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 extended to a point $P$. What is the measure 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 extended to a point $P$. What is the measure of $\angle CBP$?

Accepted Answer

**Solution:**

1. Given that the arc $ABC$ subtends an angle of $134^	ext{o}$ at the centre $O$ of the circle.

2. The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference.

3. Let $Q$ be a point on the circumference such that $\angle AQC$ is the angle subtended by the arc $ABC$ at the circumference.

4. Using the property:
   $$\angle AOC = 2 	imes \angle AQC$$
   Therefore,
   $$\angle AQC = \frac{1}{2} 	imes 134^	ext{o} = 67^	ext{o}$$

5. Since $AB$ is extended to $P$, $\angle CBP$ is an exterior angle of the cyclic quadrilateral formed by points $A$, $B$, $C$, and $Q$.

6. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle, so:
   $$\angle CBP = \angle AQC = 67^	ext{o}$$

**Answer:** $67^	ext{o}$

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