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 the measures of each angle in the triangle. The angles are given as \( 3x^\circ \), \( (2x - 7)^\circ \), and \( (4x - 11)^\circ \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write the equation:
\[ 3x + (2x - 7) + (4x - 11) = 180 \]

2. **Combine Like Terms**:
Combine the \( x \) terms and the constant terms:
\[ 3x + 2x + 4x - 7 - 11 = 180 \]
This simplifies to:
\[ 9x - 18 = 180 \]

3. **Isolate \( x \)**:
Add \( 18 \) to both sides to isolate the term with \( x \):
\[ 9x = 180 + 18 \]
Simplifying gives:
\[ 9x = 198 \]

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

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

6. **Summary of Results**:
- The value of \( x \) is \( 22 \).
- The angles of the triangle are:
- Angle A: \( 66^\circ \)
- Angle B: \( 37^\circ \)
- Angle C: \( 77^\circ \)