Points E and F are taken in the rectangle ABCD on the sides BC and AD so that AB = BE and CD = FD

Points E and F are taken in the rectangle ABCD on the sides BC and AD so that AB = BE and CD = FD a) prove that AE is the bisector of the angle BAD and CF is the bisector of the angle BCD. b) define the type of the quadrangle AECF.

The ABE triangle is isosceles, since AB = BE by condition, then the angle BAE = BEA.

The angle BEA = EAD as cross-lying angles at the intersection of parallel straight lines BC and AD of the secant AE, then the angle BAE = EAD, and therefore the segment AE is the bisector of the angle BAD. Similarly, ДF is the bisector of the ВСD angle. Q.E.D.

Right-angled triangles ABE and CDF are equal in two legs, then AE = CF. In the quadrangle AECF, the segments CE and AF are equal, since BC = AD, and BE = DF. Since in the quadrilateral AECF the opposite sides are pairwise equal, this is a parallelogram.

Answer: Quadrilateral AECF is a parallelogram.