Which of the following are NOT sufficient to prove that a quadrilateral is a parallelogram?
2 answers:
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.
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