## NCERT Solutions for Class 9 Maths Lines and Angles – Chapter 6

**Class 9 Maths Lines and Angles** : In the previous article Lines and Angles, we had discussed parallel lines and transversal. In this article, we will learn about the theorem based on the below axioms.

**Axiom 1: – If a transversal intersects two parallel lines then each pair of corresponding angles is equal.**

**Axiom 2:- Two lines are parallel to each other, if a transversal intersects those two lines such that a pair of corresponding angles is equal.**

Let us now discuss these theorem one by one:-

__Theorem 6.2__

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

<PQA=<QRC (corresponding angle axiom) equation (i)

<PQA=<BQR equation (ii)

From equ (i) and equ (ii) we can conclude

<BQR=<QRC

<AQR=<QRD

So, from the above, we can conclude that

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

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__Theorem 6.3__

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

In the below figure the transversal PS intersects lines AB and CD at points Q and R respectively.

<BQR=<PQA ……As they are vertically opposite angles (1)

<BQR=<QRC …….Given to us (2)

From 1 and 2, we conclude

<PQA=<QRC

As they are corresponding angles

AB is parallel CD (Converse of corresponding angles axiom)

__Theorem 6.4__

If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

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__Theorem 6.5__

If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Theorem 6.6

Lines which are parallel to the same line are parallel to each other

We will draw line* t* transversal for the lines* l,m, *and* n.*

It is already given to us that line *m *is parallel to the line* l* and line *n* is parallel to line* l*

So it implies

<1=<2 and <1=<3 (using the corresponding angles axiom

So <2=<3

As the < 2 and <3 are corresponding angles and are equal.

Hence with above discussion, we can conclude

Line* m* is parallel line *n*.

Hence proved.

Using the Converse of corresponding angles axiom.

The above theorem also states true for more than two lines also.

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We will discuss the angle sum property of a triangle in the next article.

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