ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. If `/_ADC = 70^@` and `/_DAB = 60^@`, then the `/_PBC + /_PCB` is

Options

A. `130^@`
B. `150^@`
C. `155^@`
D. `180^@`

Answer

To solve the problem step by step, we will use the properties of cyclic quadrilaterals and the angles formed when lines are extended.

### Step 1: Identify the angles in the cyclic quadrilateral
Given that \( \angle ADC = 70^\circ \) and \( \angle DAB = 60^\circ \).

### Step 2: Calculate the opposite angles
In a cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \).
- Therefore, we can find \( \angle ABC \) as follows:
\[ \angle ABC = 180^\circ - \angle ADC = 180^\circ - 70^\circ = 110^\circ \]
- Similarly, we can find \( \angle CDA \):
\[ \angle CDA = 180^\circ - \angle DAB = 180^\circ - 60^\circ = 120^\circ \]

### Step 3: Analyze the angles at point P
Since \( AB \) and \( CD \) are extended to meet at point P, we can analyze the angles formed:
- The angle \( \angle PBC \) is equal to \( \angle ABC \) because they are vertically opposite angles. Thus:
\[ \angle PBC = 110^\circ \]
- The angle \( \angle PCB \) is equal to \( \angle CDA \) because they are also vertically opposite angles. Thus:
\[ \angle PCB = 120^\circ \]

### Step 4: Calculate the sum of angles \( \angle PBC + \angle PCB \)
Now, we can find the sum of the angles:
\[ \angle PBC + \angle PCB = 110^\circ + 120^\circ = 230^\circ \]

### Conclusion
Thus, the sum \( \angle PBC + \angle PCB \) is \( 230^\circ \).