NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

Recall that a line which intersects two or more lines at distinct points is called a transversal (see Figure 6.18).

Figure 6.18

Line l intersects lines m and n at points P and Q respectively. Therefore, line l is a transversal for lines m and n. Observe that four angles are formed at each of the points P and Q.

Let us name these angles as ∠1, ∠2, ... ..., ∠8 as shown in Figure 6.18.

∠1, ∠2, ∠7 and ∠8 are called exterior angles, while ∠3, ∠4, ∠5 and ∠6 are called interior angles.

Recall that in the earlier classes, we have named some pairs of angles formed when a transversal intersects two lines. These are as follows:

(a) Corresponding angles :

(i) ∠1 and ∠5
(ii) ∠2 and ∠6
(iii) ∠4 and ∠8
(iv) ∠3 and ∠7

(b) Alternate interior angles :

(i) ∠4 and ∠6
(ii) ∠3 and ∠5

(c) Alternate exterior angles:

(i) ∠1 and ∠7
(ii) ∠2 and ∠8

(d) Interior angles on the same side of the transversal:

(i) ∠4 and ∠5
(ii) ∠3 and ∠6

Interior angles on the same side of the transversal are also referred to as consecutive interior angles or allied angles or co-interior angles. Further, many a times, we simply use the words alternate angles for alternate interior angles.

Now, let us find out the relationship between the angles in these pairs when line m is parallel to line n. We know that the ruled lines of your notebook are parallel to each other. So, with ruler and pencil, draw two parallel lines along any two of these lines and a transversal to intersect them as shown in Figure 6.19.

Figure 6.19

Now, measure any pair of corresponding angles and find out the relationship between them. We may find that :
∠1 = ∠5,
∠2 = ∠6,
∠4 = ∠8 and
∠3 = ∠7.

From this, we may conclude the following axiom.

Axiom 6.3

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

Axiom 6.3 is also referred to as the corresponding angles axiom.

Now, let us discuss the converse of this axiom which is as follows:

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

Does this statement hold true? It can be verified as follows:

Draw a line AD and mark points B and C on it. At B and C, construct ∠ABQ and ∠BCS equal to each other as shown in Figure 6.20 (i).

Figure 6.20 (i)

Produce QB and SC on the other side of AD to form two lines PQ and RS [see Figure 6.20 (ii)].

Figure 6.20 (ii)

We may observe that the two lines do not intersect each other. We may also draw common perpendiculars to the two lines PQ and RS at different points and measure their lengths. We will find it the same everywhere. So, we may conclude that the lines are parallel. Therefore, the converse of corresponding angles axiom is also true. So, we have the following axiom:

Axiom 6.4

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

Can we use corresponding angles axiom to find out the relation between the alternate interior angles when a transversal intersects two parallel lines? In Figure 6.21, transversal PS intersects parallel lines AB and CD at points Q and R respectively.

Figure 6.21

Is ∠BQR = ∠QRC and ∠AQR = ∠QRD?

We know that ∠PQA = ∠QRC (1) (Corresponding angles axiom)

Is ∠PQA = ∠BQR? Yes! (Why ?) (2)

So, from (1) and (2), we may conclude that

∠BQR = ∠QRC.

Similarly,

∠AQR = ∠QRD.

This result can be stated as a theorem given below:

Theorem 6.2

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

Now, using the converse of the corresponding angles axiom, can we show that the two lines are parallel if a pair of alternate interior angles is equal?

Figure 6.22

In Figure 6.22, the transversal PS intersects lines AB and CD at points Q and R respectively such that ∠BQR = ∠QRC.

Is AB ∥ CD? ( ∥ is a symbol FOR PARALLEL lines. )

∠BQR = ∠PQA (Why?) ... ... (1)

But ∠BQR = ∠QRC (Given) ... ... (2)

So, from (1) and (2), we may conclude that

\[∠PQA\ =\ ∠QRC\]

But they are corresponding angles.

So, AB ∥ CD (Converse of corresponding angles axiom)

This result can be stated as a theorem as given below:

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 a similar way, we can obtain the following two theorems related to interior angle on the same side of the transversal.

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.

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.

We may recall that we have verified all the above axioms and theorems in our earlier classes through activities. We may repeat those activities here again.