# Transversal

In geometry, a
**
transversal
**
is a line that intersects two or more other (often
parallel
) lines.

In the figure below, line $n$ is a transversal cutting lines $l$ and $m$ .

When two or more lines are cut by a transversal, the angles which occupy the same relative position are called corresponding angles .

In the figure the pairs of corresponding angles are:

$\begin{array}{l}\angle 1\text{and}\angle 5\\ \angle 2\text{and}\angle 6\\ \angle 3\text{and}\angle 7\\ \angle 4\text{and}\angle 8\end{array}$

When the lines are parallel, the corresponding angles are congruent .

When two lines are cut by a transversal, the pairs of angles on one side of the transversal and inside the two lines are called the consecutive interior angles .

In the above figure, the consecutive interior angles are:

$\begin{array}{l}\angle 3\text{and}\angle 6\\ \angle 4\text{and}\angle 5\end{array}$

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary .

When two lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles .

In the above figure, the alternate interior angles are:

$\begin{array}{l}\angle 3\text{and}\angle 5\\ \angle 4\text{and}\angle 6\end{array}$

If two parallel lines are cut by a transversal, then the alternate interior angles formed are congruent .

When two lines are cut by a transversal, the pairs of angles on either side of the transversal and outside the two lines are called the alternate exterior angles .

In the above figure, the alternate exterior angles are:

$\begin{array}{l}\angle 2\text{and}\angle 8\\ \angle 1\text{and}\angle 7\end{array}$

If two parallel lines are cut by a transversal, then the alternate exterior angles formed are congruent .

**
Example 1:
**

In the above diagram, the lines $j$ and $k$ are cut by the transversal $l$ . The angles $\angle c$ and $\angle e$ are…

A. Corresponding Angles

B. Consecutive Interior Angles

C. Alternate Interior Angles

D. Alternate Exterior Angles

The angles $\angle c$ and $\angle e$ lie on either side of the transversal $l$ and inside the two lines $j$ and $k$ .

Therefore, they are alternate interior angles.

The correct choice is $C$ .

**
Example 2:
**

In the above figure if lines $\stackrel{\leftrightarrow}{AB}$ and $\stackrel{\leftrightarrow}{CD}$ are parallel and $m\angle AXF=140\xb0$ then what is the measure of $\angle CYE$ ?

The angles $\angle AXF$ and $\angle CYE$ lie on one side of the transversal $\stackrel{\leftrightarrow}{EF}$ and inside the two lines $\stackrel{\leftrightarrow}{AB}$ and $\stackrel{\leftrightarrow}{CD}$ . So, they are consecutive interior angles.

Since the lines $\stackrel{\leftrightarrow}{AB}$ and $\stackrel{\leftrightarrow}{CD}$ are parallel, by the consecutive interior angles theorem , $\angle AXF$ and $\angle CYE$ are supplementary.

That is, $m\angle AXF+m\angle CYE=180\xb0$ .

But, $m\angle AXF=140\xb0$ .

Substitute and solve.

$\begin{array}{l}140\xb0+m\angle CYE=180\xb0\\ 140\xb0+m\angle CYE-140\xb0=180\xb0-140\xb0\\ m\angle CYE=40\xb0\end{array}$