Answer 1

(i) Given: DE || GF
Now, ∠ DEF = ∠ GFM              (Corresponding angles as DM is a transversal line)
⇒ 2 ∠ DEG = ∠ DFG         (Ray EG and ray FG are bisectors of ∠ DEF and ∠ DFM)
⇒ 2 ∠ DEG = ∠ EDF          (∵ ∠ EDF = ∠ DFG, alternate angles as DF is a transversal line)
⇒ ∠ DEG =`1/2` EDF

(ii) Given: DE || GF
∠ DEG = ∠ EGF              (Alternate angles as EG is a transversal line)
∴ ∠ GEF = ∠ EGF            (∵ ∠ DEG = ∠ GEF)
∴ EF = FG                         (Sides opposite to equal angles)