Question

Statement 1: AB II DC and AD II BC Statement 2: AB = DC and AD=BC. For a Quadrilateral ABCD is a parallelogram * 1 point Either Statement 1 or Statement 2 should be true Both Statement 1 and Statement 2 should be true Only Statement 1 is true Only Statement 2 is true​

Answer 1

Answer:

both of them them are true

Answer 2

Given :-

Statement 1 : AB II DC and AD II BC

Statement 2 : AB = DC and AD=BC .

For a Quadrilateral ABCD is a parallelogram :-

1) Either Statement 1 or Statement 2 should be true .

2) Both Statement 1 and Statement 2 should be true .

3) Only Statement 1 is true .

4) Only Statement 2 is true .

Answer :- (1)

Explanation :-

Statement 1 :- AB II DC and AD II BC .

In ∆ABD and ∆CDB we have,

→ ∠ADB = ∠CBD { AD II BC , Alternate angles. }

→ ∠ABD = ∠CDB { AB II DC , Alternate angles. }

→ BD = DB { Common. }

So,

→ ∆ABD ≅ ∆CDB { By AAS congruence rule. }

then,

→ AB = CD { By CPCT .}

→ AD = CB { By CPCT . }

since in quadrilateral ABCD opposite sides are parallel and equal therefore, we can conclude that, ABCD is a parallelogram .

Statement 2 :- AB = DC and AD = BC .

In ∆ABD and ∆CDB we have,

→ AB = CD { given .}

→ AD = CB { given. }

→ BD = DB { common. }

So,

→ ∆ABD ≅ ∆CDB { By SSS congruence rule. }

then,

→ ∠ABD = ∠CDB { By CPCT. }

since, Alternate angles are equal , therefore, AB || DC .

→ ∠ADB = ∠CBD { By CPCT .}

since, Alternate angles are equal , therefore, AD || BC .

since in quadrilateral ABCD opposite sides are parallel and equal therefore, we can conclude that, ABCD is a parallelogram .

Hence, we can conclude that, For a Quadrilateral ABCD is a parallelogram (1) Either Statement 1 or Statement 2 should be true .

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