When two lines are cut by a transversal angles on the same side of the transversal and inside the lines?

In Geometry, we might have come across different types of lines, such as parallel lines, perpendicular lines, intersecting lines, and so on. Apart from that, we have another line called a transversal. This can be observed when a road crosses two or more roads, or a railway line crosses several other lines. These give a basic idea of a transversal. Transversals play an important role in establishing whether two or more other lines in the Euclidean plane are parallel. In this article, you will learn the definition of transversal line, angles made by the transversal with parallel and non-parallel lines with an example. Also, learn more concepts of geometry here.

Table of Contents:

Transversal Meaning

A transversal is defined as a line that passes through two lines in the same plane at two distinct points in the geometry. A transversal intersection with two lines produces various types of angles in pairs, such as consecutive interior angles, corresponding angles and alternate angles. A transversal produces 8 angles and this can be observed from the figure given below:

Transversal Line Definition

A line that intersects two or more lines at distinct points is called a transversal line.

Transversal Lines

A transversal line can be obtained by intersecting two or more lines in a plane that may be parallel or non-parallel.

In the above figure, line “t” is the transversal of two non-parallel lines a and b.

Also, we can draw two transverse lines for two parallel lines and non-perpendicular lines.

Transversal and Parallel Lines

In the below-given figure, the line RS represents the Transversal of Parallel Lines EF and GH.

Transversal Angles

The angles made by a transversal can be categorized into several types such as interior angles, exterior angles, pairs of corresponding angles, pairs of alternate interior angles, pairs of alternate exterior angles, and the pairs of interior angles on the same side of the transversal. All these angles can be identified in both cases, which means parallel and non-parallel lines.

Transversal Lines and Angles

Let’s understand the angles made by transversal lines with other lines from the below table.

Interior angles	∠3, ∠4, ∠5, ∠6	Interior angles	∠3, ∠4, ∠5, ∠6
Exterior angles	∠1, ∠2, ∠7, ∠8	Exterior angles	∠1, ∠2, ∠7, ∠8
Pairs of corresponding angles	∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8	Pairs of corresponding angles	∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
Pairs of alternate interior angles	∠3 and ∠6, ∠4 and ∠5	Pairs of alternate interior angles	∠3 and ∠6, ∠4 and ∠5
Pairs of alternate exterior angles	∠1 and ∠8, ∠2 and ∠7	Pairs of alternate exterior angles	∠1 and ∠8, ∠2 and ∠7
Pairs of interior angles on the same side of the transversal	∠3 and ∠5, ∠4 and ∠6	Pairs of interior angles on the same side of the transversal	∠3 and ∠5, ∠4 and ∠6

From the above table, we can say that the angles associated with the transversal will be the same in pairs in parallel and non-parallel lines.

Click here to learn more about lines and angles.

Transversal Properties

Some of the properties of transversal lines with respect to the parallel lines are listed below.

• If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.
  
  Here,

LM is the transversal made by the parallel lines PQ and RS such that:

The pair of corresponding angles that are represented with the same letters are equal.

• If two parallel lines are cut by a transversal, each pair of alternate interior angles are equal.
  
  Here, ∠A = ∠D and ∠B = ∠C
• If two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary, i.e. they add up to 180 degrees.
  
  This property can also be written for a single transversal line.

Here,

∠3 + ∠5 = 180° and ∠4 + ∠6 = 180°

Transversal Example

Let’s have a look at the solved example given below:

Question: In the given figure, AB and CD are parallel lines intersected by a transversal PQ at L and M, respectively. If ∠CMQ = 60°, find all other angles.

Solution:

A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles.

Thus, the corresponding angles are equal.

∠ALM = ∠CMQ = 60° {given}

We know that vertically opposite angles are equal.

∠LMD = ∠CMQ = 60° {given}

And

∠ALM = ∠PLB = 60°

Here, ∠CMQ + ∠QMD = 180° are the linear pair

On rearranging, we get,

∠QMD = 180° – 60° = 120°

Also, the corresponding angles are equal.

∠QMD = ∠MLB = 120°

Now,

∠QMD = ∠CML = 120° {vertically opposite angles}

∠MLB = ∠ALP = 120° {vertically opposite angles}