Question

prove that if two lines intersect each other then the bisector of vertically opposite angle are in the same line

Answer 1

Draw two intersecting lines AB and CD intersecting at O.

Angles AOC and BOD are vertically opposite angles and hence are equal.

For the same reason, angles AOD and BOC are equal.

Draw angle bisector OE for angle AOC and angle bisector OF for angle BOD.

Angles EOC and BOF are equal , each being half of above verticaly opposite angles.

For the same reason, Angles AOE and DOF are equal.

Now sum of (angles EOC + COB+BOF) + (angles AOE+AOD+DOF) = 360 degree

Also (angles EOC + COB+BOF) = (angles AOE+AOD+DOF)

as angle EOC = ange AOE, angle COB = angle AOD, angle BOF = angleDOF

Hence each of these = 360 / 2  = 180 degree.

Hence bisectors OE and OF are in the same straight line.

SEE PICTURE ATTACHED

Answer 2

Answer:

Draw two intersecting lines AB and CD intersecting at O.

Angles AOC and BOD are vertically opposite angles and hence are equal.

For the same reason, angles AOD and BOC are equal.

Draw angle bisector OE for angle AOC and angle bisector OF for angle BOD.

Angles EOC and BOF are equal , each being half of above verticaly opposite angles.

For the same reason, Angles AOE and DOF are equal.

Now sum of (angles EOC + COB+BOF) + (angles AOE+AOD+DOF) = 360 degree

Also (angles EOC + COB+BOF) = (angles AOE+AOD+DOF)

as angle EOC = ange AOE, angle COB = angle AOD, angle BOF = angleDOF

Hence each of these = 360 / 2  = 180 degree.

Hence bisectors OE and OF are in the same straight line.