Class 9 Maths

Lines And Angles

Parallel Lines

If two lines do not meet at a point if extended to both directions, such lines are called parallel lines.

parallel lines

Lines PQ and RS are parallel lines.

The length of the common perpendiculars at different points on these parallel lines is same. This equal length is called the distance between two parallel lines.

Axiom 1

If a ray stands on a line, then the sum of two adjacent angles so formed is 180º.

Conversely if the sum of two adjacent angles is 180º, then a ray stands on a line (i.e., the non-common arms form a line).

Axiom 2

If the sum of two adjacent angles is 180º, then the non-common arms of the angles form a line. It is called Linear Pair Axiom.

Theorem 1

If two lines intersect each other, then the vertically opposite angles are equal.

Solution: Given: Two lines AB and CD intersect each other at O.

Opposite Angles

To Prove:

Ray OA stands on line CD.
Hence, ∠AOC + ∠AOD = 180° ………….equation (i) [Linear pair axiom]
Again ray OD stands on line AB
Hence, ∠AOD + ∠BOD = 180° ……………equation (ii)

From equation (i) and (ii)
∠AOC + ∠AOD = ∠AOD + ∠BOD
Or, ∠AOC + ∠AOD - ∠AOD = ∠BOD
Or, ∠AOC = ∠BOD

Now, again;
Ray OB stands on line CD
So, ∠BOC + ∠BOD = 180° …………..equation (iii) (Linear pair axiom)
Again ray OD stands on line AB
So, ∠AOD + ∠BOD = 180° …………..equation (iv)

From equation (iii) and (iv);
∠BOC + ∠BOD = ∠AOD + ∠BOD
Or, ∠BOC + ∠BOD - ∠BOD = ∠AOD
Or, ∠BOC = ∠AOD Proved