prove the theorem: if pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram
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Given: Quadrilateral ABCD in which AB = DC and AD = BC$$
To prove: ABCD is a parallelogram.
Construction: Draw diagonal AC.
Proof: Some statements and reasons for their validity are given below:
Statement Reason
AD=BC Given
AB=CD Given
AC=AC Common
△CBA≅△ADC By SSS postulate
∠DCA=∠BAC Corresponding angles of congruent triangles
DC∥AB Alternate angles are equal.
∠DAC=∠BCA Corresponding angles of congruent triangles
AD∥BC Alternate angles are equal
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