In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively,

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)

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