Three angles of a quadrilateral are in the ratio 4:6:3. If the fourth angle is `100^@`, find the ther three angles of the quadrilateral.

Options

A. `90^@,110^@ and 60^@`
B. `70^@,110^@ and 60^@`
C. `80^@,120^@ and 60^@`
D. `80^@,120^@ and 80^@`

Answer

To find the three angles of the quadrilateral given that they are in the ratio 4:6:3 and the fourth angle is 100 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that the sum of all angles in a quadrilateral is 360 degrees. We are given three angles in the ratio 4:6:3 and the fourth angle is 100 degrees.

2. **Assign Variables**: Let the three angles be represented as:
- First angle = 4x
- Second angle = 6x
- Third angle = 3x

3. **Set Up the Equation**: According to the property of quadrilaterals, the sum of the angles is:
\[ 4x + 6x + 3x + 100 = 360 \]

4. **Combine Like Terms**: Combine the terms involving x:
\[ 13x + 100 = 360 \]

5. **Isolate the Variable**: Subtract 100 from both sides to isolate the term with x:
\[ 13x = 360 - 100 \]
\[ 13x = 260 \]

6. **Solve for x**: Divide both sides by 13 to find the value of x:
\[ x = \frac{260}{13} \]
\[ x = 20 \]

7. **Calculate the Angles**: Now substitute x back into the expressions for the angles:
- First angle = \( 4x = 4 \times 20 = 80 \) degrees
- Second angle = \( 6x = 6 \times 20 = 120 \) degrees
- Third angle = \( 3x = 3 \times 20 = 60 \) degrees

8. **Final Result**: The three angles of the quadrilateral are:
- First angle = 80 degrees
- Second angle = 120 degrees
- Third angle = 60 degrees

### Summary of the Angles:
- First angle: 80 degrees
- Second angle: 120 degrees
- Third angle: 60 degrees
- Fourth angle: 100 degrees (given)