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Lowering flaps increases the wing's camber and allows the aircraft to fly at a smaller angle of attack (AOA).
Take this wing, starting in a clean configuration. When the pilot lowers the flaps, two things immediately happen: the wing camber and the AOA both increase.
The camber increases because flaps change the shape of the wing, adding more curvature. This produces more lift.
The AOA increases because the effective chord line, which runs from the leading edge of the wing to the trailing edge of the flap, pivots up. This increases the angle between the chord line and the relative wind (the AOA).
What's the result? This increase in camber and AOA produces more lift.
Since the flaps-down wing has more camber than a clean wing, it creates the same amount of lift at a smaller AOA. The flaps-down aircraft now flies at a smaller AOA to balance the four forces of flight.
Colin is a Boldmethod co-founder, pilot and graphic artist. He's been a flight instructor at the University of North Dakota, an airline pilot on the CRJ-200, and has directed development of numerous commercial and military training systems. You can reach him at email@example.com.