The angles of a quadrilateral are in the ratio `2:4:5:7.` Find the angles.

Answer

To find the angles of a quadrilateral given in the ratio \(2:4:5:7\), we can follow these steps:

### Step-by-Step Solution:

1. **Set up the angles in terms of a variable**:
Let the angles of the quadrilateral be represented as:
- First angle = \(2x\)
- Second angle = \(4x\)
- Third angle = \(5x\)
- Fourth angle = \(7x\)

2. **Use the property of the sum of angles in a quadrilateral**:
The sum of the angles in a quadrilateral is always \(360^\circ\). Therefore, we can write the equation:
\[ 2x + 4x + 5x + 7x = 360^\circ \]

3. **Combine like terms**:
Combine the terms on the left side:
\[ (2 + 4 + 5 + 7)x = 360^\circ \]
This simplifies to:
\[ 18x = 360^\circ \]

4. **Solve for \(x\)**:
To find \(x\), divide both sides of the equation by \(18\):
\[ x = \frac{360^\circ}{18} = 20^\circ \]

5. **Calculate each angle**:
Now that we have the value of \(x\), we can find each angle:
- First angle = \(2x = 2 \times 20^\circ = 40^\circ\)
- Second angle = \(4x = 4 \times 20^\circ = 80^\circ\)
- Third angle = \(5x = 5 \times 20^\circ = 100^\circ\)
- Fourth angle = \(7x = 7 \times 20^\circ = 140^\circ\)

6. **List the angles**:
The angles of the quadrilateral are:
- First angle: \(40^\circ\)
- Second angle: \(80^\circ\)
- Third angle: \(100^\circ\)
- Fourth angle: \(140^\circ\)

### Final Answer:
The angles of the quadrilateral are \(40^\circ\), \(80^\circ\), \(100^\circ\), and \(140^\circ\).