PARALLEL LINES

The blue and green lines below suggest *parallel lines*:

DEFINITION
parallel lines

Two lines are *parallel* if and only if they lie in the same plane and do not intersect.

The condition ‘in the same plane’ is critical.

Two lines in three-dimensional space can certainly
not intersect and yet not be parallel,

as the sides of the box below illustrate:

that don't intersect, but are not parallel

Lines such as those illustrated above—that don't intersect, but are not parallel—are called *skew* lines:

DEFINITION
skew lines

Two lines are *skew* if and only if they are not parallel and do not intersect.

This section explores conditions under which two lines are guaranteed to be parallel.

TRANSVERSALS AND ASSOCIATED ANGLES

First, we need an auxiliary line, called a *transversal*, that *traverses* or *cuts across*
two
coplanar lines:

This transversal brings lots of different angles into the picture, which are given names below.

Notice that the word *alternate* refers to being on *opposite (alternate) sides of the transversal*.

The word *interior* refers to being *inside* the two lines.

The word *exterior* refers to being *outside* the two lines.

*ALTERNATE INTERIOR ANGLES* are inside the lines, and on alternate sides of the transversal.

There are two pairs of alternate interior angles.

*ALTERNATE EXTERIOR ANGLES* are outside the lines, and on alternate sides of the transversal.

There are two pairs of alternate exterior angles.

*SAME-SIDE INTERIOR ANGLES* are inside the lines, and on the same side of the transversal.

There are two pairs of same-side interior angles.

*CORRESPONDING ANGLES* are in the same
‘quadrant’ formed by each line and the transversal.

In the sketch below, the ‘upper’ and ‘lower’ angles are both in ‘Quadrant I’.

There are four pairs of corresponding angles.

When two lines are cut by a transversal, the angles named above can be used to decide if the lines are parallel or not, as follows:

THEOREM
conditions for parallel lines

Suppose two lines are cut by a transversal.

Then, the following are equivalent:

Then, the following are equivalent:

- the two lines are parallel
- a pair of alternate interior angles are congruent
- a pair of alternate exterior angles are congruent
- a pair of corresponding angles are congruent
- a pair of same-side interior angles are supplementary

There is incredible power in a theorem of this sort!

Since these five statements are equivalent:

- if one of them is true, then all of them are true
- if one of them is false, then all of them are false

To prove that multiple statements are equivalent,

mathematicians often prove a chain of
implications;

the chain starts and ends at the same statement, and covers all the statements in between.

For example, to prove that [beautiful math coming... please be patient] $\,A\Leftrightarrow B\text{ and }B\Leftrightarrow C\,$, one could prove any of the following:

- [beautiful math coming... please be patient] $\,A\Rightarrow B\,$ and $\,B\Rightarrow C\,$ and $\,C\Rightarrow A\,$
- $\,B\Rightarrow A\,$ and $\,A\Rightarrow C\,$ and $\,C\Rightarrow B\,$
- $\,C\Rightarrow B\,$ and $\,B\Rightarrow A\,$ and $\,A\Rightarrow C\,$

Here's a truth table which proves that
[beautiful math coming... please be patient]
$\,(A\Leftrightarrow B \text{ and } B\Leftrightarrow C)\,$
is equivalent to
$\,(A\Rightarrow B\,$ and
$\,B\Rightarrow C\,$ and
$\,C\Rightarrow A)\,$.

(The mathematical word ‘and’ is associative, so
we can say things like ‘$\,P\text{ and }Q\text{ and }R\,$’
without
any ambiguity.

Also, the only time that ‘$\,P\text{ and }Q\text{ and }R\,$’ is true
is when all three subsentences are true.)

Since there are three individual statements ($\,A\,$, $\,B\,$, and $\,C\,$),

and two possible truth values for each (true or false),

there are $\,2^3 = 8\,$ rows in the truth table.

$A$ | $B$ | $C$ | [beautiful math coming... please be patient] $A\Leftrightarrow B$ | $B\Leftrightarrow C$ | [beautiful math coming... please be patient] $A\Leftrightarrow B\,$ and $\,B\Leftrightarrow C$ | $A\Rightarrow B$ | $B\Rightarrow C$ | $C\Rightarrow A$ | $A\Rightarrow B\,$ and $\,B\Rightarrow C$ and $\,C\Rightarrow A\,$ |

T | T | T | T | T | T | T | T | T | T |

T | T | F | T | F | F | T | F | T | F |

T | F | T | F | F | F | F | T | T | F |

T | F | F | F | T | F | F | T | T | F |

F | T | T | F | T | F | T | T | F | F |

F | T | F | F | F | F | T | F | T | F |

F | F | T | T | F | F | T | T | F | F |

F | F | F | T | T | T | T | T | T | T |

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Parallelograms and Negating Sentences

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Parallelograms and Negating Sentences

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.