Parallel Lines

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PARALLEL LINES

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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.

Click here to use Geometer's Sketchpad to explore parallel lines!

The condition "in the same plane" is critical, since two lines in three-dimensional space can certainly not intersect and yet not be parallel, as the sides of the box below illustrate. Such lines are said to be skew.

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.

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

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

Use the Geometer's Sketchpad to explore parallel lines cut by a transversal!

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

For example, to prove that A⇔B⇔C , one could prove any of the following:

A⇒B and B⇒C and C⇒A
B⇒A and A⇒C and C⇒B
C⇒B and B⇒A and A⇒C

Here's a truth table which proves that (A⇔B  and  B⇔C) is equivalent to (A⇒B and B⇒C and C⇒A) .
Notice that since there are three individual statements (A, B, and C), and two possible truth values for each (true or false), there are 2³ = 8 rows in the truth table.

A	B	C	A⇔B	B⇔C	A⇔B and B⇔C	A⇒B	B⇒C	C⇒A	A⇒B and B⇒C and C⇒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

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