From point E, taken on the side BC of the rectangle ABCD, the perpendicular EF

From point E, taken on the side BC of the rectangle ABCD, the perpendicular EF is lowered to the side AD. Prove that ABEF is a rectangle.

We will prove that the quadrilateral ABEF is a rectangle if we find out that all of its corners are right.

Angle BAF coincides with the angle BAD of rectangle ABCD and means straight line.

Likewise, angle ABE is also right.

Since EF is perpendicular to AD, the angle EFA is right.

But the sum of the angles of the quadrilateral is 360 ° and therefore

angle BEF = 360 ° – BAF – ABE – EFA = 90 °.

We have shown that all angles ABEF are straight. Hence,

ABEF is a rectangle, as required.