Question

If pairs of opposite sides of a quadrilateral are
congruent then that quadrilateral is a parallelogram​

Answer 1

Answer:

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If a quadrilateral is a rectangle, then it is a parallelogram.

Step-by-step explanation:

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Answer 2

Answer:

Take Quadrilateral ABCD, with side AB parallel and congruent to side DC. Then draw diagonal AC. Angle CAB will be congruent to angle DCA, because if two parallel lines are cut by a transversal, then alternate interior angles, which angles CAB and DCA are, must be congruent. We are given that sides AB and CD are congruent. Lastly diagonal CD is congruent to itself, CD ≅ CD, by the reflexive property of congruence.

This makes ΔDCA ≅ ΔBAC by Side Angle Side congruence theorem.

This makes angle DAC ≅ angle BCA by corresponding parts of congruent triangles are congruent.

Which makes side AD parallel to side BC by the converse of alternate interior angles, if a transversal which crosses two lines, creates two congruent alternate interior angles, then the two lines must be parallel.

This makes quadrilateral ABCD a parallelogram, because we have shown that it has two pairs of parallel sides.