A transversal is a line in a plane that intersects two (or more) other lines, which are on the same plane, at distinct points.
Transversal Angles
In the figure, the line t is intersecting two coplanar lines $l_1$ and $l_2$. The line t is called the transversal of the lines $l_1$ and $l_2$. Here, the lines $l_1$ and $l_2$ may or may not be parallel.

Transversal angles are the angles formed by a transversal and the lines cut by the transversal. If a transversal intersects two coplanar lines, it forms four angles at the intersection of each line and the transversal, as shown in the figure. These eight angles are called the transversal angles.
The transversal angles can be categorized into the following, based on their position.
  • Interior Angles: Interior angles are the angles that are formed by two lines cut by a transversal and are between the two lines. In the above figure, the angles $\angle$3, $\angle$4, $\angle$5, $\angle$6 are the interior angles.
  • Exterior Angles: Exterior angles are the angles that are formed by two lines cut by a transversal and are outside of the lines. In the above figure, the angles$\angle$1, $\angle$2, $\angle$7, and $\angle$8 are the exterior angles.
The pairs of angles formed by two lines cut by a transversal are categorized as follows:
  • Corresponding angles: Corresponding angles are pairs of angles that are formed by two lines cut by a transversal such that they are on the same side of the transversal and on the same side of the lines. That is, the corresponding angles are the transversal angles that line in the same relative positions.
In the above figure, the corresponding angles (must be in pairs) are

$\angle$1 and $\angle$5 above left
$\angle$3 and $\angle$7 below left
$\angle$2 and $\angle$6 above right
$\angle$4 and $\angle$8 below right

  • Alternate angles: Alternative angles are the angles that lie on the opposite sides of the transversal. The alternative angles can be categorized into:
Alternate Interior Angles: Alternate interior angles are pairs of interior angles that have different vertices and lie on opposite sides of the transversal.

In the above figure, the alternate interior angles are:

$\angle$3 and $\angle$6
$\angle$4 and $\angle$5

Alternate Exterior Angles: Alternate exterior angles are pairs of exterior angles that have different vertices and lie on opposite sides of the transversal.

In the above figure, the alternate exterior angles are

$\angle$1 and $\angle$8
$\angle$2 and $\angle$7

  • Consecutive Interior Angles: Consecutive Interior Angles are pairs of interior angles that are on the same side of the transversal.
In the above figure, the consecutive interior angles are

$\angle$3 and $\angle$5
$\angle$4 and $\angle$6

If two lines cut by the transversal t are parallel, that is, if $l_1$ || $l_2$, we can relate the measures of the transversal angles as mentioned hereunder:

Angles formed by Transversals

1. If two parallel lines are cut by a transversal, then the corresponding angles are congruent. That is, in the figure, we have
$\angle$1$\cong$$\angle$5
$\angle$3$\cong$$\angle$7
$\angle$2$\cong$$\angle$6
$\angle$4$\cong$$\angle$8

2. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. That is, in the figure, we have

$\angle$3$\cong$$\angle$6
$\angle$4$\cong$$\angle$5

3. If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. That is, in the figure, we have

$\angle$1$\cong$$\angle$8
$\angle$2$\cong$$\angle$7

4. If two parallel lines are cut by a transversal, then the consecutive interior angles are (the interior angles on the same side of the transversal) are supplementary. That is, from the above figure, we have
m $\angle$3+m $\angle$5=$\angle$$180^o$
m $\angle$4+m $\angle$6=$\angle$$180^o$

5. If two lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary. That is, from the above figure, we have

m $\angle$1+m $\angle$7=$\angle$$180^o$
m $\angle$2+m $\angle$8=$\angle$$180^o$

6. From the above discussion, if $l_1$$\cong$$l_2$, from the above figure, we can have

$\angle$1$\cong$$\angle$4$\cong$$\angle$5$\cong$$\angle$8
$\angle$2$\cong$$\angle$3$\cong$$\angle$6$\cong$$\angle$7

Conversely, the two lines cut by the transversal are parallel, if any of the following relationships hold between the pairs of transversal angles:

1. If the corresponding angles are congruent, then the lines $l_1$ and $l_2$ cut by the transversal are parallel. That is, $l_1$$\cong$$l_2$, if any of the following holds:

$\angle$1$\cong$$\angle$5
$\angle$3$\cong$$\angle$7
$\angle$2$\cong$$\angle$6
$\angle$4$\cong$$\angle$8

2. If the alternate interior angles are congruent. Then the two lines $l_1$ and $l_2$ cut by the transversal are parallel. That is, $l_1$$\cong$$l_2$, if any of the following holds:

$\angle$3$\cong$$\angle$6
$\angle$4$\cong$$\angle$5

3. If the alternate exterior angles are congruent. Then the two lines $l_1$ and $l_2$ cut by the transversal are parallel. That is, $l_1$$\cong$$l_2$, if any of the following holds:
$\angle$1$\cong$$\angle$8
$\angle$2$\cong$$\angle$7

4. If the consecutive interior angles are supplementary, then the two lines $l_1$ and $l_2$ cut be the transversal are parallel. That is, $l_1$$\cong$$l_2$, if any of the following holds:

m $\angle$3+m $\angle$5=$\angle$$180^o$
m $\angle$4+m $\angle$6=$\angle$$180^o$

Solved Examples

Question 1: Identity the transversal and classify each angle pair in the following figure:
 
Transversal Angles Examples
Solution:
 
TransversalCorresponding AnglesAlternate Interior Angles Alternate Exterior AnglesConsecutive Interior Angles
l
$\angle$1 and $\angle$5
$\angle$2 and $\angle$6
$\angle$4 and $\angle$7
$\angle$3 and $\angle$8
$\angle$2 and $\angle$7
$\angle$3 and $\angle$5
$\angle$1 and $\angle$8
$\angle$4 and $\angle$6
$\angle$2 and $\angle$5
$\angle$3 and $\angle$7
m
$\angle$5 and $\angle$9
$\angle$6 and $\angle$10
$\angle$7 and $\angle$12
$\angle$8 and $\angle$11
$\angle$7 and $\angle$10
$\angle$8 and $\angle$9
$\angle$5 and $\angle$11
$\angle$6 and $\angle$12
$\angle$7 and $\angle$9
$\angle$8 and $\angle$10
n
$\angle$1 and $\angle$12
$\angle$2 and $\angle$9
$\angle$3 and $\angle$10
$\angle$4 and $\angle$11
$\angle$3 and $\angle$12
$\angle$4 and $\angle$9
$\angle$1 and $\angle$10
$\angle$2 and $\angle$11
$\angle$3 and $\angle$9
$\angle$4 and $\angle$12

 

Question 2: Check whether the lines $l_1$ and $l_2$ are parallel or not, in the following figure.
 
 Transversal Angles Examples
Solution:
 
In the figure, the transversal t cuts the lines $l_1$ and $l_2$. From the figure, (3x-150)$^o$ and x$^o$ are the measures of vertical angles.  Hence, they are equal. Thus, we have
             3x-150=x
        $\rightarrow$3x-x=150
        $\rightarrow$2x=150
        $\rightarrow$x=75
Now, (2x-75)$^o$ and (3x-150)$^o$ are measures of the corresponding angles, and we have
        2x-75=2(75)-75=75
        3x-150=3(75)-150=75
Since the measures of the corresponding angles are equal, the corresponding angles are congruent. Hence, the lines $l_1$ and $l_2$ are parallel.