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Subsections

# 3 The parallel axiom

Axiom 8 (Parallel Axiom)   Given a line , and a point not on , there is exactly one line passing through and parallel to .

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.

Proof. First, since a line has infinitely many points in common with itself, it is parallel to itself; hence the relation is reflexive (this is the point of the strange definition).

Second, the definition is obviously symmetric; it is defined in terms of the two lines; not one with relation to the other.

Third, suppose is parallel to , and is parallel to . There is obviously nothing further to prove unless the three lines are distinct. Assume that and are not parallel. Since two lines are either equal, parallel, or have exactly one point in common, we must have that and have a point in common. But this contradicts Axiom 8.

## 3.1 Alternate interior angles

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.

## 3.2 Existence of parallel lines

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.

## 3.3 The sum of the angles of a triangle

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.

Theorem 3.7   Let be a parallelogram. Then ; ; ; and .

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