ABCD is a cyclic quadrilateral. Diagonals AC and BD meets at P. If `angleAPB = 110^(@)` and `angleCBD = 30^(@)`, then `angleADB` measures

Options

A. `55^(@)`
B. `30^(@)`
C. `70^(@)`
D. `80^(@)`

Answer

To find the measure of angle ADB in the cyclic quadrilateral ABCD, we can follow these steps:

### Step 1: Identify the given angles
We are given:
- Angle APB = 110°
- Angle CBD = 30°

### Step 2: Use the property of vertically opposite angles
Since diagonals AC and BD intersect at point P, we can use the property of vertically opposite angles. Therefore:
- Angle DPC = Angle APB = 110°

### Step 3: Find angle APD using the linear pair property
Since angles APB and APD form a linear pair (they are on a straight line), we can find angle APD:
- Angle APD = 180° - Angle APB
- Angle APD = 180° - 110° = 70°

### Step 4: Use the property of angles in the same segment
According to the property of angles in the same segment, we know that:
- Angle CBD = Angle CAD
Thus, we have:
- Angle CAD = Angle CBD = 30°

### Step 5: Analyze triangle ADP
Now, we will consider triangle ADP. The sum of angles in a triangle is 180°. Therefore, we can write:
- Angle ADP + Angle APD + Angle CAD = 180°

Substituting the known values:
- Angle ADP + 70° + 30° = 180°

### Step 6: Solve for angle ADP
Now, we can simplify the equation:
- Angle ADP + 100° = 180°
- Angle ADP = 180° - 100° = 80°

### Conclusion
Thus, the measure of angle ADB is:
- Angle ADB = Angle ADP = 80°

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