Two opposite sides of a quadrilateral are the same length. The diagonals of the quadrilateral are also the same length. Prove the quadrilateral is a trapezoid.

Let ABCD be the quadrilateral with
|AB| = |CD| and |AC| = |BD|.

Triangles ABC and DCB are congruent by SSS
  ==> angle ABC = angle DCB = angle BCD

Triangles BAD and CDA are congruent by SSS
  ==> angle CDA = angle BAD = angle DAB

360 = angle ABC + angle BCD +
      angle CDA + angle DAB

    = 2(angle ABC) + 2(angle DAB)

    or

180 = angle ABC + angle DAB

  ==> DA || BC

  ==> ABCD is a trapezoid.

Posted by Bractals on 2008-04-17 13:01:22