Parallel Lines and a Transversal

Parallel Lines and a Transversal

Recall 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 ∠ 1, ∠ 2, . . ., ∠8 as shown in Fig. 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, you 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
(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 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 : ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8 and ∠3 = ∠7. From this, you 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 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:
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, transversal PS intersects parallel lines AB and CD at points Q and R respectively.

Is ∠ BQR = ∠ QRC and ∠ AQR = ∠ QRD? You know that ∠ PQA = ∠ QRC            (1)
                     (Corresponding angles axiom)
Is       ∠ PQA = ∠ BQR? Yes! (Why?)             (2)
So, from (1) and (2), you may conclude that
               ∠ BQR = ∠ QRC.
Similarly,    ∠ AQR = ∠ QRD.
This result can be stated as a theorem given below:
Theorem: 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

∠ BQR = ∠ QRC.
Is AB || CD?
∠ BQR = ∠ PQA   (Why?)       (1)
But,   ∠ BQR = ∠ QRC  (Given)       (2)
So, from (1) and (2), you 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 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, you can obtain the following two theorems related to interior angles 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.
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.

Lines Parallel to the Same Line

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, ∠ 1 = ∠ 2 and ∠ 1 = ∠ 3
                (Corresponding angles axiom)
So,    ∠ 2 = ∠ 3 (Why?)
But ∠ 2 and ∠ 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:
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.

Angle Sum Property of a Triangle

In the earlier classes, you have studied through activities that the sum of all the angles of a triangle is 180°. We can prove this statement using the axioms and theorems related to parallel lines.
Theorem 6.7: The sum of the angles of a triangle is 180º.
Proof: Let us see what is given in the statement above, that is, the hypothesis and what we need to prove. We are given a triangle PQR and ∠ 1, ∠ 2 and ∠ 3 are the angles of Δ PQR (see Fig.)

We need to prove that ∠ 1 + ∠ 2 + ∠ 3 = 180°. Let us  draw a line XPY parallel to QR through the opposite vertex P, as shown in Fig. 6.35, so that we can use the properties related to parallel lines.
Now, XPY is a line.
Therefore,   ∠ 4 + ∠ 1 + ∠ 5 = 180°   (1)
But XPY || QR and PQ, PR are transversal.
So,    ∠ 4 = ∠ 2     and    ∠ 5 = ∠ 3
            (Pairs of alternate angles)

Substituting ∠ 4 and ∠ 5 in (1), we get
                                       ∠ 2 + ∠ 1 + ∠ 3 = 180°
            That is,      ∠ 1 + ∠ 2 + ∠ 3 = 180° 
Recall that you have studied about the formation of an exterior angle of a triangle in the earlier classes (see Fig. 6.36). Side QR is produced to point S, ∠ PRS is called an exterior angle of ΔPQR.
 Is     ∠ 3 + ∠ 4 = 180°? (Why?)        (1)
Also, see that
∠ 1 + ∠ 2 + ∠ 3 = 180° (Why?)         (2)
From (1) and (2), you can see that
∠ 4 = ∠ 1 + ∠ 2.
This result can be stated in the form of a theorem as given below:

Theorem: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
It is obvious from the above theorem that an exterior angle of a triangle is greater than either of its interior apposite angles.
Now, let us take some examples based on the above theorems.