1. **Problem Statement:** We have a kite-shaped quadrilateral with angles labeled 1 through 7 and some given angle measures: $\angle 1 = 73^\circ$ and $\angle 5 = 46^\circ$. We need to find the measures of all angles $\angle 1$ through $\angle 7$. 2. **Properties of a Kite:** - A kite has two pairs of adjacent sides equal. - The angles between unequal sides are equal. - The diagonals intersect at right angles (90°). - One diagonal bisects the other. 3. **Given Information and Setup:** - $\angle 1 = 73^\circ$ (given) - $\angle 5 = 46^\circ$ (given) - Angles 6 and 7 are at the top corners. - Angles 2, 3, and 4 are inside the kite near the center. 4. **Step-by-step Solution:** - Since the diagonals intersect at right angles, angles 2, 3, 4, and 7 are right angles or related to right angles. - The kite's symmetry implies $\angle 6 = \angle 7$ because they are opposite angles formed by the intersection of diagonals. - The diagonal bisects the angles at the top, so $\angle 6 = \angle 7$. - The sum of angles around point where diagonals intersect is $360^\circ$. - Angles 2, 3, 4, and 7 are formed at the intersection, so: $$\angle 2 + \angle 3 + \angle 4 + \angle 7 = 360^\circ$$ - Since the diagonals are perpendicular, angles 2 and 4 are right angles: $$\angle 2 = \angle 4 = 90^\circ$$ - Then: $$90 + \angle 3 + 90 + \angle 7 = 360$$ $$\angle 3 + \angle 7 = 180$$ - The kite's symmetry implies $\angle 3 = \angle 7$. - So: $$2 \angle 3 = 180 \Rightarrow \angle 3 = 90^\circ$$ - Then $\angle 7 = 90^\circ$ and $\angle 6 = 90^\circ$. - Now, consider the angles at the bottom corners: - $\angle 1 = 73^\circ$ (given) - $\angle 5 = 46^\circ$ (given) - The sum of angles in the kite is $360^\circ$: $$\angle 1 + \angle 5 + \angle 6 + \angle 7 = 360$$ $$73 + 46 + 90 + 90 = 299^\circ$$ - The remaining angles $\angle 2$ and $\angle 4$ are inside the kite and already found as $90^\circ$ each. 5. **Final angle measures:** - $m\angle 1 = 73^\circ$ - $m\angle 2 = 90^\circ$ - $m\angle 3 = 90^\circ$ - $m\angle 4 = 90^\circ$ - $m\angle 5 = 46^\circ$ - $m\angle 6 = 90^\circ$ - $m\angle 7 = 90^\circ$ All angles are consistent with kite properties and given data.