One angle of a quadrilateral has measure `(2pi^(c))/(5)` and the measures of other three angles are in the ratio `2:3:4` . Find their measures in radians and in degrees.

Answer

To solve the problem, we need to find the measures of the angles of a quadrilateral given one angle and the ratio of the other three angles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- One angle of the quadrilateral is \( \frac{2\pi}{5} \) radians.
- The other three angles are in the ratio \( 2:3:4 \).

2. **Sum of Angles in a Quadrilateral:**
- The sum of all angles in a quadrilateral is \( 360^\circ \) or \( 2\pi \) radians.

3. **Let the Angles be Represented:**
- Let the three angles in the ratio \( 2:3:4 \) be represented as \( 2C, 3C, \) and \( 4C \).

4. **Set Up the Equation:**
- The equation for the sum of the angles can be set up as:
\[ 2C + 3C + 4C + \frac{2\pi}{5} = 2\pi \]

5. **Combine Like Terms:**
- Combine the terms involving \( C \):
\[ 9C + \frac{2\pi}{5} = 2\pi \]

6. **Isolate \( C \):**
- Subtract \( \frac{2\pi}{5} \) from both sides:
\[ 9C = 2\pi - \frac{2\pi}{5} \]
- To combine the terms on the right, convert \( 2\pi \) into a fraction:
\[ 2\pi = \frac{10\pi}{5} \]
- Now the equation becomes:
\[ 9C = \frac{10\pi}{5} - \frac{2\pi}{5} = \frac{8\pi}{5} \]

7. **Solve for \( C \):**
- Divide both sides by 9:
\[ C = \frac{8\pi}{5 \times 9} = \frac{8\pi}{45} \]

8. **Calculate Each Angle:**
- Now we can find the measures of the three angles:
- First angle:
\[ 2C = 2 \times \frac{8\pi}{45} = \frac{16\pi}{45} \]
- Second angle:
\[ 3C = 3 \times \frac{8\pi}{45} = \frac{24\pi}{45} \]
- Third angle:
\[ 4C = 4 \times \frac{8\pi}{45} = \frac{32\pi}{45} \]

9. **List All Angles in Radians:**
- The four angles in radians are:
- \( \frac{2\pi}{5} \)
- \( \frac{16\pi}{45} \)
- \( \frac{24\pi}{45} \)
- \( \frac{32\pi}{45} \)

10. **Convert Radians to Degrees:**
- Use the conversion formula \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \):
- For \( \frac{16\pi}{45} \):
\[ \frac{16\pi}{45} \times \frac{180}{\pi} = \frac{16 \times 180}{45} = 64^\circ \]
- For \( \frac{24\pi}{45} \):
\[ \frac{24\pi}{45} \times \frac{180}{\pi} = \frac{24 \times 180}{45} = 96^\circ \]
- For \( \frac{32\pi}{45} \):
\[ \frac{32\pi}{45} \times \frac{180}{\pi} = \frac{32 \times 180}{45} = 128^\circ \]
- For \( \frac{2\pi}{5} \):
\[ \frac{2\pi}{5} \times \frac{180}{\pi} = \frac{2 \times 180}{5} = 72^\circ \]

11. **Final Angles in Degrees:**
- The angles in degrees are:
- \( 64^\circ \)
- \( 96^\circ \)
- \( 128^\circ \)
- \( 72^\circ \)