Transversal

In this chapter, you will learn about transversals, transversal lines and angles, and transversal examples to strengthen your concepts of transversals.

You check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Before getting started, let's hear Jack's story.

Jack was driving from New Delhi to Agra, while Jill was driving on the same road from Agra to New Delhi.

Both roads on the highway that go to and fro are parallel.

Now, imagine these two roads are two parallel lines. Is it possible to calculate the distance between these two lines?

In fact, you can. Parallel lines are cut by a transversal. There are plenty of math transversal examples that we will see in this chapter, but before we get there, let us first understand what transversals are.

parallel lines cut by a transversal | transversal lines and angles

Well, transversal is one of the main connecting links between any two lines (or parallel lines). In this article, we will learn about such transversals and how they help us in finding different properties of the two lines.

Lesson Plan

What are Transversals and Transverse Lines?

For two or more lines, a transversal is any line that intersects two lines at distinct points.

In the following figure, L1 and L2 are two lines that are cut by a transversal L. Here the line L is known as a transversal line. In the parallel and transversal line universe, the transversal line basically connects both parallel lines.

parallel lines cut by a transversal | transversal lines and angles

From the diagram, we can say that 'L is a transversal that cuts the lines L1 and L2, and thus line L is the transversal line.'


Construct a Transversal with Parallel Lines

The construction of a transversal is easy. First, we create two parallel lines.

How to construct a transversal with two parallel lines

Now, we construct the angle (at which we want to create the transversal) on the first line (say \(x^o\)) as shown.

How to construct a transversal with two parallel lines

Further, we extend this constructed angle up to the point it covers both the parallel lines as done here.

How to construct a transversal with two parallel lines

So now here is what we get: Our transversal on the two parallel lines at the desired angle (\(x^o\)).


What Are Transversal Angles?

When two parallel lines are cut by a transversal, a number of angles are formed by these two intersections.

parallel lines cut by a transversal | transversal lines and angles

From the given diagram, if we try to organize the angles on the basis of the relative positions they occupy, we get the following categories of angles :

Corresponding Angles

The following pairs of angles are corresponding angles:

  • \(\angle 1\) and \(\angle 5\)
  • \(\angle 2\) and \(\angle 6\)
  • \(\angle 3\) and \(\angle 7\)
  • \(\angle4\) and \(\angle 8\)

Alternate Interior Angles

The following pairs of angles are alternate interior angles:

  • \(\angle 3\) and \(\angle 6\)
  • \(\angle4\) and \(\angle 5\)

Alternate Exterior Angles

The following pairs of angles are alternate exterior angles:

  • \(\angle 1\) and \(\angle 8\)
  • \(\angle 2\) and \(\angle 7\)

Co-interior Angles

The following pairs of angles are co-interior angles:

  • \(\angle 3\) and \(\angle 5\)
  • \(\angle4\) and \(\angle 6\)
 
important notes to remember
Important Notes
  1. A transversal is any line that intersects two lines at distinct points.
  2. A transversal that cuts the lines L1 and L2, is the transversal line.
  3. A number of angles are formed by intersection caused by transversals. These are :

    a) Corresponding Angles
    b) Alternate Interior Angles
    c) Alternate Exterior Angles
    d) Co-interior Angles

Solved Examples

Let us now have a look at some solved examples on transversals.

Example 1

 

 

Consider the following figure, in which L1 and L2 are parallel lines:

Alternate interior angles - straight lines intersecting by transversal - solved examples on transversals

What is the value of \(\angle C\)?

Solution

Through C, draw a line parallel to L1 and L2, as shown below:

Co-interior angles - straight lines intersecting by transversal - solved examples on transversals

We have:

\(\angle x\) = \(\angle \beta \) = 600 (alternate interior angles)

\(\angle y\) = 1800 – 1200 (co-interior angles)

\(\angle y\) = 600

Thus, we get

\(\angle C\) = \(\angle x\) + \(\angle y\) = 1200

\(\therefore \angle C\) = 1200
Example 2

 

 

Consider the following figure, in which L1, L2, and L3 are parallel lines. The line AC is perpendicular to L3, as shown:

Three parallel straight lines intersecting by transversal - solved examples on transversals

What are the values of \(\angle x\), \(\angle y\) and \(\angle z\)?

Solution

Since AC ⊥ L3, it will be perpendicular to each of the other two parallels as well. This means that:

\(\angle x\) = 900 – 600 = 300

Now,

\(\angle y\) = 1800 – \(\angle x\) (co-interior angles)

\(\angle y\) = 1500

\(\angle z\) = \(\angle y\) = 1500 (corresponding angles)

\(\therefore \angle x\) = 30& \(\angle z\) = \(\angle y\) = 1500
Example 3

 

 

Consider the following figure, in which AX \(\parallel \) CY. What is the value of \(\angle B\)?

straight lines intersecting by transversal - solved examples on transversals

Solution

Through A and C, drawn two lines perpendicular to AX and CY. Through B, draw a line parallel to these perpendiculars, as shown (the three dotted lines are parallel):

Three straight lines intersecting by transversal - solved examples on transversals

Note how we have marked the various angles. Clearly,

\(\angle x\) = 200 (alternate interior angles)

\(\angle y\) = 500 (alternate interior angles)

Thus,

\(\angle B\) or \(\angle ABC\) = \(\angle x\) + \(\angle y\) = 700

\(\therefore \angle B\) = 700
Example 4

 

 

Three parallel lines are intersected by a transversal, as shown below:

Parallel lines intersected by transeversal - straight lines intersecting by transversal - solved examples on transversals

If \(\angle 1\):\(\angle 2\) = 2:1, what is the value of \(\angle 3\)?

Solution

Consider the following figure:

Parallel lines and corresponding angles - straight lines intersecting by transversal - solved examples on transversals

Clearly, \(\angle 1\) = \(\angle4\), because these are corresponding angles. Now,

\(\angle 1\):\(\angle 2\) = 2:1 \\ \(\therefore\) \(\angle4\):\(\angle 2\) = 2:1

Also,

\(\angle4\) + \(\angle 2\) = 1800, (linear pair)

Using these two facts, it is easy to conclude that:

\(\angle4\) = 1200, \(\angle 1\) = 600

Thus,

\(\angle 1\) = \(\angle4\) = 1200

And hence we get

\(\angle 3\) = \(\angle 1\) = 1200 (alternate interior angles)

\(\therefore \angle 3\) = 1200
Example 5

 

 

Consider the following figure, in which L1 \(\parallel \) L2:

straight lines intersecting by transversal - solved examples on transversals

The angle bisectors of \(\angle BAX\) and \(\angle ABY\) intersect at C, as shown. Find the value of \(\angle ACB\).

Solution

We note that \(\angle BAX\) and \(\angle ABY\) are co-interior angles, which means that their sum is 1800:

\(\angle BAX\) + \(\angle ABY\) = 1800

2(\(\angle 1\) + \(\angle 2\)) = 1800

\(\angle 1\) + \(\angle 2\) = 900

Now, in ΔACB, we apply the angle sum property:

\(\angle 1\) + \(\angle 2\) + \(\angle ACB\) = 1800

900 + \(\angle ACB\) = 1800

\(\angle ACB\) = 900

\(\therefore \angle ACB\) = 900
 
Challenge your math skills
Challenging Questions

Two parallel lines are cut by two different transversals at distinguished points. All the corresponding angles formed by these four intersections are found to be equal. Can you comment upon the figure formed by these four lines?

Hint: Opposite angles of the formed quadrilateral are found to be equal in the given scenario.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the 'Check Answer' button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about Transversal with the solved examples and practice questions. Now you will be able to easily solve problems on transversal examples, parallel lines cut by a transversal, transversal lines and angles, transversal and parallel lines.

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Frequently Asked Questions (FAQs)

1. Can transversal lines be straight?

Yes, in fact, transversal lines have to be straight lines that stretch throughout the given parallel lines.

2. How many angles are formed by the transversal?

There are a number of angles formed by the transversal. These are
a) Corresponding Angles
b) Alternate Interior Angles
c) Alternate Exterior Angles
d) Co-interior Angles

3. Do transversal lines have to be parallel?

No. Parallel and transversal lines are different. Transversal lines can be at any angle to the given parallel lines. Thus, any 2 transversals need not be parallel.

  
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