Question

If ABCD is a parallelogram, E and F are midpoints of AD and BC respectively. Then quadrilateral DEBF will be ?

• A. A parallelogram
• B. A trapezium
• C. A rhombus
• D. A rectangle

Answer 1

The correct option is A A parallelogram

Given:
ABCD is parallelogram, E and F are mid points of AD and BC respectively.

To prove:
DEBF is a parallelogram.

Proof:

In quadrilateral DEBF,

DE = BF------(1)

[ $\frac{1}{2}AD=\frac{1}{2}BC$ ]

BF||DE [ BC||AD ]

$⟹\angle 1=\angle 2$ -------(2)

[Alternate interior angles]

Now, In $△DEFand△BFE$,

DE = BF from (1)

$\angle DEF=\angle BFE$ from (2)

EF = EF common side

By S.A.S $△DEF\cong △BFE$

$\angle DFE=\angle BEF$

[Corresponding parts of congruent triangles]

$⟹FD||BE$

Now,

$\because$ DE||BF and FD||BE, DEBF is a parallelogram.