **Solve for x**:
To find the value of x, divide both sides of the equation by 15:
\[ x = \frac{180}{15} \]
Simplifying this gives:
\[ x = 12 \]

5. **Calculate Each Angle**:
Now that we have the value of x, we can find each angle:
- Angle A = 4x = 4 × 12 = 48 degrees
- Angle B = 5x = 5 × 12 = 60 degrees
- Angle C = 6x = 6 × 12 = 72 degrees

6. **State the Final Answer**:
The measures of the angles of the triangle are:
- 48 degrees
- 60 degrees
- 72 degrees