1. **Problem 6: Find m\angle WX** Given a circle with points W, X, Y, Z and an arc of 75°. 2. **Formula:** The measure of an inscribed angle is half the measure of its intercepted arc. 3. **Calculation:** $$m\angle WX = \frac{1}{2} \times 75^\circ = 37.5^\circ$$ --- 4. **Problem 7: Find m\angle EHSK** Given a circle with points E, F, H, S and arcs 51° and 40°. 5. **Formula:** The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs. 6. **Calculation:** $$m\angle EHSK = \frac{1}{2} (51^\circ + 40^\circ) = \frac{1}{2} \times 91^\circ = 45.5^\circ$$ --- 7. **Problem 8: Find m\angle PS** Given a circle with points P, S, T but no arc measure given explicitly. 8. **Assumption:** Since no arc measure is given, we cannot calculate m\angle PS without additional information. --- 9. **Problem 10: Find value of 6m + 12k** Given a circle with points U, V, W, X and an arc of 60°. 10. **Formula:** The measure of an inscribed angle is half the measure of its intercepted arc. 11. **Equation:** $$6m + 12k = \frac{1}{2} \times 60^\circ = 30^\circ$$ 12. **Simplify:** $$\cancel{6}m + \cancel{12}k = 30$$ $$m + 2k = 5$$ --- 13. **Problem 11: Find value of k + 30** Given arcs 54°, 110°, 34°, 130° around circle points J, K, L, M. 14. **Sum of arcs in circle:** $$54 + 110 + 34 + 130 = 328^\circ$$ 15. **Since total circle is 360°, missing arc is:** $$360 - 328 = 32^\circ$$ 16. **Assuming k + 30 corresponds to an angle related to these arcs, more info needed to solve.** --- 17. **Problem 12: Find value of 3x + 7** Given circle with points O, P, Q, R but no arc measure given. 18. **Insufficient data to solve without arc or angle measures.** --- 19. **Problem 13: Error Analysis for m\angle BC = 53^\circ** 20. **Error:** The angle m\angle BC cannot be equal to 53° if it is an inscribed angle intercepting an arc different from 106°. 21. **Correction:** The measure of an inscribed angle is half the measure of its intercepted arc, so if the intercepted arc is 106°, then: $$m\angle BC = \frac{1}{2} \times 106^\circ = 53^\circ$$ 22. **If the arc is not 106°, the angle measure must be recalculated accordingly.** --- 23. **Problem 14: Error Analysis in finding x** 24. **Given:** $$x + 78 + 65 = 180^\circ$$ $$65 + x = 180$$ $$x = 115$$ 25. **Error:** The step removing 78° from the equation is incorrect. 26. **Correction:** The correct steps are: $$x + 78 + 65 = 180$$ $$x + 143 = 180$$ $$x = 180 - 143 = 37$$ --- 27. **Problem 16: Determine if polygons can be inscribed in a circle** - a. Right triangle: Yes, any right triangle can be inscribed in a circle with the hypotenuse as diameter. - b. Kite: Not always; only if it is cyclic (opposite angles sum to 180°). - c. Rhombus: Only if it is a square (all angles 90°), otherwise no. - d. Isosceles trapezoid: Yes, all isosceles trapezoids are cyclic. --- 28. **Problem 22: Find values of x and y, then interior angles of polygon ABCD** Given angles: $$6y^\circ, 2x^\circ, 4x^\circ$$ 29. **Sum of interior angles of quadrilateral:** $$360^\circ$$ 30. **Equation:** $$6y + 2x + 4x + \text{fourth angle} = 360$$ 31. **Assuming fourth angle is known or equal to 0 (not given), cannot solve uniquely without more info.** --- **Final answers:** - m\angle WX = 37.5° - m\angle EHSK = 45.5° - m\angle PS: Insufficient data - 6m + 12k = 30°, so m + 2k = 5 - k + 30: Insufficient data - 3x + 7: Insufficient data - Error in m\angle BC corrected by using half intercepted arc - x corrected to 37° - Polygon inscribability: right triangle (yes), kite (sometimes), rhombus (only square), isosceles trapezoid (yes) - Values of x, y in polygon ABCD: insufficient data