ABCD is a cyclic quadrilateral. PBQ is a tangent drawn from the point B of the circle, if `/_DBP = 65^(@)`, then measures of `/_BCD` is

Options

A. `65^(@)`
B. `90^(@)`
C. `110^(@)`
D. `115^(@)`

Answer

To solve the problem, we need to find the measure of angle BCD in the cyclic quadrilateral ABCD, given that angle DBP is 65 degrees.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- ABCD is a cyclic quadrilateral.
- PBQ is a tangent to the circle at point B.
- Angle DBP = 65 degrees.

2. **Draw the Figure**:
- Draw the cyclic quadrilateral ABCD.
- Mark the tangent PBQ at point B.
- Label the angles as given in the problem.

3. **Identify Angles**:
- Since PBQ is a tangent to the circle at point B, the radius OB is perpendicular to the tangent PB. Therefore, angle OBD + angle DBP = 90 degrees.
- Let angle OBD be denoted as \( \theta \). Then:
\[ \theta + 65^\circ = 90^\circ \]
- Solving for \( \theta \):
\[ \theta = 90^\circ - 65^\circ = 25^\circ \]

4. **Use Isosceles Triangle Properties**:
- In triangle OBD, OB = OD (both are radii of the circle).
- Therefore, triangle OBD is isosceles, which means angles OBD and ODB are equal.
- Let angle OBD = angle ODB = \( \theta = 25^\circ \).

5. **Find Angle DOB**:
- The sum of angles in triangle OBD is:
\[ \text{Angle OBD} + \text{Angle ODB} + \text{Angle DOB} = 180^\circ \]
- Substituting the known angles:
\[ 25^\circ + 25^\circ + \text{Angle DOB} = 180^\circ \]
- Therefore:
\[ \text{Angle DOB} = 180^\circ - 50^\circ = 130^\circ \]

6. **Relate Angle DAB to Angle DOB**:
- Since angle DAB subtends the same arc as angle DOB at the circumference, we have:
\[ \text{Angle DAB} = \frac{1}{2} \times \text{Angle DOB} = \frac{1}{2} \times 130^\circ = 65^\circ \]

7. **Use the Cyclic Quadrilateral Property**:
- In cyclic quadrilateral ABCD, the opposite angles are supplementary:
\[ \text{Angle DAB} + \text{Angle BCD} = 180^\circ \]
- Substituting the known value:
\[ 65^\circ + \text{Angle BCD} = 180^\circ \]
- Therefore:
\[ \text{Angle BCD} = 180^\circ - 65^\circ = 115^\circ \]

### Final Answer:
The measure of angle BCD is \( 115^\circ \).