In a quadrilateral ABCD, the bisectors of `/_A` and `/_B` meet at O. If `/_C = 70^@ and /_D= 130^@`, then measure of `/_AOB `is

Options

A. `40^@`
B. `60^@`
C. `80^@`
D. `100^@`

Answer

To find the measure of angle AOB in quadrilateral ABCD, where the angle bisectors of angles A and B meet at point O, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Angles**:
We know that:
- Angle C = 70°
- Angle D = 130°

2. **Use the Angle Sum Property of Quadrilaterals**:
The sum of the interior angles of a quadrilateral is 360°. Therefore, we can express the relationship as:
\[ \angle A + \angle B + \angle C + \angle D = 360° \]

3. **Substitute Known Values**:
Substitute the values of angles C and D into the equation:
\[ \angle A + \angle B + 70° + 130° = 360° \]

4. **Simplify the Equation**:
Combine the known angles:
\[ \angle A + \angle B + 200° = 360° \]
Now, isolate the sum of angles A and B:
\[ \angle A + \angle B = 360° - 200° = 160° \]

5. **Use the Angle Bisector Theorem**:
The angle bisectors of angles A and B meet at point O. The measure of angle AOB can be calculated using the formula:
\[ \angle AOB = \frac{\angle C + \angle D}{2} \]

6. **Substitute Values for C and D**:
Now substitute the values of angles C and D:
\[ \angle AOB = \frac{70° + 130°}{2} \]

7. **Calculate the Result**:
Add the angles and divide by 2:
\[ \angle AOB = \frac{200°}{2} = 100° \]

### Final Answer:
The measure of angle AOB is **100°**.

---