The measures of the four angles of a quadrilateral are in the ratio of 1:2:3: 4. What is the measure of the greatest angle of quadrilateral?

Options

A. `144^(@)`
B. `135^(@)`
C. `125^(@)`
D. `150^(@)`

Answer

To find the measure of the greatest angle of a quadrilateral where the angles are in the ratio of 1:2:3:4, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles in Terms of a Variable:**
Let the angles of the quadrilateral be represented as follows based on the given ratio:
- Angle A = 1x
- Angle B = 2x
- Angle C = 3x
- Angle D = 4x

2. **Set Up the Equation for the Sum of Angles:**
We know that the sum of the angles in a quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360 \]
Substituting the expressions for the angles:
\[ x + 2x + 3x + 4x = 360 \]

3. **Combine Like Terms:**
Combine the terms on the left side of the equation:
\[ 10x = 360 \]

4. **Solve for x:**
To find the value of x, divide both sides of the equation by 10:
\[ x = \frac{360}{10} = 36 \]

5. **Calculate Each Angle:**
Now that we have the value of x, we can find each angle:
- Angle A = 1x = 1(36) = 36 degrees
- Angle B = 2x = 2(36) = 72 degrees
- Angle C = 3x = 3(36) = 108 degrees
- Angle D = 4x = 4(36) = 144 degrees

6. **Identify the Greatest Angle:**
Among the angles calculated, the greatest angle is:
\[ \text{Angle D} = 144 \text{ degrees} \]

### Final Answer:
The measure of the greatest angle of the quadrilateral is **144 degrees**.