Question

Prove that if the sum of two adjacent angles is ${180}^o$, then the non-common arms are two opposite rays.

Answer 1

Given: Two adjacent angles are $\angle AOC$ and $\angle BOC$ and $\angle AOC+\angle BOC={180}^o$
To prove: OA and OB are two opposite rays.
Construction: Let OA and OB are not two opposite rays.
Then, draw a ray OE opposite to OA such that AOE is a straight line.
Proof:
$\angle AOC+\angle BOC={180}^o$ .....(1)
$\angle AOC+\angle COE={180}^o$ ....(2) (Linear pair)
$\angle AOC+\angle EOC=\angle AOC+\angle BOC$ (From equation 1 and 2)
$\therefore \angle EOC=\angle BOC$
This is possible only if OE and OB coincide. Hence, OA and OB are two opposite rays.