1. The problem asks for the central angle of the missing puzzle piece that completes a circle and a semicircle. 2. Recall that a full circle has a total central angle of $$360^\circ$$ and a semicircle has a total central angle of $$180^\circ$$. 3. The given sector angles are $$27^\circ$$, $$124^\circ$$, $$80^\circ$$, $$63^\circ$$, and one quarter circle sector which is $$90^\circ$$. 4. First, sum the known angles of the sectors that form the circle: $$27^\circ + 124^\circ + 80^\circ + 63^\circ + 90^\circ = 384^\circ$$ 5. Since the total for a circle is $$360^\circ$$, the sum $$384^\circ$$ exceeds this, so these sectors must be split between the circle and the semicircle. 6. The problem states the pieces should make a circle and a semicircle, so we separate the sectors into two groups: one group sums to $$360^\circ$$ (circle), the other sums to $$180^\circ$$ (semicircle). 7. Let's check the sum of the four sectors excluding the quarter circle (90°): $$27^\circ + 124^\circ + 80^\circ + 63^\circ = 294^\circ$$ 8. Adding the quarter circle (90°) to 294° gives 384°, which is more than 360°, so the quarter circle must belong to the semicircle group. 9. The semicircle has a total of $$180^\circ$$, and the quarter circle sector is $$90^\circ$$, so the missing piece in the semicircle is: $$180^\circ - 90^\circ = 90^\circ$$ 10. The circle group has sectors with angles $$27^\circ$$, $$124^\circ$$, $$80^\circ$$, and $$63^\circ$$, summing to $$294^\circ$$. 11. The missing piece in the circle is: $$360^\circ - 294^\circ = 66^\circ$$ 12. Among the answer choices, the closest to $$66^\circ$$ is $$63^\circ$$. 13. Therefore, the central angle of the missing piece is $$63^\circ$$. Final answer: **63°**