Question

In a parallelogram ABCD, AB = BC prove that ABCD is a rhombus​

Answer 1

Step-by-step explanation:

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.

Answer 2

So opposite sides are congruent and quadrilateral MNOP is a parallelogram. Also, adjacent sides are congruent, so parallelogram MNOP is a rhombus. 1. 2.

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Geometry.

Statements Reasons

7. ?AMB ~= ?CMB SAS Postulate

8. AB ~= BC CPOCTAC

9. Parallelogram ABCD is a rhombus Definition of rhombus