09M0601
Corresponding Angles Axiom
Consider the following figure. t l 1 2
P 4
3
5
6 Q
m 8
7
The figure shows a transversal t, which intersects the lines l and m at points P and Q respectively. In this way, four angles 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. t
2
1
l 4
3 P
5
6 Q
7
m 8
Thus, in this case, each pair of corresponding angles is equal. ? 2 = 6 1 = 5 3 = 7 4 = 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. Lines And Angles
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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.
t
85º
m
95º
95º
n
p
In the figure, we observe that the corresponding angles formed at the points, where transversal m intersects lines n and p, 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 when a 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 = 100q, then find CYX.
E
A
100º
B X
Y
C
D
F
Solution: Here, AXE and CYX are corresponding angles. Now using corresponding angles axiom, we obtain: CYX = AXE = 100q Thus, the measure of CYX is 100q.
Lines And Angles
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Example 2: Check whether the lines l and m are parallel or not. t
l 130째
m
100째
Solution: In the given figure, the corresponding angles formed by lines l and m are not equal. Hence, the lines l and m are not parallel.
Lines And Angles
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Alternate Interior Angles Axiom
09M0602
When a transversal intersects two lines, four angles are formed at each of the point of intersection as shown below. t l 1 2 4
3
5
6
m 8
7
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 6 The first two pairs of angles are called exterior alternate angles and the last two pairs are called interior alternate angles. 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. t
2
1
l 4
3 P
5
6 7
Q
m 8
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 lines are parallel.
Lines And Angles
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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? t
t
l 120º
l
100º
80º
m
m
120º
(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, the lines 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 = 120q. Show that AB is parallel to EF. B
A 120º
C D
E
F
Solution: It is given that ABD = 120q. AB is parallel to CD. ? BDC = ABD (Pair of alternate interior angles) BDC = 120q … (1) Also, CD is parallel to EF. ? BFE = BDC (Pair of corresponding angles) BFE = 120q … (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 and CD at points P and Q respectively. If APE = 110q, then find all the angles formed at points P and Q. E 1
A 4
P 6
5
C
2
8
B
3 D
7
Q F
Solution: In the given figure: 3 = 1 = 110q 5 = 3 = 110q 7 = 5 = 110q
(Vertically opposite angles) (Alternate interior angles) (Vertically opposite angles)
Here, 1 and 2 form a linear pair. ? 1 + 2 = 180q 110q + 2 = 180q ( 1 = 110q) 2 = 70q Now, 4 = 2 = 70q 6 = 4 = 70q 8 = 6 = 70q
(Vertically opposite angles) (Alternate interior angles) (Vertically opposite angles)
? 1 = 3 = 5 = 7 = 110q and 2 = 4 = 6 = 8 = 70q
Lines And Angles
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