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Question

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

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Solution

## ABCD is a quadrilateral with four equal sidesIf the sides are equal they will produce four equal anglesHence each angle will be 90 degree (Sum of interior angles is 360 degree)Hence ABCD is considered as a SQUAREAC and BD are joined which meet at OAC and BD are diagonals , which bisect interior angles.Angle BAD is bisected by ACAngles BAC = DACAngle BCD is bisected by AC Angles ACD = BCASince, BAD = BCD = 90°Angles BAC = DAC = Angles ACD = BCABAC = ACDThey are also interiorly alternate to each other.So by converse theorem,AB is parallel to CDSimilarly Angles ABD = CBD = ADB = CDB (when BD is the bisector)CBD = ADBThey are also interiorly alternate to each other.So by converse theorem,AD is parallel to BCHence opposite pairs of sides are parallel to each other

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