The angles of a quadrilateral are in the ratio `1:2:4:5`. Find the measure of each angle.

Answer

To solve the problem of finding the angles of a quadrilateral in the ratio of 1:2:4:5, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles**:
Let the angles of the quadrilateral be represented as:
- First angle = \( x \)
- Second angle = \( 2x \)
- Third angle = \( 4x \)
- Fourth angle = \( 5x \)

2. **Set Up the Equation**:
The sum of the angles in any quadrilateral is always equal to \( 360^\circ \). Therefore, we can write the equation:
\[ x + 2x + 4x + 5x = 360^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side of the equation:
\[ (1x + 2x + 4x + 5x) = 12x \]
Thus, the equation simplifies to:
\[ 12x = 360^\circ \]

4. **Solve for \( x \)**:
To find the value of \( x \), divide both sides of the equation by 12:
\[ x = \frac{360^\circ}{12} = 30^\circ \]

5. **Calculate Each Angle**:
Now that we have the value of \( x \), we can find each angle:
- First angle = \( x = 30^\circ \)
- Second angle = \( 2x = 2 \times 30^\circ = 60^\circ \)
- Third angle = \( 4x = 4 \times 30^\circ = 120^\circ \)
- Fourth angle = \( 5x = 5 \times 30^\circ = 150^\circ \)

6. **Conclusion**:
The measures of the angles in the quadrilateral are:
- First angle: \( 30^\circ \)
- Second angle: \( 60^\circ \)
- Third angle: \( 120^\circ \)
- Fourth angle: \( 150^\circ \)

### Final Answer:
The angles of the quadrilateral are \( 30^\circ, 60^\circ, 120^\circ, \) and \( 150^\circ \).

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