1. **Problem statement:** Find the missing angles for problems 22 to 32. 2. **Formula and rules:** - The sum of angles in a triangle is always $$180^\circ$$. - The sum of angles in a quadrilateral is always $$360^\circ$$. 3. **Problem 22:** Triangle with angles 58°, ? and ?. - Let the unknown angles be $$x$$ and $$y$$. - Since one angle is a right angle (90°), the sum is $$58^\circ + 90^\circ + x = 180^\circ$$. - Calculate $$x$$: $$x = 180^\circ - 58^\circ - 90^\circ = 32^\circ$$. 4. **Problem 23:** Triangle with angles 65°, 65°, and ?. - Let the unknown angle be $$x$$. - Sum of angles: $$65^\circ + 65^\circ + x = 180^\circ$$ - Calculate $$x$$: $$x = 180^\circ - 130^\circ = 50^\circ$$. 5. **Problem 24:** Angles 129°, 14°, and ? in a triangle. - Let the unknown angle be $$x$$. - Sum of angles: $$129^\circ + 14^\circ + x = 180^\circ$$ - Calculate $$x$$: $$x = 180^\circ - 143^\circ = 37^\circ$$. 6. **Problem 25:** Angles 37°, 56°, and ? in a triangle. - Let the unknown angle be $$x$$. - Sum of angles: $$37^\circ + 56^\circ + x = 180^\circ$$ - Calculate $$x$$: $$x = 180^\circ - 93^\circ = 87^\circ$$. 7. **Problem 26:** Angles 23°, 67°, and ? in a triangle. - Let the unknown angle be $$x$$. - Sum of angles: $$23^\circ + 67^\circ + x = 180^\circ$$ - Calculate $$x$$: $$x = 180^\circ - 90^\circ = 90^\circ$$. 8. **Problem 27:** Quadrilateral with angles 103°, 98°, 70°, and ?. - Let the unknown angle be $$x$$. - Sum of angles in quadrilateral: $$103^\circ + 98^\circ + 70^\circ + x = 360^\circ$$ - Calculate $$x$$: $$x = 360^\circ - 271^\circ = 89^\circ$$. 9. **Problem 28:** Quadrilateral with angle 61° and three unknown angles. - Let unknown angles be $$x, y, z$$. - Sum: $$61^\circ + x + y + z = 360^\circ$$ - Without more info, cannot find unique values. 10. **Problem 29:** Rhombus with all angles unknown. - Rhombus has opposite equal angles and adjacent angles supplementary. - Let one angle be $$x$$, then adjacent angle is $$180^\circ - x$$. - Sum of all angles: $$x + (180^\circ - x) + x + (180^\circ - x) = 360^\circ$$ - This is always true, so angles depend on shape. 11. **Problem 30:** Quadrilateral with angles 89°, 124°, 59°, and ?. - Let unknown angle be $$x$$. - Sum: $$89^\circ + 124^\circ + 59^\circ + x = 360^\circ$$ - Calculate $$x$$: $$x = 360^\circ - 272^\circ = 88^\circ$$. 12. **Problem 31:** Quadrilateral with angles 79°, 57°, 100°, and ?. - Let unknown angle be $$x$$. - Sum: $$79^\circ + 57^\circ + 100^\circ + x = 360^\circ$$ - Calculate $$x$$: $$x = 360^\circ - 236^\circ = 124^\circ$$. 13. **Problem 32:** Quadrilateral with angles 125°, 61°, 118°, and ?. - Let unknown angle be $$x$$. - Sum: $$125^\circ + 61^\circ + 118^\circ + x = 360^\circ$$ - Calculate $$x$$: $$x = 360^\circ - 304^\circ = 56^\circ$$. **Final answers:** - 22: 32° - 23: 50° - 24: 37° - 25: 87° - 26: 90° - 27: 89° - 28: Cannot determine unique angles - 29: Angles depend on rhombus shape - 30: 88° - 31: 124° - 32: 56°