Class 9 Maths

# Lines and Angles

## Parallel Lines And A Transversal

### Axiom 3

If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. Here, Exterior angles are ∠1, ∠2, ∠7 and ∠8
Interior angles are ∠3, ∠4, ∠5 and ∠6
Corresponding angles are ∠
(i) ∠1 and ∠5
(ii) ∠2 and ∠6
(iii) ∠4 and ∠8
(iv) ∠3 and ∠7

### Axiom 4

If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Thus, (i) ∠1 = ∠5, (ii) ∠2 = ∠6, (iii) ∠4 = ∠8 and (iv) ∠3 = ∠7
Alternate Interior Angles: (i) ∠4 and ∠6 and (ii) ∠3 and ∠5
Alternate Exterior Angles: (i) ∠1 and ∠7 and (ii) ∠2 and ∠8
If a transversal intersects two parallel lines then each pair of alternate interior and exterior angles are equal.
Alternate Interior Angles: (i) ∠4 = ∠6 and (ii) ∠3 = ∠5
Alternate Exterior Angles: (i) ∠1 = ∠7 and (ii) ∠2 = ∠8
Interior angles on the same side of the transversal line are called the consecutive interior angles or allied angles or co-interior angles. They are as follows: (i) ∠4 and ∠5, and (ii) ∠3 and ∠6

### Theorem 2

If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Solution: Given: Let PQ and RS are two parallel lines and AB be the transversal which intersects them on L and M respectively.

To Prove: ∠PLM = ∠SML
And ∠LMR = ∠MLQ Proof: ∠PLM = ∠RMB ………….equation (i) (Corresponding ngles)
∠RMB = ∠SML ………….equation (ii) (vertically opposite angles)
From equation (i) and (ii)
∠PLM = ∠SML

Similarly, ∠LMR = ∠ALP ……….equation (iii) (corresponding angles)
∠ALP = ∠MLQ …………equation (iv) (vertically opposite angles)
From equation (iii) and (iv)
∠LMR = ∠MLQ Proved

#### Theorem 3

If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

Solution: Given: - A transversal AB intersects two lines PQ and RS such that
∠PLM = ∠SML

To Prove: PQ ||RS
Use same figure as in Theorem 2.
Proof: ∠PLM = ∠SML ……………equation (i) (Given)
∠SML = ∠RMB …………equation (ii) (vertically opposite angles)
From equations (i) and (ii);
∠PLM = ∠RMB

But these are corresponding angles.
We know that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines ate parallel to each other.
Hence, PQ║RS Proved.