In a circle with centre $O$, an arc $ABC$ subtends an angle of $134^ ext{o}$ at the centre of the circle. The chord $AB$ is produced to a point $P$. Find the value of $\angle CBP$.

Question

In a circle with centre $O$, an arc $ABC$ subtends an angle of $134^	ext{o}$ at the centre of the circle. The chord $AB$ is produced to a point $P$. Find the value of $\angle CBP$.

Accepted Answer

**Solution:**

1. Given that the arc $ABC$ subtends an angle of $134^\circ$ at the centre $O$ of the circle.

2. By the circle theorem, the angle subtended by the same arc at the circumference is half the angle subtended at the centre. So,
   $$\angle AQC = \frac{1}{2} 	imes \angle AOC = \frac{1}{2} 	imes 134^\circ = 67^\circ$$

3. Consider the cyclic quadrilateral formed by points $A, B, C,$ and $Q$ on the circle.

4. The exterior angle $\angle CBP$ at vertex $B$ (formed by extending chord $AB$ to $P$) is equal to the interior opposite angle $\angle AQC$ of the cyclic quadrilateral.

5. Therefore,
   $$\angle CBP = \angle AQC = 67^\circ$$

**Answer:** $67^\circ$

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