Question

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

Answer 1

ABCD is a quadrilateral with four equal sides
If the sides are equal they will produce four equal angles
Hence each angle will be 90 degree (Sum of interior angles is 360 degree)
Hence ABCD is considered as a SQUARE
AC and BD are joined which meet at O
AC and BD are diagonals , which bisect interior angles.
Angle BAD is bisected by AC
Angles BAC = DAC
Angle BCD is bisected by AC 
Angles ACD = BCA
Since, BAD = BCD = 90°
Angles BAC = DAC = Angles ACD = BCA
BAC = ACD
They are also interiorly alternate to each other.
So by converse theorem,
AB is parallel to CD
Similarly Angles ABD = CBD = ADB = CDB (when BD is the bisector)
CBD = ADB
They are also interiorly alternate to each other.
So by converse theorem,
AD is parallel to BC
Hence opposite pairs of sides are parallel to each other