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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:

1  and  5 2  and  6 3  and  7 4  and  8

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:

3  and  6 4  and  5

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:

3  and  5 4  and  6

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:

2  and  8 1  and  7

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 A B  and 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 E F and inside the two lines A B and C D . So, they are consecutive interior angles.

Since the lines A B and 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.

140 ° + m C Y E = 180 ° 140 ° + m C Y E 140 ° = 180 ° 140 ° m C Y E = 40 °

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