1. **State the problem:** We are given a kite with two numbered angles, $\angle 1 = 18^\circ$ and $\angle 2 = 60^\circ$, and we need to find the measures of these angles. 2. **Recall kite properties:** A kite is a quadrilateral with two pairs of adjacent sides equal. It has one axis of symmetry, and the angles between unequal sides are equal. 3. **Analyze given angles:** Since $\angle 1$ and $\angle 2$ are given, and the kite is symmetric, these angles correspond to specific vertices. 4. **Check angle sum:** The sum of interior angles in any quadrilateral is $360^\circ$. 5. **Calculate remaining angles:** Let the other two angles be $x$ and $y$. Using symmetry, if $\angle 1 = 18^\circ$ and $\angle 2 = 60^\circ$, then the other two angles are also $18^\circ$ and $60^\circ$ respectively. 6. **Verify sum:** $18 + 60 + 18 + 60 = 156^\circ$, which is less than $360^\circ$. This suggests the problem only asks for the given angles, which are already provided. **Final answer:** $$m\angle 1 = 18^\circ$$ $$m\angle 2 = 60^\circ$$