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A diagonal of a parallelogram bisects one of its angle. Prove that it will bisect its opposite angle also.

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Let us draw the figure as per given condition (Fig.). In it, AC is a diagonal which bisects ∠BAD of the parallelogram ABCD, i.e., it is given that ∠BAC = ∠DAC. We need to prove that ∠BCA = ∠DCA. 

AB || CD and AC is a transversal. 

Therefore, ∠BAC = ∠DCA (Alternate angles) (1)

Similarly, ∠DAC = ∠BCA (From AD || BC) (2) 

But it is given that ∠BAC = ∠DAC (3) 

Therefore, from (1), (2) and (3), we have 

∠BCA = ∠DCA

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