1. **Problem Statement:** We are given a kite-shaped quadrilateral EFGD with diagonals intersecting at H. The angle at vertex D, specifically $\angle GDE$, measures 59°. We need to find the missing angle values $m\angle GDE$, $m\angle DEH$, and $m\angle DGH$. 2. **Properties of a Kite:** A kite has two pairs of adjacent sides equal and its diagonals intersect at right angles. The diagonal connecting the vertices where the pairs meet is the axis of symmetry. 3. **Key Formula:** The diagonals of a kite are perpendicular, so $$\angle DEH = \angle DGH = 90^\circ.$$ 4. **Given:** $m\angle GDE = 59^\circ$. 5. **Find:** - $m\angle DEH$ and $m\angle DGH$ are the angles formed by the diagonals intersecting at H. 6. **Solution:** - Since diagonals intersect at right angles in a kite, $$m\angle DEH = 90^\circ$$ $$m\angle DGH = 90^\circ$$ - The given $m\angle GDE = 59^\circ$ is already known. 7. **Final answers:** $$m\angle GDE = 59^\circ$$ $$m\angle DEH = 90^\circ$$ $$m\angle DGH = 90^\circ$$ These values satisfy the properties of the kite and the given information.