Dear Student,

13 ) From given diagram we know : BC | | DE

So,

Height of $∆$ BCD =  Height of $∆$ BCE  =  h     (  As both triangles lies between same parallel lines " DE | |BC " by taking base ' BC ' )

We know :  Area of triangle  = $\frac{1}{2}$ $×$ Base $×$ Height , So

Area of $∆$ BCD = $\frac{1}{2}$ $×$ BC $×$ h                                            --- ( 1 )

And

Area of $∆$ BCE  = $\frac{1}{2}$ $×$ BC $×$ h                                            --- ( 2 )

From equation 1 and 2 we get :

Area of $∆$ BCD = Area of $∆$ BCE

Now we subtract ' Area of $∆$ BOC ' on both hand side we get :

Area of $∆$ BCD - Area of $∆$ BOC =  Area of $∆$ BCE - Area of $∆$ BOC

We know from Third Axiom of Euclid " If equals be subtracted from equals, the remainders are equal. " So

Area of $∆$ BOD =  Area of $∆$ COE                                                          ( Hence proved )