Ray OE bisects ∠AOB and OF is the ray opposite OE. Show that ∠FOB =∠FOA.

Ray OE bisects ∠BOA
.^.. ∠BOE = ∠AOE          ----------------------(1)
∠FOB  and ∠BOE  form a linear pair.
⇒ ∠FOB + ∠BOE = 180° (Linear Pair Axiom)
         ⇒ ∠FOB  = 180° - ∠BOE  ----------------- (2)
Similarly, ∠FOA and ∠AOE form a linear pair.
⇒ ∠FOA + ∠AOE = 180°   
          ⇒ ∠FOA = 180° - ∠AOE  ----------------- (3)
From (1) ∠BOE = ∠AOE
   180° - ∠BOE =180° - ∠AOE  
       .^..  ∠FOB  = ∠FOA   (From 2 and 3) .