Angles of a triangle are `(3x)^(@),(2x-7) and (4x-11)^(@)` . Find the measure of x and each angle of the triangle

Answer

To solve the problem, we need to find the value of \( x \) and each angle of the triangle given the angles as \( 3x^\circ \), \( 2x - 7^\circ \), and \( 4x - 11^\circ \).

### Step-by-Step Solution:

1. **Write the equation for the sum of angles in a triangle**:
The sum of the angles in any triangle is always \( 180^\circ \). Therefore, we can write the equation:
\[ 3x + (2x - 7) + (4x - 11) = 180 \]

2. **Combine like terms**:
Now, we will combine the terms involving \( x \) and the constant terms:
\[ 3x + 2x + 4x - 7 - 11 = 180 \]
This simplifies to:
\[ 9x - 18 = 180 \]

3. **Isolate \( x \)**:
Next, we will isolate \( x \) by adding \( 18 \) to both sides of the equation:
\[ 9x = 180 + 18 \]
Simplifying this gives:
\[ 9x = 198 \]

4. **Solve for \( x \)**:
Now, divide both sides by \( 9 \):
\[ x = \frac{198}{9} \]
Calculating this gives:
\[ x = 22 \]

5. **Calculate each angle**:
Now that we have \( x \), we can find each angle:
- For angle \( A \):
\[ A = 3x = 3 \times 22 = 66^\circ \]
- For angle \( B \):
\[ B = 2x - 7 = 2 \times 22 - 7 = 44 - 7 = 37^\circ \]
- For angle \( C \):
\[ C = 4x - 11 = 4 \times 22 - 11 = 88 - 11 = 77^\circ \]

6. **Final result**:
The values of \( x \) and the angles are:
- \( x = 22 \)
- Angle \( A = 66^\circ \)
- Angle \( B = 37^\circ \)
- Angle \( C = 77^\circ \)

### Summary:
- The measure of \( x \) is \( 22 \).
- The angles of the triangle are \( 66^\circ \), \( 37^\circ \), and \( 77^\circ \).