In figure, AB and CD are straight lines and OP and OQ are respectively the bisectors of a ∠ BOD and ∠ AOC.  Show that the rays OP and OQ are opposite rays.
        

Given: AB and CD are straight lines and OP and OQ are respectively the bisectors of angles BOD 
and AOC. 
To prove: The rays OP and OQ are in the same line.
Proof:  OQ is the bisector of ∠ AOC
.^.. ∠ AOQ = 1/2 ∠ AOC
Similarly, OP is the bisector of ∠ DOB,
.^.. ∠ DOP = 1/2 ∠ DOB
∠ AOQ + ∠ AOD + ∠ DOP = 1/2 ∠ AOC + 1/2 ∠ DOB + ∠ AOD
2 (∠ AOQ + ∠ AOD + ∠ DOP) = ∠ AOC + ∠ DOB + 2 ∠ AOD
                                           = ∠ AOC + ∠ DOB + ∠ AOD + ∠ COB           ( ∠ AOD = ∠ COB vertically opp. angles )                            
                                           = 360° (Angles around a point 360°)
∴ ∠ AOQ + ∠ AOD + ∠ DOP = 360° /2 = 180°
.^..  Points Q, O, P are collinear.
or OP and OQ are opposite rays.