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 20080417 13:01:22 