## 9th Class Maths – Lines and Angles | Maths NCERT Solutions Class 9

**9th Class Maths – Chapter 6 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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