Prove that a quadrilateral with one pair of opposite sides equal and parallel, is a parallelogram.

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Solution

Let ABCD be the given quadrilateral with AB = DC and AB || DC.

Consider triangle ABC and CDA.

AB = CD (Given)

∠CAB = ∠ACD (Alternate angles)

AC = CA (common)

Therefore, by SAS congruency, the triangle are congruent.

Thus ∠DAC = BCA (Corresponding angles)

Hence, AD|| BC. Thus, ABCD is a parallelogram

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Similar questions

$(1)$ Rectangle	(p) A quadrilateral having its opposite sides equal and parallel.
$(2)$ Square	(q) A parallelogram having its opposite sides equal and each of the angle is a right angle.
$(3)$ Parallelogram	(r) A parallelogram having all sides equal and each of the angle is a right angle.
$(4)$ Rhombus	(s) A quadrilateral in which a pair of opposite sides are parallel.
$(5)$ Trapezium	(t) A parallelogram having all sides equal.

(1)	Rectangle	(p)	A quadrilateral having its opposite sides equal and parallel.
(2)	Square	(q)	A parallelogram having its opposite sides equal and each of the angle is a right angle.
(3)	Parallelogram	(r)	A parallelogram having all sides equal and each of the angle is a right angle.
(4)	Rhombus	(s)	A quadrilateral in which a pair of opposite sides are parallel.
(5)	Trapezium	(t)	A parallelogram having all sides equal.

90^{\circ}

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