Let (AB) and (CD) be 2 parallel lines by a transversal at E and F respectively.The bisector of AEF and BEF cut (CD) at M and N respectively.The bisector of CFE cuts [ME] at S, and the bisector of DFE cuts [Ne] at T.Show that: ETFS is a rectangle

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  1. The sum of angles AEF and BEF is 180 degrees. Therefore the sum of the bisectors of those angles (MEF and FEN) is 90 degrees. That sum angle (MEN or SET) is one of the angles of quadrilateral SETF. Angles BEF and EFM are equal because of the way EF intercpts parallel lines. Therefore EN and SF are parallel. SE and FT are parallel for the same reason.

    Therefore EFTS is a parallelogram. Since we have shown it has one right angle, all angles are right angles, so it is a rectangle.

    You will have to draw the figure, as I did, to make sense of all that.

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