In a quadrilateral ABCD, OA and OB are the angle bi- sectors of `angleDABandangleCBA`. If`∠ADC=70 and ∠ BCD= 80 then find the angle AOB

Options

A. `70^(@)`
B. `80^(@)`
C. `90^(@)`
D. `75^(@)`

Answer

To find the angle AOB in the quadrilateral ABCD, where OA and OB are the angle bisectors of angles DAB and CBA respectively, we can follow these steps:

### Step 1: Understand the given angles
We are given:
- ∠ADC = 70°
- ∠BCD = 80°

### Step 2: Define the angles at points A and B
Let:
- ∠DAB = 2x (since OA is the angle bisector)
- ∠CBA = 2y (since OB is the angle bisector)

### Step 3: Use the property of quadrilaterals
The sum of all angles in a quadrilateral is 360°. Therefore, we can set up the equation:
\[ \angle DAB + \angle ABC + \angle BCD + \angle ADC = 360° \]
Substituting the angles we have:
\[ 2x + 2y + 80° + 70° = 360° \]

### Step 4: Simplify the equation
Combine the known angles:
\[ 2x + 2y + 150° = 360° \]
Subtract 150° from both sides:
\[ 2x + 2y = 210° \]
Dividing the entire equation by 2 gives:
\[ x + y = 105° \]

### Step 5: Find angle AOB
In triangle AOB, we know that:
\[ \angle AOB = 180° - (∠DAB + ∠CBA) = 180° - (2x + 2y) \]
Substituting \(x + y = 105°\):
\[ \angle AOB = 180° - 2(x + y) = 180° - 2(105°) = 180° - 210° = -30° \]
This indicates an error in the calculation. Let's correct it.

### Correct Calculation for angle AOB
Since we need to find \( \angle AOB \):
\[ \angle AOB = 180° - (x + y) \]
Substituting \(x + y = 105°\):
\[ \angle AOB = 180° - 105° = 75° \]

### Final Answer
Thus, the angle AOB is:
\[ \angle AOB = 75° \]

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