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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 $m$ .

Math diagram

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:

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

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:

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

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:

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

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:

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

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

Example 1:

Math diagram

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

A. Corresponding Angles

B. Consecutive Interior Angles

C. Alternate Interior Angles

D. Alternate Exterior Angles

The angles $∠ c$ and $∠ e$ 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:

Math diagram

In the above figure if lines $\overset{\leftrightarrow}{A B}$ and $\overset{\leftrightarrow}{C D}$ are parallel and $m ∠ A X F = 140 °$ then what is the measure of $∠ C Y E$ ?

The angles $∠ A X F$ and $∠ C Y E$ lie on one side of the transversal $\overset{\leftrightarrow}{E F}$ and inside the two lines $\overset{\leftrightarrow}{A B}$ and $\overset{\leftrightarrow}{C D}$ . So, they are consecutive interior angles.

Since the lines $\overset{\leftrightarrow}{A B}$ and $\overset{\leftrightarrow}{C D}$ are parallel, by the consecutive interior angles theorem , $∠ A X F$ and $∠ C Y E$ are supplementary.

That is, $m ∠ A X F + m ∠ C Y E = 180 °$ .

But, $m ∠ A X F = 140 °$ .

Substitute and solve.

$\begin{array}{l} 140 ° + m ∠ C Y E = 180 ° \\ 140 ° + m ∠ C Y E - 140 ° = 180 ° - 140 ° \\ m ∠ C Y E = 40 ° \end{array}$