Question

A quadrilateral ABCD has exactly one axis of symmetry, which is not a diagonal. Show that the quadrilateral is an isosoeles trapezium.

Answer 1

Consider a quadrilateral ABCD.
Let m be a line which is not a diagonal of quad. ABCD.
Let the quad. ABCD have this line only as its axis of symmetry.
Then A, B and D, C are the two pairs of points each pair symmetric w.r.t., line m.
∴ Line m is the perpendicular bisector of AB as well as DC.
⇒ AB || DC      ...(1)
Also AD and BC are symmetric w.r.t., line m.
∴ AD = BC     ...(2)
In view of (1) and (2), quadrilateral ABCD is an isosceles trapezium.
Hence proved.