$\left[\begin{array}{l}\text { The inscribed angle is equal to half the central angle } \\ \text { [Central angle. Arc length and circumference }]\end{array}\right.$

$A B$ and $A C$ are chords of a circle; $\cup A B=110^{\circ}, \cup A C=40^{\circ}$. Find the angle $B A C$.
Consider two possible arrangements of points $A, B$, and $C$ on a circle.

## Solution

In one of the possible cases, the arc $B C$, not containing point $A$, is $360^{\circ}-\left(110^{\circ}+40^{\circ}\right)=210^{\circ}$, and in the other $\cup B C=110^{\circ}-40^{\circ}=70^{\circ}$. Therefore, the desired angle is either $105^{\circ}$ or $35^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_02603d12f3437ea52c37g-12.jpg?height=383&width=440&top_left_y=732&top_left_x=452)

Fig. 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_02603d12f3437ea52c37g-12.jpg?height=335&width=372&top_left_y=770&top_left_x=1252)

Answer

$105^{\circ}, 35^{\circ}$.