Question

Which of the following are NOT sufficient to prove that a quadrilateral is a parallelogram?

Answer 1

Answer with explanation:

We can prove that a quadrilateral is a parallelogram , if

I. Two pairs of opposite angles congruent.

Consider, a Quadrilateral ABCD, in which , ∠A=∠C,and ∠B=∠D.

Now, ∠A+∠B+∠C+∠D=360°

which gives, ∠A+∠B=180 and, ∠A+∠D=180°

Showing, AB║CD, and AD║BC.

Join ,either, AC or BD.

So, By AAS,we can prove that ,the two triangles are congruent.

So, Quadrilateral ABCD is a Parallelogram.

III.Both pairs of opposite sides are congruent.→Axiom

V.Both pairs of opposite sides are parallel. →→Proved in I

VI.One pair of opposite sides are both parallel and congruent.

In Quadrilateral ,ABCD

AB║CD,AB=CD

Join AC

In Δ ABC and ΔADC

∠BAC=∠DCA→Alternate interior  Angles

AC is common.

AB=CD→[given]

Δ ABC ≅ ΔADC→→[SAS]

VIII.The diagonals bisect each other.

In ΔAOB and ΔCOD

AO=OC

BO=OD

∠AOB=∠COD→[Vertically opposite angles]

→∠AOB≅∠COD[SAS]

which gives, AB=CD.By taking other two triangles, ∠AOD=∠COB, we can prove, AD=BC.

So, when opposite sides are equal in a Quadrilateral it is a parallelogram.

The Condition which are not Sufficient to prove that a quadrilateral is a parallelogram:

II: A pair of adjacent angles are supplementary.

IV: A pair of opposite angles congruent and a pair of opposite sides congruent.

VI: A pair of opposite sides parallel and the other pair of opposite sides congruent.

Answer 2

The answers are
I. Two pairs of opposite angles congruent. 
II. A pair of adjacent angles are supplementary.  
VIII. The diagonals bisect each other.