In figure, AB and CD are straight lines and
OP and OQ are respectively, the bisectors of a LBoD and LAOC show that the rays op and oq are opposite Rays
Answers
In order to prove that OP and OQ are in the same line , it is sufficient to prove that \angle∠ POQ = 180°.
Now, OP is the bisectors of \angle∠ BOD
\implies⟹ \angle∠ 1 = \angle∠ 6 .............. (1)
And, OQ is the bisectors of \angle∠ AOC
\therefore∴ \angle∠ 3 = \angle∠ 4 ................ (2)
Clearly, \angle∠ 2 and \angle∠ 5 are vertically opposite angles
\therefore∴ \angle∠ 2 = \angle∠ 5 ................ (3)
_________________
We know that the sum of angles formed at a point is 360°.
\therefore∴ \angle∠ 1 + \angle∠ 2 + \angle∠ 3 + \angle∠ 4 + \angle∠ 5 + \angle∠ 6 = 360°
\implies⟹ (\angle∠ 1 + \angle∠ 6) + (\angle∠ 3 + \angle∠ 4) + (\angle∠ 2 + \angle∠ 5) = 360°
\implies⟹ 2\angle∠ 1 + 2\angle∠ 3 + 2\angle∠ 2 = 360°
\implies⟹ 2 (\angle∠ 1 + \angle∠ 3 + \angle∠ 2) = 360°
\imples\imples \angle∠ 1 + \angle∠ 3 + \angle∠ 2 = 180°
\implies⟹ \angle∠ POQ = 180°
Answer:
In order to prove that OP and OQ are in the same line , it is sufficient to prove that \angle∠ POQ = 180°.
Now, OP is the bisectors of \angle∠ BOD
\implies⟹ \angle∠ 1 = \angle∠ 6 .............. (1)
And, OQ is the bisectors of \angle∠ AOC
\therefore∴ \angle∠ 3 = \angle∠ 4 ................ (2)
Clearly, \angle∠ 2 and \angle∠ 5 are vertically opposite angles
\therefore∴ \angle∠ 2 = \angle∠ 5 ................ (3)
_________________
We know that the sum of angles formed at a point is 360°.
\therefore∴ \angle∠ 1 + \angle∠ 2 + \angle∠ 3 + \angle∠ 4 + \angle∠ 5 + \angle∠ 6 = 360°
\implies⟹ (\angle∠ 1 + \angle∠ 6) + (\angle∠ 3 + \angle∠ 4) + (\angle∠ 2 + \angle∠ 5) = 360°
\implies⟹ 2\angle∠ 1 + 2\angle∠ 3 + 2\angle∠ 2 = 360°
\implies⟹ 2 (\angle∠ 1 + \angle∠ 3 + \angle∠ 2) = 360°
\imples\imples \angle∠ 1 + \angle∠ 3 + \angle∠ 2 = 180°
\implies⟹ \angle∠ POQ = 180°
Hope you understood