Question

Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Answer 1

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 360ï½.∴ 1 + 2 + 3 + 4 + 5 + 6 = 360ï½⇒ 1 + 2 + 3 + 3 + 2 + 1 = 360ï½ (Using (1), (2) and (3))⇒ 21 + 22 + 23 = 360ï½⇒ 2(1 + 2 + 3) = 360ï½⇒ DOY + AOD + AOX = 180ï½⇒ XOY = 180ï½∴ The bisectors of pair of vertically opposite angles are on the same straight line.HOPE IT HELPS..!

Answer 2

Answer:

statement: AB and CD are straight lines intersecting at O. OP and OQ are respectively the bisectors of ∠BOD and ∠AOC.

given: OP and OQ are the bisectors o thier respective angles

to prove: OP and OQ lie on the same line.

FIGURE is given below.

PROOF: now OP is the bisector of ∠BOD

         →∠1=∠6            .....1

        and, OQ is the bisector of∠AOC

       ∴∠3=∠4               .....2

       Clearly, ∠2 and ∠5 are vertically opposite angle

         ∠2=∠5                     ....3

we know that the sum of the angles formed at a point is 360°.

∴∠1+∠2+∠3+∠4+∠5+∠6=360°

→(∠1+∠6)+(∠3+∠4)+(∠2+∠5)=360°

→2∠1+2∠3+2∠2=360°

→2(∠1+∠3+∠2)=360°    [using 1,2,3]

→∠1+∠2+∠3=180°

→∠POQ=180°

HENCE,OP AND OQ ARE IN THE SAME STRAIGHT LINE.

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