[ Inscribed and Circumscribed Circles $]$ [ Inscribed Angle Equals Half of Central Angle ]

Let $O$ be the center of the circle circumscribed around triangle $A B C$. Find the angle $O A C$, if: a) $\angle B=50^{\circ}$; b) $\angle B=126^{\circ}$.
The inscribed angle is equal to half of the corresponding central angle.

## Solution

a) The angular measure of the arc $A C$, not containing point $B$, is $100^{\circ}$. Therefore, $\angle A O C=100^{\circ}, \angle O A C=40^{\circ}$.

b) The angular measure of the arc $A B C$ is $360^{\circ}-2 \cdot 126^{\circ}=108^{\circ}$. Therefore, $\angle A O C=108^{\circ}, \angle O A C=36^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_845c09db4774ba28be39g-12.jpg?height=446&width=446&top_left_y=1715&top_left_x=314)

Fig. 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_845c09db4774ba28be39g-12.jpg?height=417&width=437&top_left_y=1724&top_left_x=796)

A

![](https://cdn.mathpix.com/cropped/2024_05_06_845c09db4774ba28be39g-12.jpg?height=412&width=480&top_left_y=1735&top_left_x=1278)

Fig. 2.

## Answer

a) $40^{\circ} ;$ b) $36^{\circ}$.