Step 1: Identify the given angle. The angle $\hat{S}_1$ is given as $48^\circ$. From the diagram, this angle is $\angle RST$. Step 2: Apply the Tangent-Chord Theorem. 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: $$ \angle RQS = \angle RST $$ $$ \hat{Q}_2 = 48^\circ $$ Reason: Tangent-Chord Theorem. Step 3: Use the given relationship between $\hat{Q}_1$ and $\hat{Q}_2$. The problem states that $\hat{Q}_1 = \hat{Q}_2$. Since $\hat{Q}_2 = 48^\circ$, then: $$ \hat{Q}_1 = 48^\circ $$ Reason: Given. The size of $\hat{Q}_1$ is $\boxed{\text{48}^\circ}$.