# 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.