Identify the given angle.

The angle $\hat{S}_1$ is given as $48^\circ$. From the diagram, this angle is $\angle RST$.

The line ST is a tangent to the circle at point S (implied by the diagram and the angle given in this configuration). The chord is RS.
According to the Tangent-Chord Theorem, the angle between a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment.
The angle in the alternate segment subtended by chord RS is $\angle RQS$.
Therefore, we have:

Reason: Tangent-Chord Theorem.

The problem states that $\hat{Q}_1 = \hat{Q}_2$.
Since $\hat{Q}_2 = 48^\circ$, then:

Reason: Given.

The size of $\hat{Q}_1$ is $\boxed{\text{48}^\circ}$.