$A B$ and $A C$ are equal chords, $M A N$ is a tangent, the angular measure of arc $B C$, not containing point $A$, is $200^{\circ}$. Find the angles $M A B$ and $N A C$.
Consider all possible arrangements of points $M, A, C$, and apply the theorem about the angle between a tangent and a chord.

## Solution

Note that

$$
\cup C A B=360^{\circ}-200^{\circ}=160^{\circ}, \cup A B=\cup A C=\frac{160^{\circ}}{2}=80^{\circ}
$$

If angle $\angle M A B$ is acute, then

$$
\angle M A B=\frac{\cup A B}{2}=40^{\circ}
$$

Then $\angle N A C=40^{\circ}$.

If angle $\angle M A B$ is obtuse, then

$$
\angle M A B=180^{\circ}-40^{\circ}=140^{\circ}.
$$

Then $\angle N A C=140^{\circ}$.

## Answer

$\angle M A B=\angle N A C=40^{\circ}$ or $\angle M A B=\angle N A C=140^{\circ}$.