1. **Problem 31:** Given triangle ABC with angles $m\angle A = X$, $m\angle B = 2X$, and $m\angle C = 2X + 30^\circ$. Find $m\angle B$. 2. The sum of angles in any triangle is $180^\circ$. So, $$X + 2X + (2X + 30) = 180$$ 3. Simplify the equation: $$X + 2X + 2X + 30 = 180$$ $$5X + 30 = 180$$ 4. Subtract 30 from both sides: $$5X + \cancel{30} - \cancel{30} = 180 - 30$$ $$5X = 150$$ 5. Divide both sides by 5: $$\frac{5X}{\cancel{5}} = \frac{150}{\cancel{5}}$$ $$X = 30$$ 6. Find $m\angle B = 2X = 2 \times 30 = 60^\circ$. --- 7. **Problem 32:** Triangle SAD with $m\angle S = 2x$, $m\angle A = x - 23$, $m\angle D = x - 17$. Find $m\angle S$. 8. Sum of angles: $$2x + (x - 23) + (x - 17) = 180$$ 9. Simplify: $$2x + x - 23 + x - 17 = 180$$ $$4x - 40 = 180$$ 10. Add 40 to both sides: $$4x - \cancel{40} + \cancel{40} = 180 + 40$$ $$4x = 220$$ 11. Divide both sides by 4: $$\frac{4x}{\cancel{4}} = \frac{220}{\cancel{4}}$$ $$x = 55$$ 12. Find $m\angle S = 2x = 2 \times 55 = 110^\circ$. --- 13. **Problem 33:** Right triangle XYZ with right angle at X, $m\angle Y = x + 5$, $m\angle Z = x - 7$. Find $m\angle Z$. 14. Sum of angles in triangle: $$90 + (x + 5) + (x - 7) = 180$$ 15. Simplify: $$90 + x + 5 + x - 7 = 180$$ $$2x + 88 = 180$$ 16. Subtract 88: $$2x + \cancel{88} - \cancel{88} = 180 - 88$$ $$2x = 92$$ 17. Divide by 2: $$\frac{2x}{\cancel{2}} = \frac{92}{\cancel{2}}$$ $$x = 46$$ 18. Find $m\angle Z = x - 7 = 46 - 7 = 39^\circ$. --- 19. **Problem 34:** Triangle QRS is equilateral (all sides congruent), so all angles equal. 20. Sum of angles: $$m\angle Q + m\angle R + m\angle S = 180$$ 21. Since all equal: $$3 \times m\angle Q = 180$$ 22. Divide by 3: $$m\angle Q = \frac{180}{3} = 60^\circ$$ 23. So, $$m\angle Q = m\angle R = m\angle S = 60^\circ$$ --- 24. **Problem 35:** Triangle WXY with $m\angle W = x - 22$, $m\angle X = 3x + 19$, $m\angle Y = x - 17$. Find $m\angle X$. 25. Sum of angles: $$(x - 22) + (3x + 19) + (x - 17) = 180$$ 26. Simplify: $$x - 22 + 3x + 19 + x - 17 = 180$$ $$5x - 20 = 180$$ 27. Add 20: $$5x - \cancel{20} + \cancel{20} = 180 + 20$$ $$5x = 200$$ 28. Divide by 5: $$\frac{5x}{\cancel{5}} = \frac{200}{\cancel{5}}$$ $$x = 40$$ 29. Find $m\angle X = 3x + 19 = 3 \times 40 + 19 = 120 + 19 = 139^\circ$. --- **Final answers:** - $m\angle B = 60^\circ$ - $m\angle S = 110^\circ$ - $m\angle Z = 39^\circ$ - $m\angle Q = m\angle R = m\angle S = 60^\circ$ - $m\angle X = 139^\circ$