prove that the bisectors of vertically opposite lie on the same line
Answers
Answer:
Step-by-step explanation:
AB and CD are straight lines intersecting at O. OX the bisector of angles ∠AOC and OY is the OY is the bisector of ∠BOD.
OY is the bisector of ∠BOD.
∴∠1=∠6 ……..(1)
OX is the bisector of ∠AOC
∴∠3=∠4 …….(2)
∠2=∠5 ……….(3) (vertically opposite angles)
We know that, the sum of the angles formed at a point is 360
o
.
∴∠1+∠2+∠3+∠4+∠5+∠6=360
o
⇒∠1+∠2+∠3+∠3+∠2+∠1=360
o
(using 1,2 and 3)
⇒2∠1+2∠2+2∠3=360
o
2(∠1+∠2+∠3)=360
o
⇒∠DOY+∠AOD+∠AOX=180
o
∠XOY=180
o
∴ The bisectors of pair of vertically opposite angles are on the same straight lineAB and CD are straight lines intersecting at O. OX the bisector of angles ∠AOC and OY is the OY is the bisector of ∠BOD.
OY is the bisector of ∠BOD.
∴∠1=∠6 ……..(1)
OX is the bisector of ∠AOC
∴∠3=∠4 …….(2)
∠2=∠5 ……….(3) (vertically opposite angles)
We know that, the sum of the angles formed at a point is 360
o
.
∴∠1+∠2+∠3+∠4+∠5+∠6=360
o
⇒∠1+∠2+∠3+∠3+∠2+∠1=360
o
(using 1,2 and 3)
⇒2∠1+2∠2+2∠3=360
o
2(∠1+∠2+∠3)=360
o
⇒∠DOY+∠AOD+∠AOX=180
o
∠XOY=180
o
∴ The bisectors of pair of vertically opposite angles are on the same straight line