1. **State the problem:** We need to find the measure of angle $\angle 3$ in a kite-shaped quadrilateral where one angle is given as $27^\circ$.\n\n2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal. The diagonals intersect at right angles ($90^\circ$). The angles between unequal sides are equal.\n\n3. **Given:** The angle at the top-right corner adjacent to the horizontal diagonal is $27^\circ$. The diagonals intersect at $90^\circ$.\n\n4. **Identify $\angle 3$:** $\angle 3$ is the angle adjacent to the $27^\circ$ angle on the right side of the kite, formed by the diagonal and the side with two tick marks.\n\n5. **Use the right angle property:** Since the diagonals intersect at $90^\circ$, the sum of the two adjacent angles at the intersection on the right side must be $90^\circ$.\n\n6. **Calculate $\angle 3$:** $$\angle 3 = 90^\circ - 27^\circ = 63^\circ.$$\n\n**Final answer:** $\boxed{63^\circ}$