Question

Draw two intersecting line segment AB and CD intersecting at O. Measure the size of each pair of vertically opposite angles. Verify that each pair of vertically opposite angles are equal.

Answer 1

Angles vertically opposite to each other are equal.

Explanation:

There are two lines AB and CD intersecting at point O making angles 1 ,2,3 and 4.

The angles 1 and3 , angles 2 and 4 are vertically opposite angles.

Now consider the line AB

<1 + <2= 180 degrees--------- eq A

Now consider the line CD

<2+ <3= 180 degrees-----------eq B

As both equations are equal

< 1+ <2= <2+ <3 = 180 degrees

or

< 1+ <2= <2+ <3 or

< 1= <3 subtracting <2 from both sides

Now consider the line BA

<3+ <4= 180 degrees--------- eq C

Now consider the line DC

<4+ <1= 180 degrees-----------eq D

As both equations are equal

< 3+ <4= <4+ <1 = 180 degrees

or

< 3+ <4= <4+ <1

or

< 3= <1 subtracting <4 from both sides

This holds true for all vertically opposite angles that they are equal.

These can be measured with a protractor.

This can also be proved that if we add angles 1,2,3 and 4 we will get 360

and if we subtract angles 1 & 2 from 360 we will get 180 degrees which angles 3&4 make together.

<1+<2+<3+<4= 360 degrees

But <1 +<2= 180

360 - [ <1+<2] = 180 degrees

<3 +<4= 180

<1 +<2= <3 +<4= 180