In parallelogram ABCD, point E is the midpoint of side AB. It is known that EC = ED

In parallelogram ABCD, point E is the midpoint of side AB. It is known that EC = ED. Prove that the given parallelogram is a rectangle.

Consider triangles BCE and DAE: BE = AE (E is the middle of AB), CE = DE (by condition), BC = AD (by the property of a parallelogram, the opposite sides of the parallelogram are equal).

Therefore, triangles BCE and DАЕ are equal (on three sides). This means that their angles are also equal.

The CBE angle is equal to the DAE angle.

The sum of two adjacent angles of a parallelogram is 180 degrees, so the angle CBE = DAE = 180 °: 2 = 90 °. This means that the two remaining angles of the parallelogram are also 90 ° each.

A parallelogram with right angles is a rectangle.