Q. 355.0( 3 Votes )

# 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.

Answer :

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

.

Similarly, OP is the bisector of ∠ 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°

.

or OP and OQ are opposite rays.

and AOC.

To prove: The rays OP and OQ are in the same line.

Proof: OQ is the bisector of ∠ AOC

.

^{.}. ∠ AOQ = 1/2 ∠ AOCSimilarly, 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.

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