Question

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

I. Two pairs of opposite angles congruent.
II. A pair of adjacent angles are supplementary.
III. Both pairs of opposite sides are congruent.
IV. A pair of opposite angles congruent and a pair of opposite sides congruent.
V. Both pairs of opposite sides are parallel.
VI. A pair of opposite sides parallel and the other pair of opposite sides congruent.
VII. One pair of opposite sides are both parallel and congruent.
VIII. The diagonals bisect each other.

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

Quadrilaterals are known to be shapes that have four sides.

The statements that are NOT sufficient to prove that a quadrilateral is a parallelogram are:

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.

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

Parallelogram

Parallelogram is known to be a quadrilateral with four sides. Some examples of parallelogram are:

• Rhombus
• Square
• Rectangle

Some of the properties of parallelogram are:

1. Opposite angles of a parallelogram are equal.
2. The opposite sides are equal and also parallel.
3. The diagonals actually bisect each other.
4. The sum of any two adjacent angles of a parallelogram is supplementary (i.e 180°).

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