Question

Prove that if a ray stands on a line, then the sum of two adjacent angles so formed is ${180}^o$.

Answer 1

Given: A ray OC stands on line AB then, adjacent angle $\angle AOC$ and $\angle BOC$ are formed.
To prove: $\angle AOC+\angle BOC={180}^o$
Construction: Draw a ray $OE\perp AB$.
Proof: $\angle AOC=\angle AOE+\angle EOC$ ....(1)
$\angle BOC=\angle BOE-\angle EOC$ ....(2)
Adding equation 1 and 2
$\angle AOC+\angle BOC=\angle AOE+\angle EOC+\angle BOE-\angle EOC$
$\Rightarrow \angle AOC+\angle BOC=\angle AOE+\angle BOE$
$\Rightarrow \angle AOC+\angle BOC={90}^o+{90}^o(OE\perp AB)$
$\Rightarrow \angle AOC+\angle BOC={180}^o$
Hence, proved.