Lines Cut by a Transversal

A transversal is any line that cuts across two or more lines. For our study we have limited the number to 2 lines. The transversal can form as many as 8 different angles when cutting across two lines. In some cases, however, this number can be limited depending on the way the cut is made and how many points of intersection exist. For example, look at Figure 1:

Figure 1

Lines a, b, and c are all transversals because each of them cuts two lines. Notice that each line forms 8 angles along its length over the two lines that have been cut. What can you tell about these angles? What is the relationship among them?

Look at line b as the transversal and the eight angles it forms over lines a and c. Pairs of corresponding angles would be: 5, 10; 8, 9; 6, 11;

7, 12. Pairs of alternate interior angles would be: 8, 11 and 7, 10; pairs of alternate exterior angles would be 5, 12 and 6, 9.

Question: Which angles in the diagram must be congruent? Name the pairs in the same way as above. Why?

Now look at a set of lines a  and b that are parallel and cut by transversal line c. Again, the same types of angles form over the lines and the transversal as above.

Figure 2

However, there is a new relationship because we have parallel lines: the corresponding angles are congruent with equal measures. This relationship also tells us that the alternate interior and exterior angles are congruent because of the corresponding angles and vertical angles in Figure 2.  Can you find other interesting relationships, such as angles that are not linear pairs that are supplementary?