Transversal Definition

A line or line segment cutting across a set of lines is called a transversal. If the lines cut by the transversal are parallel, there are certain relationships between the angles formed by the line segments.

The figure below shows a transversal cutting across a set of parallel lines. At each point of intersection of the transversal and the line there are four angles formed.

A set of parallel lines cut by a transversal

When a transversal cuts a set of two parallel lines then there are three sets of angles formed. They are:

Corresponding angles
Alternate interior angles and
Co-interior angles

These angles are formed whether the lines being cut are parallel or not. If the lines are parallel then there is a relationship between the angles. But if the lines are not parallel then such relationships will not exist.

Properties of angles formed by Transversal cutting parallel lines

If two lines intersect there are four angles formed at the intersection. So if a transversal cuts a set of parallel lines then there will be two intersections and a total of eight angles will be formed.

There are three rules governing the angles formed when a set of two parallel line is cut by a transversal

Rule 1 : The angles formed at the corresponding positions of each intersection are called corresponding angles. Pairs of corresponding angle are equal.

The corresponding angles are marked in similar colors.

Corresponding Angles

The corresponding angles are

Angles 1 and Angle 5

Angles 2 and Angle 6

Angles 3 and Angle 7

Angles 4 and Angle 8

So these pairs of angles are equal.

Converse of Rule 1: If two lines are cut by a transversal and if a pair of corresponding angles is equal, then the lines are parallel

Rule 2: The angles formed in between the lines on either side of the transversal are called alternate interior angles. Pairs of alternate interior angle are equal.

In the diagram below alternate interior angles are marked in the same color

Alternate Interior Angles

The alternate interior angles are

Angles 3 and Angle 5

Angles 4 and Angle 6

Converse of Rule 2: If two lines are cut by a transversal and if a pair of alternate interior angles is equal, then the lines are parallel

Rule 3: The angles formed in between the lines on the same side of the transversal are called co interior angles. Pairs of co-interior angles are supplementary..

In the diagram below co-interior angles are marked in the same color

Co-interior Angles

Co-interior angles are:

Angles 3 and Angle 6

Angles 4 and Angle 5

Converse of Rule 3: If two lines are cut by a transversal and if a pair of co-interior angles is supplementary, then the lines are parallel

To summarize when a transversal cuts a set of two parallel lines:

There are four pairs of corresponding angles formed and they are equal.
There are two pairs of alternate interior angles formed and they are equal.
There are two pairs of co interior angles formed and they are supplementary.

The above rules will be valid only if the lines GH and IJ are parallel. If they are not parallel then none of the rules are valid.

The converse of these rules are also true and they can be extended to the case when there  are more than two parallel lines.

Also note that at each intersection the vertical angles are equal.

These rules will help us to calculate all the eight angles if we know one angle.

Exercise:

Prob 1: Draw the figure of three parallel lines cut by a transversal. Mark all the equal angles and when will all the angles be equal?

Ans :

The required diagram is as below. Equal angles are marked in the same color.

Three parallel lines cut by a transversal

Angles in different colors are supplementary.

All angles are equal means each angle has to be 90o. This happens when the transversal is perpendicular to the parallel lines.

Prob 2: A set of two parallel lines is cut by a transversal. one of the top angles is 500. Can you find all the other angles?

Ans:

Look at the above figure. One top angles is given to be 50o. Let this be angle 1

Angle 2 = 180 – angle 1 being angle in a straight line

Angle 2 = 180 – 50 = 130o

Angle 4 = Angle 2 = 130o

Angle 3= Angle 1 = 50o being Vertical angles

Angle 5 = Angle 1 = 50o being corresponding angles

Angle 6 = Angle 2 = 130o being corresponding angles

Angle 7 = Angle 3 = 50o being corresponding angles

Angle 8 = Angle 4 = 130o being corresponding angles

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Prob 3: Two parallel lines are cut by a transversal and one co-interior angle is 80o greater than the other. Find all the angles?

Ans:

Look at the above figure. One co-interior angles is given to be 80o.. Let this be angle 3

Angle 4 = 180 – angle 3 being angle in a straight line

Angle 4 = 180 – 80 = 100o

Angle 1= Angle 3 = 80o being Vertical angles

Angle 2= Angle 4 = 100o being Vertical angles

Angle 6 = 180 – Angle 3 being co-interior angles

Angle 6 = 180 – 80 = 100o

Angle 5 = 180 – Angle 4 being co-interior angles

Angle 5 = 180 – 100 = 80o

Angle 7 = Angle 5 = 80o being vertical angles

Angle 8 = Angle 6 = 100o being vertical angles