1. **Problem 5: Find the measures of angles 1, 2, 3, and 4 in the triangle figure.** 2. The triangle has angles 59°, 22°, and the angles labeled 1, 2, 3 inside it, plus an exterior angle 4. 3. Recall the Triangle Angle Sum Theorem: The sum of interior angles in a triangle is $$180^\circ$$. 4. Given angles: one base angle is $$59^\circ$$, the top angle is split into $$22^\circ$$ and angle 1. 5. Calculate angle 1: Since the top angle is split into $$22^\circ$$ and angle 1, and the total top angle plus base angles must sum to $$180^\circ$$, first find the top angle. 6. The sum of the two base angles is $$59^\circ + 22^\circ = 81^\circ$$. 7. Therefore, the top angle (angle 1 + 22°) is $$180^\circ - 81^\circ = 99^\circ$$. 8. So, angle 1 is $$99^\circ - 22^\circ = 77^\circ$$. 9. Angle 2 and angle 3 are interior angles adjacent to angle 1 and 59°, likely complementary or supplementary depending on the figure. Assuming angle 2 is adjacent to 59° and angle 3 adjacent to 22°. 10. Since angles 2 and 3 are interior angles on a straight line with angles 59° and 22°, respectively, use the Linear Pair Theorem: angles on a straight line sum to $$180^\circ$$. 11. Calculate angle 2: $$m\angle 2 = 180^\circ - 59^\circ = 121^\circ$$. 12. Calculate angle 3: $$m\angle 3 = 180^\circ - 22^\circ = 158^\circ$$. 13. Angle 4 is an exterior angle adjacent to angle 3, so it equals the sum of the two non-adjacent interior angles (exterior angle theorem). 14. Using the exterior angle theorem: $$m\angle 4 = m\angle 1 + m\angle 2 = 77^\circ + 121^\circ = 198^\circ$$. **Final answers:** $$m\angle 1 = 77^\circ$$ $$m\angle 2 = 121^\circ$$ $$m\angle 3 = 158^\circ$$ $$m\angle 4 = 198^\circ$$