Parallel Lines and a Transversal
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.
