Question

In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively, of $\Delta ABC$. Show that $\angle EDF=\angle A,\angle DEF=\angle Band\angle DFE=\angle C$.

Answer 1

$△ABC$ is shown below.

D, E and F are the midpoints of sides BC, CA and AB, respectively.
As F and E are the mid points of sides AB and AC of $△ABC$
$\therefore FE\parallel BC$ (By mid point theorem)
Similarly, $DE\parallel FBandFD\parallel AC$
Therefore, AFDE, BDEF and DCEF are all parallelograms.
In parallelogram AFDE, we have:
$\angle A=\angle EDF$
(Opposite angles are equal)
In parallelogram BDEF, we have:
$\angle B=\angle DEF$
(Opposite angles are equal)
In parallelogram DCEF, we have:
$\angle C=\angle DFE$
(Opposite angles are equal)