Question

ABCD is a quadrilateral in which all four sides are equal show that both pairs of opposite sides are parallel

Answer 1

Given: A quadrilateral with all sides equal.

To prove that the opposite pairs of sides are parallel.

Construction. Draw a rhombus ABCD. Join BD.

Proof: In triangles ABD and BCD

AD=CD [given]

AB=BC [given]

BD is common

So triangles ABD and BCD are congruent.

<ADB = <CBD and so alternate angles. Therefore, BD is a transversal and so AD is parallel to CB.

Again, <ABD = <CDB and so alternate angles. Therefore, BD is a transversal and so AB is parallel to CD.

Hence opposite pairs of sides are parallel.

QED.