We need to find the measure of angle $MON$ in the given diagram. We know that $AM$ and $AN$ are tangent to the circle at points $M$ and $N$, respectively, and angle $MAN$ is $60^\circ$.

1. Problem Description

We need to find the measure of angle $MON$ in the given diagram. We know that $AM$ and $AN$ are tangent to the circle at points $M$ and $N$, respectively, and angle $MAN$ is $60^\circ$.

2. Solution Steps

Since $AM$ and $AN$ are tangents to the circle at points $M$ and $N$, respectively, $OM$ is perpendicular to $AM$ and $ON$ is perpendicular to $AN$. This means that $\angle AMO = 90^\circ$ and $\angle ANO = 90^\circ$.

Consider quadrilateral $AMON$. The sum of the angles in a quadrilateral is $360^\circ$. Therefore,

$\angle MAN + \angle AMO + \angle ANO + \angle MON = 360^\circ$

We are given that $\angle MAN = 60^\circ$, and we know that $\angle AMO = 90^\circ$ and $\angle ANO = 90^\circ$. Substituting these values into the equation:

$60^\circ + 90^\circ + 90^\circ + \angle MON = 360^\circ$

$240^\circ + \angle MON = 360^\circ$

$\angle MON = 360^\circ - 240^\circ$

$\angle MON = 120^\circ$

3. Final Answer

The measure of angle $MON$ is $120^\circ$.