1. **Problem statement:** We have a kite-shaped quadrilateral with angles 50° at the top-left vertex and 80° at the lower-left interior angle near the diagonal intersection. We need to find the values of angles $x$ and $y$. 2. **Properties of a kite:** A kite has two pairs of adjacent equal sides, and its diagonals intersect at right angles (90°). The diagonal intersection creates four right angles. 3. **Using the diagonal intersection:** Since the diagonals intersect at 90°, the angles around the intersection point sum to 360°, with each angle being 90°. 4. **Finding $x$:** The angle $x$ is part of the right angle at the right vertex. Since the kite's diagonal creates a right angle, and $x$ is part of it, we use the fact that the sum of angles around a point is 360° and the kite's symmetry. 5. **Finding $y$:** The angle $y$ is adjacent to the 80° angle at the lower-left side. Since the sum of angles in a triangle is 180°, and the kite's diagonal splits the kite into triangles, we use this to find $y$. 6. **Calculations:** - At the lower-left vertex, the two angles adjacent to the diagonal are 80° and $y$. Since the diagonal is a straight line, these two angles sum to 180°: $$80^\circ + y = 180^\circ$$ $$y = 180^\circ - 80^\circ = 100^\circ$$ - At the right vertex, the diagonal creates a right angle (90°). The angle $x$ is part of this right angle, and the other part is 50° (top-left angle). Since the kite is symmetric, the angle at the right vertex is equal to the angle at the top-left vertex plus $x$: $$50^\circ + x = 90^\circ$$ $$x = 90^\circ - 50^\circ = 40^\circ$$ 7. **Final answers:** $$x = 40^\circ$$ $$y = 100^\circ$$