Question

In parallelogram ABCD, E and F are mid points of the sides AB and CD respectively. The line segment AF and BF meet the line segment ED and EC at point G and H respectively. Prove that:
(i) Triangle HEB and FHC are congruent
(ii) GEHF is a parallelogram.

Answer 1

Hint:

Draw the diagram first. Given that, in parallelogram ABCD, E and F are mid points of the sides AB and CD respectively; i.e. AE = BE = CF = DF (since AB = CD). Now we can easily show $\vartriangle BEH\; \cong \vartriangle FHC$ by AAS rule of congruence.
For the second proof, note that the quadrilateral AECF and BEDF are both parallelograms, since one pair of their sides are equal and parallel. Therefore, $EH\parallel GF$ and $EG\parallel HF$. So both of the pairs of sides of the quadrilateral GEHF are parallel. Therefore quadrilateral GEHF is a parallelogram.

Complete step-by-step answer:

In parallelogram ABCD, E and F are mid points of the sides AB and CD respectively.
$\Rightarrow$ AE = BE and CF = DF
$\Rightarrow$ AE = BE = CF = DF … (since AB = CD)
Given that, the line segments AF and BF meet the line segments ED and EC at point G and H respectively.
Now, in $\vartriangle BEH$ and $\vartriangle FHC$,
BE = CF
$\angle BEH = \angle CHF$ (vertically opposite angles)
$\angle BEH = \angle HCF$ (alternate angles; since $AB\parallel CD$, EC transversal)
Therefore, $\vartriangle BEH\; \cong \vartriangle FHC$ (by AAS rule of congruence)
Proved (i)
Again, note that the quadrilateral AECF is a parallelogram, since one pair of its sides are equal and parallel.
$\Rightarrow$ $EC\parallel AF$
$\Rightarrow$ $EH\parallel GF$
Similarly, quadrilateral BEDF is also a parallelogram.
$\Rightarrow$ $ED\parallel BF$
$\Rightarrow$ $EG\parallel HF$
So both of the pairs of opposite sides of the quadrilateral GEHF are parallel.

Therefore, quadrilateral GEHF is a parallelogram.

Note:
The four rules of congruency are as follows:
SSS: When three sides of two different triangles are equal in length.
SAS: When two sides are equal, and the angle between them is also the same in measure.
AAS: When any two angles and a side is equal.
RHS: When the hypotenuse and any one side of two right angled triangles are equal in length.
Also, note the following properties of a parallelogram.
Both the pairs of opposite sides are equal and parallel to each other.
Both the pairs of opposite angles are equal.
Any two consecutive angles (or same-side interior angles) are supplementary.
The diagonals are equal in length.
The diagonals intersect each other.
The diagonal divides the parallelogram in two congruent triangles.