Question

2 in the following quadrilateral abcd, write o if the conditions make quadrilateral abcd a parallelogram and × if the conditions do not make quadrilateral abcd a parallelogram in the .
(1) ( ab = 3 mathrm { cm }, bc = 7 mathrm { cm }, cd = 7 mathrm { cm }, ) and ( da = 3 mathrm { cm } )
(2) ( angle b = 60 ^ { circ }, angle c = 120 ^ { circ }, ab = 4 mathrm { cm }, ) and ( cd = 4 mathrm { cm } )
(3) ( angle a = 75 ^ { circ }, angle c = 105 ^ { circ }, bc = 3 mathrm { cm }, ) and ( cd = 3 mathrm { cm } )
(4) let o be the intersection point of diagonals ac and bd
( a o = 7 mathrm { cm }, c o = 7 mathrm { cm }, b o = 4 mathrm { cm }, ) and ( d o = 4 mathrm { cm } )

Explanation:

(1)

Step1: Check the property of parallelogram

In a parallelogram, opposite sides are equal. Here \(AB = DA = 3\mathrm{cm}\) and \(BC=CD = 7\mathrm{cm}\). But in a parallelogram \(AB = CD\) and \(BC = DA\). So this does not satisfy the property of a parallelogram.

(2)

Step1: Check the property of parallelogram

We know that \(\angle B+\angle C=60^{\circ}+ 120^{\circ}=180^{\circ}\), so \(AB\parallel CD\) (co - interior angles are supplementary). Also \(AB = CD = 4\mathrm{cm}\). If one pair of opposite sides is parallel and equal, then the quadrilateral is a parallelogram.

(3)

Step1: Check the property of parallelogram

\(\angle A = 75^{\circ}\), \(\angle C=105^{\circ}\), \(BC = CD = 3\mathrm{cm}\). In a parallelogram, opposite angles are equal and opposite sides are equal. Here \(\angle A
eq\angle C\) and we don't know about the relationship of other sides and angles. So it is not a parallelogram.

(4)

Step1: Check the property of parallelogram

If the diagonals of a quadrilateral bisect each other (\(AO = CO = 7\mathrm{cm}\), \(BO=DO = 4\mathrm{cm}\)), then the quadrilateral is a parallelogram.

Answer:

(1) \(×\)
(2) \(○\)
(3) \(×\)
(4) \(○\)