1. **Problem Statement:** We have a rhombus with diagonals intersecting inside it, creating four smaller angles numbered 1, 2, 3, and 4. One of the outer angles of the rhombus measures 59°. 2. **Key Properties of a Rhombus:** - All sides are equal. - Opposite angles are equal. - Diagonals bisect each other at right angles (90°). - The diagonals split the rhombus into four right triangles. 3. **Using the property that diagonals intersect at 90°:** The four angles formed at the intersection of the diagonals are right angles split into two pairs of equal angles because the diagonals bisect each other. 4. **Let angles 1 and 3 be equal, and angles 2 and 4 be equal:** Since the diagonals intersect at 90°, we have: $$\angle 1 + \angle 2 = 90^\circ$$ $$\angle 3 + \angle 4 = 90^\circ$$ And since $\angle 1 = \angle 3$ and $\angle 2 = \angle 4$, we focus on $\angle 1$ and $\angle 2$. 5. **Using the given outer angle of 59°:** The outer angle adjacent to one vertex is 59°, so the interior angle at that vertex is: $$180^\circ - 59^\circ = 121^\circ$$ 6. **Relationship between interior angle and diagonal angles:** Each interior angle of the rhombus is split by the diagonal into two angles, which correspond to angles 1 and 2 (or 3 and 4) at the intersection. 7. **Set up the equation:** Since the diagonal bisects the interior angle, angles 1 and 2 add up to 121°: $$\angle 1 + \angle 2 = 121^\circ$$ But from step 4, we have: $$\angle 1 + \angle 2 = 90^\circ$$ This is a contradiction, so the diagonal does not bisect the interior angle but bisects the rhombus into right triangles. 8. **Correct approach:** The diagonals intersect at 90°, so angles 1, 2, 3, and 4 satisfy: $$\angle 1 + \angle 2 = 90^\circ$$ $$\angle 3 + \angle 4 = 90^\circ$$ And since the rhombus has interior angles 121° and 59° (opposite angles), the angles at the intersection relate to half of these angles. 9. **Calculate angles 1 and 2:** Since the diagonals bisect the angles, we have: $$\angle 1 = \frac{121^\circ}{2} = 60.5^\circ$$ $$\angle 2 = \frac{59^\circ}{2} = 29.5^\circ$$ 10. **Verify sum at intersection:** $$60.5^\circ + 29.5^\circ = 90^\circ$$ This matches the right angle formed by the diagonals. 11. **Therefore, the measures of the numbered angles are:** $$\angle 1 = \angle 3 = 60.5^\circ$$ $$\angle 2 = \angle 4 = 29.5^\circ$$ **Final answer:** Angles 1 and 3 measure 60.5°, and angles 2 and 4 measure 29.5° each.