section3

PARALLEL AND PERPENDICULAR LINES.

In this section we are going to come back to straight objects and spend some time studying the properties of lines that are located in the same plane.
Two lines a and b are called parallel if they lie in the same plane and have no points of intersection. We are going to denote the fact that a and b are parallel lines by
As we continue our discussion of parallel lines we again encounter a statement that will be accepted as a true one without a proof (such statements are called axioms).

Axiom 2. Given an arbitrary line m and a point A that does not belong to m, we can construct no more than one line n that will pass through A and will be parallel to the given line m.

Let's consider now three lines m, n and l such that l intersects with both m and n. Such a line l that intersects with two (or more) other lines is called a transversal.

Please note that lines n and m can be parallel or not parallel in the definition of a transversal l.

A transversal l that intersects with m and n creates various types of angles.

Angles ACD and CFG and also angles ACB and CFE are called corresponding angles.

Angles BCF and CFG as well as angles DCF and CFE are called alternate interior angles.

Angles ACD and EFH and also angles ACB and GFH are called alternate exterior angles.

Finally, angles DCF and CFG and also angles BCF and CFE are called one-sided interior angles.

Now that we have discussed the main issues connected with the case when two lines have no points of intersection, we are going to spend some time studying the second important case. Namely, let's consider two lines m and n that intersect at some point O.

We get four angles with vertex O: AOC, COB, BOD and DOA. From these angles there are two pairs of vertical angles: AOC and DOB, AOD and COB. We also have four pairs of supplementary angles: AOC and COB, COB and BOD, BOD and DOA, DOA and AOC. From the theorem 2.1 it follows that vertical angles are equal:

By definition of supplementary angles we also know that

Therefore if one of four angles with vertex O is equal to 90 degrees then all the rest angles are right angles as well. In this case we say that lines m and n are perpendicular and denote this fact by

Theorem 3.1 Given a point A and a line m that contains A we can always construct a line n such that it will be perpendicular to m and that will pass through A. Moreover, such line n will be unique.

Proof Let m1 be one of the rays with the endpoint A. By axiom 1 we can always construct an angle equal to 90 degrees given the ray m1 and choosing one of the half-planes (e.g. the right one). Then the line n containing the side of this angle other than the ray m1 will be perpendicular to m1 (and to m). By the same axiom such a line will be unique.

In order to finish our discussion of parallel and perpendicular lines we are going to give an important way to check whether two lines are parallel or not.

Theorem 3.2 Suppose that we have lines m and n and their transversal l. If any pair of alternate interior angles consists of equal angles or if the sum of any one-sided interior angles is equal to 180 degrees then m and n are parallel lines.

We will not give the proof of theorem 3.2 since this proof will not help us a lot in our understanding of the matter. We would rather formulate a converse theorem 3.2* and an important corollary from the theorem 3.2.

Theorem 3.2* If lines m and n are parallel and l is their transversal then any pair of alternate interior angles consists of equal angles and the sum of any one-sided interior angles is equal to 180 degrees.

Corollary If two lines m and n are perpendicular to the same line l then m and n are parallel.

EXERCISE SET #3

Problem 1. Prove the following theorem:

Theorem 3.3 Suppose that we have lines m and n and their transversal l. If any pair of corresponding angles consists of equal angles then m and n are parallel lines.

Problem 2. Suppose that angles CBE and BEF are interior one-sided angles and lines CB and EF are parallel. If the degree measure of CBE is two times larger than the degree measure of BEF what will be the degree measure of CBE?