1. **Stating the problem:** We have a quadrilateral inscribed in a circle with points T, U, S, V, and R inside the circle. We want to find expressions for the measures of angles \(\angle T\) and \(\angle V\), their sum, and classify their relationship. 2. **Key fact:** Opposite angles of a quadrilateral inscribed in a circle are supplementary, meaning their measures add up to 180°. 3. **Expressions for angles:** - \(m \angle T = \frac{1}{2} m \overset{\frown}{UV}\) (angle measure is half the intercepted arc) - \(m \angle V = \frac{1}{2} m \overset{\frown}{TS}\) 4. **Sum of angles:** \[ m \angle T + m \angle V = \frac{1}{2} m \overset{\frown}{UV} + \frac{1}{2} m \overset{\frown}{TS} = \frac{1}{2} (m \overset{\frown}{UV} + m \overset{\frown}{TS}) \] 5. **Since arcs \(UV\) and \(TS\) are opposite arcs of the circle, their measures add up to 360° minus the other two arcs, but for inscribed quadrilateral opposite arcs sum to 180°:** \[ m \overset{\frown}{UV} + m \overset{\frown}{TS} = 180^ \] 6. **Therefore:** \[ m \angle T + m \angle V = \frac{1}{2} \times 180 = 90^ \] 7. **Classification:** Since \(m \angle T + m \angle V = 90^ \), angles \(\angle T\) and \(\angle V\) are complementary. **Final answers:** - \(m \angle T = \frac{1}{2} m \overset{\frown}{UV}\) - \(m \angle V = \frac{1}{2} m \overset{\frown}{TS}\) - \(m \angle T + m \angle V = 90^ \) - \(\angle T\) and \(\angle V\) are complementary angles.