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**Subsections**

We remark that the point of the axiom is not the existence of the
parallel, but the uniqueness. We will see below that existence
actually follows from what we already know.

It is sometimes convenient to think of a line as being parallel to itself,
so we make the following formal definition. Two lines are not parallel if
they have exactly one point in common; otherwise they are parallel.

**Theorem 3.1**
*In the set of all lines in the plane, the relation of being parallel is an
equivalence relation. *

We will meet the following situation
some number of times. We are given two lines and , and a third
line , where crosses at and crosses at
. Choose a point
on , and choose a point
on , where and lie on opposite sides of the line
. Then
and
are referred to as
*alternate interior angles*.

In any given situation, there are two distinct pairs of alternate interior
angles. That is, let be some point on , where and lie
on opposite sides of , and let be some point on , where
and lie on opposite sides of . Then one could also regard
and
as being alternate interior
angles. However, observe that
and
. It follows that one pair of
alternate interior angles are equal if and only if the other pair of
alternate interior angles are equal.

**Proposition 3.2**
*If the alternate interior angles are equal, then the lines
and are parallel. *
*Proof*.
Suppose not. Then the lines

and

meet at some point

. If
necessary, we interchange the roles of the

and the

so that

is an exterior angle of

. Then

and

lie on the same side of

, so

. By
the exterior angle inequality,

so we have reached a contradiction.

Let be a line, and let
be a point not on . Pick some point on and draw the line
through and . By Axiom 7, we can find a line through
so that the alternate interior angles are equal. Hence we can find a
line through parallel to .

**Theorem 3.3** (alternate interior angles equal)

*Two lines and are parallel if and only if the alternate
interior angles are equal. *
*Proof*.
To prove the forward direction, construct the line

through

,
where there is a point

on

, with

and

on the same
side of

, so that

. Then, by
Prop.

3.2,

is a line through

parallel to

. Axiom

8 implies

. Hence

, and the desired conclusion follows.

The other direction is just Prop. 3.2,
restated as part of this theorem for convenience.

**Theorem 3.4**
*The sum of the measures of the angles of a triangle is equal to .*

*Proof*.
Consider

, and let

be the line passing through

and
parallel to

. Let

and

be two points on

, on opposite sides of

, where

and

lie on opposite sides of the line

. Then

and

lie on opposite sides of

.

**Exercise 3.1**
*Use alternate interior angles to complete the proof of this theorem.*

A *quadrilateral* is a region bounded by four line segments, so it
has four verticies on its boundary.

**Corollary 3.5**
*The sum of the measures of the angles of a quadrilateral is .*
*Proof*.
Cut the triangle into two triangles, and do the obvious computation.

A *rectangle* is a quadrilateral in which all four angles are right
angles.

**Theorem 3.6**
*If is a rectangle, then is parallel to , and
. Similarly, is parallel to and . *

**Exercise 3.2**
*Prove this theorem.
*
- i.
- Prove that opposite pairs of sides are parallel.
- ii.
- Now cut the rectangle into two triangles; prove that these two
triangles are congruent. Conclude that opposite sides of the rectangle have
equal length.

Somewhat more generally, a *parallelogram* is a quadrilateral
in which opposite sides are parallel; that is, is parallel to , and
is parallel to .
A rectangle with all four sides of equal length is a *square*; a
parallelogram with all four sides of equal length is a *rhombus*.

**Exercise 3.3**
*Prove this theorem. (Hint: Draw a diagonal.)*

**Theorem 3.8**
*If is a quadrilateral in which and , then
is a parallelogram. *

**Exercise 3.4**
*Prove this theorem.*

**Theorem 3.9**
*Let be a parallelogram with diagonals of equal length (that is,
). Then is a rectangle. *

**Exercise 3.5**
*Prove this theorem.*

** Next:** 4 Lengths, areas and
** Up:** MAT 200 Course Notes on
** Previous:** 2 Triangles and congruence
Scott Sutherland
2002-12-18