Parallel Lines Cut By a Transversal

April 24th, 2014 

Parallel lines are my personal favorite kind of lines. They blow perpendicular lines right out of the water for me. I first learned about parallel lines in depth in ninth grade Honors Geometry, and they quickly became one of my favorite topics. Parallel lines, for anyone who is unsure about what they are, are lines on the same geometrical plane that do not intersect or cross at ANY point on the plane. They are always the same distance apart from each other on the plane that they are located on. 

Image found here. 

(Perpendicular lines are lines on the same place that intersect at one point to form four 90 degree angles. Not nearly as exciting.) 

Sometimes, a pair of parallel lines is intersected by a third line, which is called a transversal. 

Image found here.

When this happens, several kinds of angles are formed. These angles are: corresponding angles, alternate interior angles, alternative exterior angles, same-side interior angles, same-side exterior angles, and vertical angles. 

Before I define these angles, we need to understand that when two angles are congruent, they have the same measurement, and when two angles are supplementary, their measurements add up to 180 degrees. 

Corresponding angles are angles that are located at the same relative position at an intersection, and are congruent. In the image above, angles 1 and 5, angles 2 and 6, angles 3 and 7, and angles 4 and 8 are corresponding angle pairs. 

Alternate interior angles are angles that are located on opposite sides of the transversal and inside of the parallel lines. They are congruent. In the image above, angles 3 and 6 and angles 4 and 5 are alternate interior angle pairs. 

Alternate exterior angles are angles that are located on opposite sides of the transversal line and outside of the parallel lines. They are congruent. In the image above, angles 1 and 8 and angles 2 and 7 are alternate exterior angle pairs. 

Same-side interior angles are angles that are located on the same side of the transversal line and inside of the parallel lines. They are supplementary. In the image above, angles 3 and 5 and angles 4 and 6 are same-side interior angle pairs. 

Same-side exterior angles are angles that are located on the same side of the transversal line and outside of the parallel lines. They are supplementary. In the image above, angles 1 and 7 and angles 2 and 8 are same-side exterior angle pairs. 

Vertical angles are angles that are on opposite sides of two intersecting lines (in this case, a parallel line and the transversal line). They are congruent. In the image above, angles 1 and 4, angles 2 and 3, angles 5 and 8, and angles 6 and 7 are vertical angle pairs. 

Oftentimes when there is a pair of parallel lines cut by a transversal, we can figure out the measure of each of the angles if we know the measurement of one angle. Let's do an example using the image below (the same image above). 

Let's say we are given that the measure of angle 1 is 135 degrees. Angle 5 is congruent to angle 1 because they are corresponding angles, so we know the measure of angle 5 is 135 degrees. Angle 4 is congruent to angle 5 because they are alternate interior angles, so we know the measure of angle 4 is 135 degrees. Angle 8 is congruent to angle 4 because they are corresponding angles, so we know the measure of angle 8 is 135 degrees. 

Angle 7 is supplementary to angle 1 because they are same-side exterior angles. 180 - 135 = 45, so we know the measure of angle 7 is 45 degrees. Angle 2 is congruent to angle 7 because they are alternate exterior angles, so we know the measure of angle 2 is 45 degrees. Angle 6 is congruent to angle 2 because they are corresponding angles, so we know the measure of angle 6 is 45 degrees. Angle 3 is congruent to angle 6 because they are alternate interior angles, so we know the measure of angle 3 is 45 degrees. 

We were able to determine the measurements of each of the angles using the measurement of just angle 1. 

Remember, if two angles that are supposed to be congruent are NOT congruent, then at some point the two "parallel" lines cross, and they aren't really parallel. The same is true when two angles that are supposed to be supplementary, are NOT supplementary. 

If you still struggle with the concept of parallel lines cut by a transversal, check out this website.