Question

If ABCD is a quadrilateral where AB=CD and AD=BC then prove that it is a parallelogram​

Answer 1

Answer:

It is proved only.

Step-by-step explanation:

If the opposite sides of a quadrilateral are equal then it's parallelogram itself.

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Answer 2

GIVEN :-

• ABCD is a quadrilateral.
• AB = CD , AD = BC.

TO PROVE :-

• The quadrilateral ABCD is a parallelogram.

PROOF :-

✭ In ∆ ABD & CDB,

➠ AD = BC [ Given ]

➠ AB = CD [ Given ]

➠ BD = BD [ Common ]

So, by SSS Congruency criteria ∆ ABD ⩭ CDB.

Now we can say that ,

➠ ∠ABD = ∠CDB [ C.P.C.T ]. ...Eq(1)

Here , ∠ABD = ∠CDB are alternate interior angles .SO CD || AB.

Similarly,

➠ ∠ADB = ∠DBC [ C.P.C.T ]. ...Eq(2)

Here , ∠ADB = ∠DBC are alternate interior angles .SO AD || BC.

Now,

➠ ∠A = ∠C [ C.P.C.T ]

➠ ∠D = ∠ADB + ∠CDB

➠ ∠D = ∠ADB + ∠ABD. [ From Eq(1) ]. ...Eq(3)

Similarly,

➠ ∠B = ∠ABD + ∠DBC

➠ ∠B = ∠ABD + ∠ADB. [ From eq(2) ]. ...Eq(4)

Now , R.H.S of Equation 3 and 4 are equal . So L.H.S are also equal i.e,

➠ ∠B = ∠D

Now, in quadrilateral ABCD we have,

• ∠A = ∠C. [ Opposite angles of || gm ]
• ∠B = ∠D. [ Opposite angles of || gm ]
• CD || AB. [ Opposite Sides of || gm ]
• BC || AD. [ Opposite Sides of || gm ]

Hence , quadrilateral ABCD is a"Parallelogram".

Hence Proved.