To solve this problem:

Step 1: Identify given information

1. $\overrightarrow{UV}$ is tangent to the circle at $U$.
2. $\angle UVT = 42^\circ$.

Step 2: Key geometric property

In a circle, the angle formed between a radius and a tangent at the point of tangency is $90^\circ$. Therefore, $\angle UTU = 90^\circ$.

Step back

To solve the problem:

1. Identify the Geometry:

   • $\overline{UV}$ is tangent to the circle at $U$.
   • $T$ is the center of the circle.
   • $\angle UVT = 42^\circ$.

2. Key Properties:

   • The tangent to a circle at a point (here, $U$) is perpendicular to the radius at that point.
   • Hence, $\angle UTU = 90^\circ$.

3. Determine $\angle T$:

   • The triangle $\triangle UVT$ has three angles: $\angle UVT$, $\angle T$, and $\angle UTV$.
   • The sum of the angles in a triangle is $180^\circ$.

   $\angle UVT + \angle T + \angle UTV = 180^\circ$

   Substituting: $42^\circ + 90^\circ + \angle T = 180^\circ$

   Simplify: $\angle T = 180^\circ - 132^\circ = 48^\circ$

Final Answer:

$\boxed{48^\circ}$

Let me know if you'd like a detailed breakdown of the solution or have further questions! Here are related questions to consider:

1. How do tangents relate to radii in a circle?
2. Can you prove that the tangent is always perpendicular to the radius at the point of tangency?
3. How would this problem change if the tangent angle were different?
4. What are other properties of angles in a triangle that can simplify geometry problems?
5. Can this technique extend to solving problems involving multiple tangents?

Tip: Always double-check angle relationships in problems involving circles and tangents!