In the diagram, OQR is a tangent to the circle at Q. Find the values of m and n.

The diagram shows a circle with a tangent line $OQR$ at point $Q$. Points $S$, $Q$, and $T$ are on the circumference.

• The angle $\angle QST = 34^\circ$.
• The right angle symbol at $Q$ indicates that $\angle SQT = 90^\circ$.
• Angle $m$ is $\angle OQS$.
• Angle $n$ is $\angle RQT$.

Since $\triangle SQT$ is a right-angled triangle ($\angle SQT = 90^\circ$), the sum of its angles is $180^\circ$.

$$\angle QST + \angle SQT + \angle QTS = 180^\circ$$ $$34^\circ + 90^\circ + \angle QTS = 180^\circ$$ $$124^\circ + \angle QTS = 180^\circ$$ $$\angle QTS = 180^\circ - 124^\circ$$ $$\angle QTS = 56^\circ$$

The Tangent-Chord Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
For the tangent $QR$ and chord $QT$, the angle between them is $n = \angle RQT$. The angle in the alternate segment subtended by chord $QT$ is $\angle QST$.

$$n = \angle RQT = \angle QST$$ $$n = 34^\circ$$

For the tangent $OQ$ and chord $SQ$, the angle between them is $m = \angle OQS$. The angle in the alternate segment subtended by chord $SQ$ is $\angle QTS$.

$$m = \angle OQS = \angle QTS$$ $$m = 56^\circ$$

The angles $m$, $\angle SQT$, and $n$ lie on the straight line $OQR$. Their sum should be $180^\circ$.

$$m + \angle SQT + n = 56^\circ + 90^\circ + 34^\circ = 180^\circ$$

The values are consistent.

The values are:

$$m = \boxed{56^\circ}$$ $$n = \boxed{34^\circ}$$