Question

From a point 'O' in a line 'AB', rays 'OC' and 'OD' are drawn on opposite sides of 'AB' such that angle BOC = angle AOD. Prove that 'OC' and 'OD' are opposite rays.
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Answer 1

From a point 'O' in a line 'AB', rays 'OC' and 'OD' are drawn on opposite sides of 'AB' such that angle BOC = angle AOD.

Given,

From a point 'O' in a line 'AB', rays 'OC' and 'OD' are drawn on opposite sides of 'AB' such that angle BOC = angle AOD.

Consider the attached figure, while going through the following steps.

From given, it's clear that,

OA is opposite to OB (as AB is line)

OD = OC (radii of same circle)

OA = OB  ( as "O" is the center of line AB)

∠ BOC = ∠ AOD

∴ Δ BOC ≅ Δ AOD (using SAS theorem)

as, OA is opposite to OB

Therefore, OC should be opposite to OD.

Hence it is proved that,  'OC' and 'OD' are opposite rays.