To solve for $QT$, we analyze the diagram and apply the necessary geometric principles. The figure shows a cyclic quadrilateral with points $Q, R, S, T$ inscribed in a circle, and angles labeled at specific vertices.

Key Observations:

1. In a cyclic quadrilateral, the opposite angles sum to $180^\circ$.
2. Angles subtended by the same arc are equal.
3. The central angle and the inscribed angle subtended by the same arc relate as: $\text{Central angle} = 2 \times \text{Inscribed angle}$.

Given Data:

• $\angle SRT = 46^\circ$
• $\angle RST = 71^\circ$

We aim to find $\angle QTS$, which subtends arc $QS$.

Step-by-Step Calculation:

1. Calculate $\angle RTS$: Using the triangle sum property in $\triangle RST$, $\angle RTS = 180^\circ - (\angle SRT + \angle RST)$ Substituting the values: $\angle RTS = 180^\circ - (46^\circ + 71^\circ) = 63^\circ.$

2. Determine $\angle QTS$: $\angle QTS$ subtends the same arc as $\angle RTS$. Therefore, by the property of angles subtended by the same arc: $\angle QTS = \angle RTS = 63^\circ.$

Final Answer:

$\boxed{\angle QTS = 63^\circ}$

Would you like me to expand on this explanation or solve related questions? Here are some questions to extend this topic:

1. How can we calculate the remaining angles in the cyclic quadrilateral?
2. What are the properties of cyclic quadrilaterals used in this problem?
3. Can we find the measure of $\angle QRS$ and verify the cyclic quadrilateral condition?
4. How do central angles help in calculating unknown inscribed angles?
5. How do we calculate the arc lengths subtended by each angle?

Tip: Always check cyclic quadrilateral properties when a figure is inscribed in a circle.