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