Find the measure of angle V in a triangle given the lengths of its sides.

1. Identify the problem type: This is a geometry problem involving a triangle where we need to find an angle given the lengths of all three sides. This suggests using the Law of Cosines.

1. Recall the Law of Cosines: The Law of Cosines states that for any triangle with sides $a$, $b$, and $c$, and the angle $C$ opposite side $c$, the following relationship holds: $c^2 = a^2 + b^2 - 2ab \cos(C)$. We can rearrange this formula to solve for the angle: $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.

3. Assign side lengths: In the given triangle $TUV$, we want to find the measure of angle $V$. Let's denote the side opposite angle $V$ as $t$, the side opposite angle $T$ as $u$, and the side opposite angle $U$ as $v$. From the diagram, we have:
- Side $TU$ (opposite $V$) = $t = 5$
- Side $UV$ (opposite $T$) = $u = 8$
- Side $VT$ (opposite $U$) = $v = 11$

1. Apply the Law of Cosines to find angle V: We will use the form of the Law of Cosines that solves for an angle. To find angle $V$, we use the formula: $v^2 = t^2 + u^2 - 2tu \cos(V)$. Rearranging to solve for $\cos(V)$: $\cos(V) = \frac{t^2 + u^2 - v^2}{2tu}$.

5. Substitute the values: Substitute the known side lengths into the formula:
$$\cos(V) = \frac{5^2 + 8^2 - 11^2}{2 \cdot 5 \cdot 8}$$

6. Calculate the terms in the numerator and denominator:
- $5^2 = 25$
- $8^2 = 64$
- $11^2 = 121$
- $2 \cdot 5 \cdot 8 = 80$

7. Substitute the calculated values back into the equation:
$$\cos(V) = \frac{25 + 64 - 121}{80}$$

8. Perform the addition and subtraction in the numerator:
$$25 + 64 = 89$$
$$89 - 121 = -32$$

9. Complete the calculation for cos(V):
$$\cos(V) = \frac{-32}{80}$$

10. Simplify the fraction:
$$\cos(V) = -\frac{32}{80} = -\frac{2 \cdot 16}{5 \cdot 16} = -\frac{2}{5} = -0.4$$

11. Find the angle V by taking the inverse cosine (arccos):
$$m\angle V = \arccos(-0.4)$$

12. Calculate the value of the angle: Using a calculator, find the arccosine of -0.4.
$$m\angle V \approx 113.578178^\circ$$

13. Round the answer to the nearest tenth: The problem asks for the answer as an integer or a decimal rounded to the nearest tenth.
$$m\angle V \approx 113.6^\circ$$