1. **Problem Statement:** We are given a circle with center at point $T$. Lines $TU$, $TV$, $TW$, and $TX$ divide the circle into sectors. Two sector angles at $T$ are given as $75^\circ$ and $55^\circ$. We need to find the measure of $\angle UTW$. 2. **Understanding the problem:** Since $T$ is the center of the circle, the angles formed by the radii at $T$ add up to $360^\circ$ because a full circle is $360^\circ$. 3. **Given angles:** - $\angle UTV = 75^\circ$ - $\angle VTW = 55^\circ$ 4. **Find $\angle UTW$:** This angle is the sum of $\angle UTV$ and $\angle VTW$ because $UT$, $TV$, and $TW$ are consecutive radii dividing the circle. 5. **Calculation:** $$\angle UTW = \angle UTV + \angle VTW = 75^\circ + 55^\circ = 130^\circ$$ 6. **Answer:** The measure of $\angle UTW$ is $130^\circ$. Therefore, the correct choice is A) 130°.