Math, asked by dilipram2006, 7 months ago

prove that the bisectors of vertically opposite lie on the same line

Answers

Answered by sushganesh02
1

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

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