Question

in the figure attached the
angle 1 = angle2
CE bisects BF
BD = DF

prove that angle C = angle E

Answer 1

in the figure,In triangles BCD and DEF,
 side BD = side DF  (given)
angle CDB = angle FDE (vertically opposite angles)
angle CBD = angle EFD (linear pair of angles)
therefore by SAA test triangle BCD is congruent to triangle DEF
THEREFORE ANGLE C = ANGLE E  (CORRESPONDING ANGLES OF CONGRUENT TRIANGLES)

Answer 2

According to the figure,
Taking ΔBCD and ΔDEF,
side BD = side DF  (given)
angle CDB = angle FDE (because they are vertically opposite angles)
angle CBD = angle EFD (because they linear pair of angles) 
.·. According to SAA (side angle angle)
ΔBCD ≡ ΔDEF 
As they are congruent
Angle C = Angle E  (Hence, proved)