Transversal Lines

Transversal Lines in geometry is defined as a line that intersects two lines at distinct points in a plane. The transversal line intersecting a pair of parallel lines is responsible for the formation of various types of angles that, include alternate interior angles, corresponding angles, and others. A transversal can intersect at least two lines that can be parallel or non-parallel.

Here, in this article, we learn about Parallel Lines and Transversals, Angle Relationship Between Parallel Lines and Transversals, and others in detail.

What is a Transversal Line?

A transversal line in geometry is a straight line that intersects two or more lines. When a transversal line intersects two lines, it creates pairs of corresponding angles, alternate interior angles, and alternate exterior angles. Understanding transversal lines is very important for the proper knowledge of the geometry. it helps to better grasp other concepts of geometry.

Transversal Lines Definition

The lines that intersect a pair of parallel line is called the transversal line. In the above added figure line 1 and 2 are parallel line and the line line which cut both 1 and 2 is called the transversal line.

Transversal Lines and Parallel Lines

The lines that never meet each other in a plane are called the parallel line and the line that intersect these parallel line is called the transversal line let's learn about them in detail.

• Parallel Lines
• Transversal Lines

Parallel Lines

Parallel lines in Geometry are defined as a pair of lines that never meet each other in the same plane. For example, if we observe a train track the two tracks always run with each other but never intersect, this is an example of the parallel line. The image of parallel lines is added below,

Transversal Lines

Transversal or Transversal Lines is any line that cuts two parallel or nonparallel lines to two different point and in this process, it creates 8 angles that are characterized into different categories that includes,

• Corresponding Angles
• Alternate Interior Angles
• Alternate Exterior Angles
• Vertical Opposite Angles

The image added below shows parallel and transversal lines.

The various Types of Angles that are formed by the intersection of transversal and parallel lines are,

• Corresponding Angles: ∠a and ∠p, ∠d and ∠s, ∠b and ∠q, ∠c and ∠r
• Alternate Interior Angles: ∠d and ∠q, and ∠c and ∠p 
• Alternate Exterior Angles: ∠b and ∠s, and ∠a and ∠r 
• Vertically Opposite Angles: ∠a and ∠c, ∠b and ∠d, ∠p and ∠r, ∠q and ∠s

Transversal Lines and Angles

Various angles that are formed by the intersection of Parallel Lines and Transversal Lines are,

• Corresponding angles
• Alternate Interior Angles
• Alternate Exterior Angles
• Vertical Angles

Let's learn about the same in detail.

Corresponding Angles

The following pairs of angles that are corresponding angles from the figure added above are,

• ∠a = ∠p
• ∠b = ∠q
• ∠d = ∠s
• ∠c = ∠r

Alternate Exterior Angles

The following pairs of angles that are alternate exterior angles from the figure added above are,

• ∠a = ∠r
• ∠b = ∠s

Alternate Interior Angles

The following pairs of angles that are alternate interior angles from the figure added above are,

• ∠d = ∠q
• ∠c = ∠p

Vertically Opposite Angles

The following pairs of angles that are vertically opposite angles from the figure added above are,

• ∠a = ∠c
• ∠b = ∠d
• ∠p = ∠r
• ∠q = ∠s

Constructing a Transversal on Parallel Lines

Constructing a transversal line on a parallel line is very simple. To construct a transversal line between two parallel lines follow the the steps added below,

Step 1: Take a pair of Parallel Lines (say l and m)

Step 2: Draw a line (t) that cuts the first line.

Step 3: Now extend the line and then make it cut the second line (m) such that it also cuts line (m) and then the line so obtained is the transversal line.

Summary of Transversal Lines

The table added below shows all types of angles that are formed by a pair of parallel lines and a transversal.

Read More,

• Properties of Parallel Lines
• Parallel and Intersecting Lines

Transversal Line Examples

Example 1: In Figure, if PQ || RS, ∠ MXQ = 135° and ∠ MYR = 40°, find ∠ XMY.

Solution:

Let's construct a line AB parallel to line PQ, through point M. 

Now, AB || PQ and PQ || RS 

AB || RS || PQ

∠ QXM + ∠ XMB = 180°...(Interior Angles on the Same Side of Transversal are Supplementary)

∠ QXM = 135°...(given)

135° + ∠ XMB = 180°

∠ XMB = 45°

∠ BMY = ∠ MYR...(Alternate Angles)

∠ BMY = 40° 

∠ XMB + ∠ BMY = 45° + 40°

Therefore, ∠ XMY = 85°

Example 2: In Figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠ BEF = 55°, find the values of x, y, and z.

Solution: 

Since,

AB || CD and CD || EF

AB || CD || EF

EB and AE are Transversal

y + 55° = 180°...(Interior Angles on the Same Side of Transversal are Supplementary)

y = 180° – 55° = 125°

x = y (Corresponding Angles)

x = y = 125°

Now, ∠ EAB + ∠ FEA = 180°...(Interior Angles on the Same Side of Transversal are Supplementary)

90° + z + 55° = 180°

Hence, z = 35°

Example 3: In Figure, find the values of x and y and then show that AB || CD.

Solution: 

Here,

x + 50° = 180° (Linear Pair is equal to 180°)

x = 130°

y = 130°  (Vertically Opposite Angles are Equal)

x = y = 130° 

In two parallel lines, the alternate interior angles are equal, and ∠x = ∠y

Hence, this proves that alternate interior angles are equal and so, AB || CD

Example 4: In Figure, if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y.

Solution:

Here, ∠APQ = ∠PQR...(Alternate Interior Angles)

x = 50°

∠APR = ∠PRD...(Alternate Interior Angles)

∠APQ + ∠QPR = 127° 

127° = 50°+ y

y = 77°

Hence, x = 50° and y = 77°