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09M0601Consider the following figure.

1

4

2

3

6 5

87

l

t

m

Q

P

The figure shows a transversal t, which intersects the lines l and m at points P and Q respectively. In this way, fourangles are formed at each of the points P and Q. Let us name these angles as 1, 2 … 8 as shown in the figure.

Let us consider some pairs of angles 2 and 6, 1 and 5, 3 and 7, and 4 and 8.

These pairs of angles have been given a special name and are called corresponding angles.According to the Corresponding Angles Axiom:

When a transversal crosses two parallel lines, each pair of corresponding angles is equal.

The following figure shows a pair of parallel lines l and m, which are intersected by the transversal t.

6 5

87

2 1

43

l

t

m

Q

P

Thus, in this case, each pair of corresponding angles is equal.2 = 6

1 = 53 = 74 = 8

In fact, the converse of the corresponding angles axiom is also true, which states that:

If a pair of corresponding angles is equal, then the lines are parallel to each other.

Corresponding Angles Axiom

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The following figure shows four lines t, m, n, and p. Here, we have to tell which pair of lines is parallel.

m

t

n p

85º

95º95º

In the figure, we observe that the corresponding angles formed at the points, where transversal m intersects lines n andp, are equal in magnitude (95°). Since the corresponding angles formed by transversal m with lines n and p are equal,lines n and p are parallel.

Using the corresponding angles axiom and its converse, we can prove many properties of angles that are formed whena transversal intersects two parallel lines.

Let us solve some more examples to understand the use of corresponding angles axiom.

Example 1:In the given figure, AB and CD are parallel lines, which are intersected by transversal EF at points X and Y.If AXE = 100 , then find CYX.

100º

E

BA

C D

F

X

Y

Solution:Here, AXE and CYX are corresponding angles. Now using corresponding angles axiom, we obtain:

CYX = AXE = 100Thus, the measure of CYX is 100 .

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Example 2:Check whether the lines l and m are parallel or not.

l

m

130°

100°

t

Solution:In the given figure, the corresponding angles formed by lines l and m are not equal. Hence, the lines l and m are notparallel.

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Alternate Interior Angles Axiom

When a transversal intersects two lines, four angles are formed at each of the point of intersection as shown below.

1

4

2

3

6 5

87

l

t

m

Let us name these angles as 1, 2 … 8.Consider the following pairs of angles:

1 and 7, 2 and 8, 3 and 5, and 4 and 6The first two pairs of angles are called exterior alternate angles and the last two pairs are called interior alternateangles.

Is there any relation between the alternate interior angles, when the lines l and m are parallel?Consider the figure in which lines l and m are parallel to each other.

6 5

8Q

2 1

43

l

t

m

7

P

Here, each pair of alternate angles is equal, i.e., 1 = 7, 2 = 8, 3 = 5, and 4 = 6.

This relation between the alternate angles is known as alternate interior angles axiom and can be stated as follows:If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

The converse of this axiom is also true, which states that:If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two linesare parallel.

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Let us see some examples to understand the use of alternate angles axiom.

Example 1:In which of the following figures, are the lines l and m parallel to each other?

l

m

t

120º

120º

l

m

t

80º

100º

(i) (ii)

Solution:(i) In this figure, the alternate interior angles are not equal. Therefore, l is not parallel to m.

(ii) Here, the exterior alternate angles are equal. Thus, the alternate interior angles will also be equal. Therefore, thelines l and m are parallel to each other.

Example 2:In the given figure, AB is parallel to CD and CD is parallel to EF. It is given that ABD = 120 . Show thatAB is parallel to EF.

AB

CD

E F

120º

Solution:It is given that ABD = 120 .AB is parallel to CD.

BDC = ABD (Pair of alternate interior angles)BDC = 120 … (1)

Also, CD is parallel to EF.BFE = BDC (Pair of corresponding angles)BFE = 120 … (2)

Now, ABD and BFE form a pair of alternate interior angles with respect to lines AB and EF.

Thus, using the converse of alternate interior angles axiom, we obtain AB || EF.

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Example 3:In the following figure, AB and CD are parallel to each other and EF is a transversal which intersects AB andCD at points P and Q respectively. If APE = 110 , then find all the angles formed at points P and Q.

1

4 3

5 6

2

78Q

P

E

F

A

C

B

D

Solution:In the given figure:

3 = 1 = 110 (Vertically opposite angles)5 = 3 = 110 (Alternate interior angles)7 = 5 = 110 (Vertically opposite angles)

Here, 1 and 2 form a linear pair.1 + 2 = 180

110 + 2 = 180 ( 1 = 110 )2 = 70

Now, 4 = 2 = 70 (Vertically opposite angles)6 = 4 = 70 (Alternate interior angles)8 = 6 = 70 (Vertically opposite angles)

1 = 3 = 5 = 7 = 110 and 2 = 4 = 6 = 8 = 70

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