Q7 of 33 Page 108

Let us prove that, if the two opposite angles and two opposite sides of a quadrilateral are equal, then the quadrilateral will be a parallelogram.


Consider quadrilateral ABCD as shown where


AD = BC and AB = CD


∠ADC = ∠ABC = x


∠DAB = ∠BCD = y


For quadrilateral ABCD to be a parallelogram we need to prove that opposite sides are parallel i.e. AB || DC and AD || BC


Sum of all angles of a quadrilateral is 360°


⇒ ∠ADC + ∠ABC + ∠DAB + ∠BCD = 360°


⇒ x + x + y + y = 360°


⇒ 2x + 2y = 360°


⇒ x + y = 180°


Thus AB || DC and AD || BC


As opposite sides are congruent and parallel quadrilateral ABCD is a parallelogram


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