1. Problem II.A: Given in circle OD, $m\angle TE = 40^\circ$, find the measure of each arc UL, TY, LE, UET. 2. Since $m\angle TE = 40^\circ$ is an inscribed angle, the arc it intercepts is twice the angle measure. 3. Therefore, the measure of the intercepted arc $m\overset{\frown}{UL} = 2 \times 40 = 80^\circ$. 4. For arcs TY, LE, and UET, additional information is needed to find their measures. Since the problem does not provide it, we cannot determine these arcs without more data. 5. Problem II.B: In circle OE, $\angle LET \cong \angle TEL$, $m\angle OEV = 5x$, $m\angle VEI = x + 42$, and OI and VL are diameters. 6. Since $\angle LET \cong \angle TEL$, triangle LET is isosceles with $\angle LET = \angle TEL$. 7. Angles $OEV$ and $VEI$ are adjacent and on a straight line, so their sum is $180^\circ$. 8. Write the equation: $5x + (x + 42) = 180$. 9. Simplify: $6x + 42 = 180$. 10. Subtract 42: $6x = 138$. 11. Divide by 6: $x = 23$. 12. Find $m\angle OEV = 5x = 5 \times 23 = 115^\circ$. 13. Find $m\angle VEI = x + 42 = 23 + 42 = 65^\circ$. 14. Since OI and VL are diameters, $m\angle TEI$ and $m\angle ITI$ relate to semicircles or right angles, but without explicit angle definitions, we assume $m\angle TEI = 90^\circ$ and $m\angle ITI = 90^\circ$ as diameters subtend right angles. 15. Problem III: Given percentages of trash types, find the measure of each central angle. 16. Use formula: $m = \text{percentage} \times 360^\circ / 100$. 17. Food waste: $14\% \Rightarrow 14 \times 3.6 = 50.4^\circ \approx 50^\circ$. 18. Glass: $2\% \Rightarrow 2 \times 3.6 = 7.2^\circ \approx 7^\circ$. 19. Metals: $12\% \Rightarrow 12 \times 3.6 = 43.2^\circ \approx 43^\circ$. 20. Paper: $40\% \Rightarrow 40 \times 3.6 = 144^\circ$. 21. Plastic: Not given explicitly, so cannot calculate. 22. Yard waste: $11\% \Rightarrow 11 \times 3.6 = 39.6^\circ \approx 40^\circ$. 23. Others: $5\% \Rightarrow 5 \times 3.6 = 18^\circ$. Final answers: - $m\overset{\frown}{UL} = 80^\circ$ - $x = 23$ - $m\angle OEV = 115^\circ$ - $m\angle VEI = 65^\circ$ - $m\angle TEI = 90^\circ$ - $m\angle ITI = 90^\circ$ - Food waste: $50^\circ$ - Glass: $7^\circ$ - Metals: $43^\circ$ - Paper: $144^\circ$ - Yard waste: $40^\circ$ - Others: $18^\circ$