`color{red} ♦`Parallel Lines and a Transversal

`color{red} ♦`Lines Parallel to the Same Line

`color{red} ♦`Lines Parallel to the Same Line

As we know that a line which intersects two or more lines at distinct points is called a `"transversal"` (see Fig. 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 `angle 1, angle 2, . . ., angle 8` as shown in Fig. 6.18.

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

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

(a) Corresponding angles :

(i) `angle 1` and `angle 5` (ii) `angle 2` and `angle 6`

(iii) `angle 4` and `angle 8` (iv) `angle 3` and `angle 7`

(b) Alternate interior angles :

(i) `angle 4 `and `angle 6` (ii) `angle 3 ` and `angle 5`

(c) Alternate exterior angles:

(i) `angle 1` and `angle 7` (ii) `angle 2 ` and `angle 8`

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

(i) `angle 4` and `angle 5` (ii) `angle 3` and `angle 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 relation between the angles in these pairs when line m is parallel to line n. You 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 Fig. 6.19.

Now, measure any pair of corresponding angles and find out the relation between them. You may find that : `angle 1 = angle 5, angle 2 = angle 6, angle 4 = angle 8` and `angle 3 = angle 7`. From this, you may conclude the following axiom.

`color{blue} text(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 `angle ABQ` and `angle BCS ` equal to each other as shown in Fig. 6.20 (i).

Produce QB and SC on the other side of AD to form two lines PQ and RS [see Fig. 6.20 (ii)]. You may observe that the two lines do not intersect each other.

You may also draw common perpendiculars to the two lines PQ and RS at different points and measure their lengths.

You will find it the same everywhere. So, you may conclude that the lines are parallel. Therefore, the converse of corresponding angles axiom is also true. So, we have the following axiom:

`color{blue} text(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 Fig. 6.21, transveral PS intersects parallel lines AB and CD at points Q and R respectively.

Is `angle BQR = angle QRC` and `angle AQR = angle QRD` ?

You know that `angle PQA = angle QRC` (1)

(Corresponding angles axiom)

Is `angle PQA = angle BQR` ? Yes! (Why ?) (2)

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

`angle BQR = angle QRC`.

Similarly, `angle AQR = angle QRD.`

This result can be stated as a theorem given below:

`color {blue} text(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 the two lines parallel if a pair of alternate interior angles is equal? In Fig. 6.22, the transversal PS intersects lines AB and CD at points Q and R respectively such that `angle BQR = angle QRC`.

Is `AB || CD`?

`angle BQR = angle PQA` (1)

But, `angle BQR = angle QRC` (Given) (2)

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

`angle PQA = angle QRC`

But they are corresponding angles.

So, `AB || CD` (Converse of corresponding angles axiom)

This result can be stated as a theorem given below:

` color{blue} text(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, you can obtain the following two theorems related to interior angles on the same side of the transversal.

` color{blue} text(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.

` color{blue} text(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.

You may recall that you have verified all the above axioms and theorems in earlier classes through activities. You may repeat those activities here also.

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 `angle 1, angle 2, . . ., angle 8` as shown in Fig. 6.18.

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

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

(a) Corresponding angles :

(i) `angle 1` and `angle 5` (ii) `angle 2` and `angle 6`

(iii) `angle 4` and `angle 8` (iv) `angle 3` and `angle 7`

(b) Alternate interior angles :

(i) `angle 4 `and `angle 6` (ii) `angle 3 ` and `angle 5`

(c) Alternate exterior angles:

(i) `angle 1` and `angle 7` (ii) `angle 2 ` and `angle 8`

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

(i) `angle 4` and `angle 5` (ii) `angle 3` and `angle 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 relation between the angles in these pairs when line m is parallel to line n. You 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 Fig. 6.19.

Now, measure any pair of corresponding angles and find out the relation between them. You may find that : `angle 1 = angle 5, angle 2 = angle 6, angle 4 = angle 8` and `angle 3 = angle 7`. From this, you may conclude the following axiom.

`color{blue} text(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 `angle ABQ` and `angle BCS ` equal to each other as shown in Fig. 6.20 (i).

Produce QB and SC on the other side of AD to form two lines PQ and RS [see Fig. 6.20 (ii)]. You may observe that the two lines do not intersect each other.

You may also draw common perpendiculars to the two lines PQ and RS at different points and measure their lengths.

You will find it the same everywhere. So, you may conclude that the lines are parallel. Therefore, the converse of corresponding angles axiom is also true. So, we have the following axiom:

`color{blue} text(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 Fig. 6.21, transveral PS intersects parallel lines AB and CD at points Q and R respectively.

Is `angle BQR = angle QRC` and `angle AQR = angle QRD` ?

You know that `angle PQA = angle QRC` (1)

(Corresponding angles axiom)

Is `angle PQA = angle BQR` ? Yes! (Why ?) (2)

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

`angle BQR = angle QRC`.

Similarly, `angle AQR = angle QRD.`

This result can be stated as a theorem given below:

`color {blue} text(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 the two lines parallel if a pair of alternate interior angles is equal? In Fig. 6.22, the transversal PS intersects lines AB and CD at points Q and R respectively such that `angle BQR = angle QRC`.

Is `AB || CD`?

`angle BQR = angle PQA` (1)

But, `angle BQR = angle QRC` (Given) (2)

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

`angle PQA = angle QRC`

But they are corresponding angles.

So, `AB || CD` (Converse of corresponding angles axiom)

This result can be stated as a theorem given below:

` color{blue} text(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, you can obtain the following two theorems related to interior angles on the same side of the transversal.

` color{blue} text(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.

` color{blue} text(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.

You may recall that you have verified all the above axioms and theorems in earlier classes through activities. You may repeat those activities here also.

If two lines are parallel to the same line, will they be parallel to each other? Let us check it.

See Fig. 6.23 in which line `m | |` line `l` and line `n | |` line `l`.

Let us draw a line `t` transversal for the lines, `l, m` and `n`. It is given that line `m | |` line `l` and line `n | |` line `l`.

Therefore, `angle 1 = angle 2` and `angle 1 = angle 3`

(Corresponding angles axiom)

So, `angle 2 = angle 3`

But `angle 2` and `angle 3` are corresponding angles and they are equal.

Therefore, you can say that

Line `m | |` Line `n`

(Converse of corresponding angles axiom)

This result can be stated in the form of the following theorem:

`color {blue} text( Theorem 6.6) :`

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

Note : The property above can be extended to more than two lines also.

Now, let us solve some examples related to parallel lines.

See Fig. 6.23 in which line `m | |` line `l` and line `n | |` line `l`.

Let us draw a line `t` transversal for the lines, `l, m` and `n`. It is given that line `m | |` line `l` and line `n | |` line `l`.

Therefore, `angle 1 = angle 2` and `angle 1 = angle 3`

(Corresponding angles axiom)

So, `angle 2 = angle 3`

But `angle 2` and `angle 3` are corresponding angles and they are equal.

Therefore, you can say that

Line `m | |` Line `n`

(Converse of corresponding angles axiom)

This result can be stated in the form of the following theorem:

`color {blue} text( Theorem 6.6) :`

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

Note : The property above can be extended to more than two lines also.

Now, let us solve some examples related to parallel lines.

Q 3210467319

In Fig. `6.24`, if `PQ || RS , angle MXQ = 135°` and `angle MYR = 40°`, find `angle XMY`.

Class 9 Chapter 6 Example 4

Class 9 Chapter 6 Example 4

Here, we need to draw a line `AB` parallel to line `PQ`, through point `M` as shown in Fig. `6.25`. Now, `AB || PQ ` and `PQ || RS`.

Therefore, `AB || RS` (Why ?)

Now, `angle QXM + angle XMB = 180°`

(`AB || PQ`, Interior angles on the same side of the transversal `X M`)

But `angle QXM = 135°`

So, `135° + angle XMB = 180°`

Therefore, `angle XMB = 45°` (1)

Now, `angle BMY = angle MYR` (`AB || RS`, Alternate angles)

Therefore, `angle BMY = 40°` (2)

Adding (1) and (2), you get

`angle XMB + angle BMY = 45° + 40°`

That is, `angle XMY = 85°`

Q 3210567410

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

Class 9 Chapter 6 Example 5

Class 9 Chapter 6 Example 5

In Fig. 6.26, a transversal `AD` intersects two lines `PQ` and `RS` at points `B`

and `C` respectively. Ray `BE` is the bisector of `angle ABQ` and ray `CG` is the bisector of

`angle BCS ;` and `BE || CG`.

We are to prove that `PQ || RS`.

It is given that ray `BE` is the bisector of `angle ABQ`.

Therefore, `angle ABE = 1/2 angle ABQ` (1)

Similarly, ray `CG` is the bisector of `angle BCS`.

Therefore, `angle BCG = 1/2 angle BCS` (2)

But `BE || CG ` and `AD` is the transversal.

Therefore, `angle ABE = angle BCG`

(Corresponding angles axiom) (3)

Substituting (1) and (2) in (3), you get

` 1/2 angle ABQ = 1/2 angle BCS`

That is, `angle ABQ = angle BCS`

But, they are the corresponding angles formed by transversal `AD` with `PQ` and `RS`;

and are equal.

Therefore, `PQ || RS`

(Converse of corresponding angles axiom)

Q 3220567411

In Fig. `6.27, AB || CD` and `CD || EF`. Also `EA bot AB`. If `angle BEF = 55°`, find the values of `x, y` and `z`.

Class 9 Chapter 6 Example 6

Class 9 Chapter 6 Example 6

`y + 55° = 180°`

(Interior angles on the same side of the transversal ED)

Therefore, `y = 180º – 55º = 125º`

Again `x = y`

(`AB || CD`, Corresponding angles axiom)

Therefore `x = 125º`

Now, since `AB || CD` and `CD || EF`, therefore, `AB || EF`.

So, `angle EAB + angle FEA = 180°` (Interior angles on the same

side of the transversal EA)

Therefore, `90° + z + 55° = 180°`

Which gives ` z = 35°`