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Question What type of quadrilateral is one in which the opposite sides are equal? Justify your answer.
Hint: Draw a diagonal and check for congruent triangles.
Let ABCD be a quadrilateral in which opposite sides are equal. Join AC. In ∆ADC and ∆CBA, we have AD = CB, DC = BA, and AC is common. ∴ By the SSS condition, AADC and ACBA are congruent. ∴ ∠1 = ∠3 and ∠2 = ∠4 AC is a transversal of lines AB and DC, and alternate angles ∠1 and ∠3 are equal. ∴ Lines AB and DC are parallel. AC is a transversal of lines AD and BC, and alternate angles ∠2 and ∠4 are equal. ∴ Lines AD and BC are parallel. ∴ By definition, the quadrilateral ABCD is a parallelogram.