# Transversals

A line which intersects two or more lines at distinct points are known as Transversals.

## Examples of Transversal line

Example 01
Transversal T intersecting two lines

In the above image you can observe;
L & M are the two non intersecting lines.
T is a transversal intersecting the lines M & N at points a & b

Example 02
Transversal T intersecting three lines

In the above figure you will observe;
L, M & N are three lines intersected by transversal T at three distinct points a, b & c

Example 03
Examples of Non-Transversal lines

In the above example, T is not a transversal because it intersects the lines M & N at the same point O.
To qualify as a transversal, the intersection should be at distinct points.

In the above example, T is not a transversal as it intersect only one line.
To qualify as transversal, it should intersect atleast two points.

## Angles made by transversal

Here we will discuss the angle formation when the transversal intersects two points

In the above figure, transversal intersects two line M & N at two distinct points.
Below are some angle properties:

(a) Interior Angles
Angles which lie inside the lines M & N are called Interior angles

Here angles 3, 4, 5 and 6 are the interior angles

(b) Exterior Angle
Angles which are located on the external sides of line M & N are called Exterior Angle

(c) Corresponding angles
Angles with the same matching corners with respect to transversal are called corresponding angles

Below are pairs of corresponding angles:
\angle 1\ and\ \angle 5
\angle 2\ and\ \angle 6
\angle 4\ and\ \angle 8
\angle 3\ and\ \angle 7

(d) Alternate Interior Angle
Alternate Interior angles are located between the two line and opposite side of the transversal

Below are alternate interior angle pairs
\angle 4\ and\ \angle 6
\angle 3\ and\ \angle 5

(e) Alternate Exterior Angle
These angles are positioned externally on the two line and placed opposite side of the transversal

Below are alternate exterior angle pairs
\angle 1\ and\ \angle 7
\angle 2\ and\ \angle 8

## Angle Property when transversal intersect parallel lines

When the two parallel lines M & N are intersected by transversal T, then following angle property is observed.

Corresponding Angles are equal
\angle 1\ =\ \angle 5
\angle 2\ =\ \angle 6
\angle 4\ =\ \angle 8
\angle 3\ =\ \angle 7

Alternate Interior Angles are equal
\angle 3\ =\ \angle 5
\angle 4\ =\ \angle 6

Alternate Exterior Angles are equal
\angle 1\ =\ \angle 7
\angle 2\ =\ \angle 8

Same Side Interior Angles are supplementary
\angle 4\ +\ \angle 5 = 180
\angle 3\ +\ \angle 6 = 180

## Questions on Transversals

(01) Is given line T transversal?

Solution
NO!!
As line T intersects M & N at common point O.

The transversal should intersect lines at distinct points.

(02) In the below figure, is line B transversal?

Solution

Yes, line B is a transversal as it intersects lines L, M & N at distinct points

(03) L & M are parallel lines and T is a transversal.
Study the figure below and find value of ∠a

Solution

Since L & M are parallel lines.
\angle b\ = \angle c\ {Corresponding Angles}
\angle b\ = 70 degree

Since L is a straight line, the sum of angles will be 180 degree
\angle b\ + \angle a\ = 180
70 + \angle b\ = 180
\angle a\ = 110 degree

(04) Two lines M & N are intersected by transversal T.
Find the alternate exterior angle of A

Solution

Alternate Exterior Angle is located at external side of the two lines and opposite end of the transversal

Here angle B is the alternate exterior angle pair of angle A

(05) In the below figure, L & M are two parallel lines intersected by transversal T
Find the value of angle B

Solution

\angle A\ = 106 degree

Angle A & B are same side interior angles.
When lines are parallel these angles are supplementary

\angle A\ + \angle B\ = 180
106 + \angle B\ = 180
\angle B\ = 180 – 106
\angle B\ = 74 degree

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