7. in a quadrilateral ABCD, AB = CD, and ZABC = ZBCD. Prove that AD and BC are parallel.
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In quadrilateral ABCD, we have AB=CD and AD=BC
⇒ Now, join segment AC.
⇒ In △ABC and △ACD, we have
⇒ AB = CD [Given]
⇒ AD = BC [Given]
⇒ AC = AC [Common side]
So, by SSS criteria,
⇒ △ABC≅△ACD
∴ ∠DAC = ∠BCA [By CPCT]
∴ CD∥AB [∵ Alternate angles are equal] ---- ( 1 )
⇒ ∠DCA=∠CAB [By CPCT]
∴ AD∥BC [∵ Alternate angles are equal] --- ( 2 )
From ( 1 ) and ( 2 ) we get that opposite sides of quadrilateral are parallel.
∴ ABCD is a parallelogram.
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