Question:
In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively, of ∆ABC. Show that ∠EDF = ∠A, ∠DEF = ∠B and ∠DEF = ∠C. Figure
Solution:
∆ 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.
∴ FE ∣∣ BC (By mid point theorem)
Similarly, DE ∣∣ FB and FD ∣∣ AC.
Therefore, AFDE, BDEF and DCEF are all parallelograms.
In parallelogram AFDE, we have:
∠A = ∠EDF (Opposite angles are equal)
In parallelogram BDEF, we have:
∠B = ∠DEF (Opposite angles are equal)
In parallelogram DCEF, we have:
∠ C = ∠ DFE (Opposite angles are equal)