Segments $PR$ and $PS$ are tangent to the circle. The measure of $\angle RPS = 66^{\circ}$. We need to find the measure of $\angle RQS$.

1. Problem Description

Segments $PR$ and $PS$ are tangent to the circle. The measure of $\angle RPS = 66^{\circ}$. We need to find the measure of $\angle RQS$.

2. Solution Steps

Since $PR$ and $PS$ are tangent to the circle at points $R$ and $S$ respectively, we know that $\angle QRP = 90^{\circ}$ and $\angle QSP = 90^{\circ}$.

Consider quadrilateral $QRPS$. The sum of the interior angles of a quadrilateral is $360^{\circ}$. Therefore,

$$\angle RQS + \angle QRP + \angle RPS + \angle QSP = 360^{\circ}$$

$$\angle RQS + 90^{\circ} + 66^{\circ} + 90^{\circ} = 360^{\circ}$$

$$\angle RQS + 246^{\circ} = 360^{\circ}$$

$$\angle RQS = 360^{\circ} - 246^{\circ}$$

$$\angle RQS = 114^{\circ}$$

3. Final Answer

The measure of $\angle RQS$ is $114^{\circ}$.