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1.6 Transversal across two parallel lines

Corresponding Angles : If a transversal cuts two parallel lines the corresponding angles are equal ( figure 1.26 )

Figure 1.26

l & m are two parallel lines cut by a transversal n to form angles 1 to 8.

Axiom : Corresponding angles are equal in measure if a transversal cuts parallel lines.

m Ð 1 = m Ð 5

m Ð 2 = m Ð 6

m Ð 3 = m Ð 7

m Ð 4 = m Ð 8 .

Alternate interior angles : If a transversal cuts two parallel lines the alternate interior angles are equal in measure. In figure 1.26 m Ð 4 = m Ð 6 and m Ð 3 = m Ð 5. This can be proved as follows :

m Ð 3 = m Ð 4 are supplementary and so also

m Ð 6 = m Ð 7 are supplementary.

Since the sums of their measures are 1800 in both cases.

m Ð 3 + m Ð 4 = m Ð 6 + m Ð 7.

However m Ð 3 = m Ð 7 as they are corresponding angles formed by a transversal across parallel lines.

Therefore, m Ð 3 + m Ð 4 = m Ð 6 + m Ð 3 i.e. m Ð 4 = m Ð 6.

Similarly it can be shown that m Ð 3 = m Ð 5 .

Alternate exterior angles : If a transversal cuts two parallel lines the alternate exterior angles are equal in measure. In figure 1.26 m Ð 1= m Ð 7 and m Ð 2 = m Ð 8. This can be easily shown as follows :

m Ð 1 = m Ð 3 (vertical angles)

m Ð 3 = m Ð 7 (corresponding angles on parallel lines)

\ mÐ 1 = m Ð 7

Consecutive Interior and Consecutive Exterior Angles : If a transversal cuts two parallel lines the consecutive interior angles and consecutive exterior angles are supplementary. In figure 1.26 m Ð 4 and m Ð 5 are supplementary and also m Ð 3 and m Ð 6 are supplementary. This is proven as :

m Ð 1 and m Ð 4 are supplementary

but m Ð 1 = m Ð 5 (corresponding angles on parallel lines )

\ m Ð 4 and m Ð 5 are supplementary.

Since consecutive exterior angles are supplementary. m Ð 1 and m Ð 8 so also m Ð 2 and m Ð 7 should be supplementary.

Proof : m Ð 1 and m Ð 4 are supplementary but

m Ð 4 = m Ð 8 (corresponding angles on parallel lines )

\ m Ð 1 and m Ð 8 are supplementary.

 

Index

Introduction

1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles made by a Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions for Parallelism

Chapter 2

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