Question

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In the figure D and E are mid - points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram.​

Answer 1

$\underline {\boxed {\bold {Question}}}$

• In the figure D and E are mid - points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram.​

$\underline {\boxed {\bold {Given}}}$

• D is midpoint of AB
• E is midpoint of BC
• DF is parallel to BC

$\underline {\boxed {\bold {Important \: concepts \: to \:be \: learned }}}$

• (1) The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

• (2) The converse of this theorem states: If a line is drawn through the mid-point of a side of a triangle parallel to the second side, it will bisect the third side.

• (3) If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

$\underline {\boxed {\bold {Answer}}}$

So by converse of Mid-point theorem (2),

D is the mid point of AB and DF ∥ BC

So, F is the mid point of AC

By Mid point theorem (1)

DF = ½ BC = BE

Now by last Concept (3)

Conditions fulfilled are :

• DF = BE  {Opposite sides are equal}

• DF || BE  {Opposite sides are parallel}

So Quadrilateral DFEB is a parallelogram.

HENCE PROVED.

Answer 2

Answer:

In the figure D and E are mid - points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram.

\underline {\boxed {\bold {Given}}}

Given

D is midpoint of AB

E is midpoint of BC

DF is parallel to BC

\underline {\boxed {\bold {Important \: concepts \: to \:be \: learned }}}

Importantconceptstobelearned

(1) The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

(2) The converse of this theorem states: If a line is drawn through the mid-point of a side of a triangle parallel to the second side, it will bisect the third side.

(3) If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

\underline {\boxed {\bold {Answer}}}

Answer

So by converse of Mid-point theorem (2),

D is the mid point of AB and DF ∥ BC

So, F is the mid point of AC

By Mid point theorem (1)

DF = ½ BC = BE

Now by last Concept (3)

Conditions fulfilled are :

DF = BE {Opposite sides are equal}

DF || BE {Opposite sides are parallel}

So Quadrilateral DFEB is a parallelogram.

HENCE PROVED.