# In parallelogram ABCD, perpendiculars BE and DF are dropped from

In parallelogram ABCD, perpendiculars BE and DF are dropped from the vertices of obtuse angles B and D onto the diagonal AC. Prove that the quadrilateral BEDF is a parallelogram.

Consider triangles ABE and CDF. The angles E and F of the triangles are right, since BE and DF are perpendicular to AC. The hypotenuses of the triangles are equal, since these are opposite sides of the ABCD parallelogram.

The angle BAE is equal to the angle FCD, as lying crosswise at the intersection of parallel lines AB and CD of the secant AC.

Therefore, triangles ABE and CFD are equal in hypotenuse and acute angle.

This means that the legs BE and DF are equal and parallel, since they are perpendicular to one segment of the AC.

If two opposite sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.

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