In the given diagram, line $TQ$ is tangent to a circle with center $S$ at point $Q$, and chord $RQ$ is a diameter of the circle. We are asked to find $m \angle PQT$, given that $x = 28^\circ$.

Key points:

1. $TQ$ is a tangent to the circle at point $Q$.
2. $RQ$ is a diameter of the circle.
3. The angle between a tangent and a chord through the point of tangency is equal to the inscribed angle on the opposite side of the chord.

This means that $m \angle PQT = m \angle PRQ$ (since they are both subtended by the same arc $PQ$).

Solution:

• We know that $x = 28^\circ$.
• The inscribed angle $\angle PRQ$ subtended by the diameter of a circle is a right angle, meaning $\angle PRQ = 90^\circ$.

Using the property of angles around point $S$, we know: $m \angle PRQ = m \angle PQT = 90^\circ - x$ Substituting $x = 28^\circ$: $m \angle PQT = 90^\circ - 28^\circ = 62^\circ$

Thus, the correct answer is $\boxed{62^\circ}$ (Option A).

Let me know if you would like further details or have any questions!

Related Questions:

1. What is the relationship between a tangent and the radius at the point of tangency?
2. How can you determine the measure of an angle subtended by a chord?
3. What is the property of the angles subtended by a diameter of a circle?
4. How does the exterior angle theorem apply in this context?
5. How would the result change if $x$ had a different value?

Tip:

Always remember that the angle formed by a tangent and a chord through the point of tangency is equal to the angle subtended by the same chord on the opposite side of the circle.