parallel postulate (Euclid’s fifth postulate) In his

famous work T

HE

E

LEMENTS

, the geometer E

UCLID

(ca. 300

B

.

C

.

E

.) proposed five basic postulates that

describe the principles of geometry on a flat

PLANE

. His

fifth postulate states:

If two lines Land Msubtended from a com-

mon line segment have interior angles sum-

ming to less than 180°, then those lines, if

extended, will meet to form a

TRIANGLE

.

One can consequently conclude that if those angles sum

to more than 180°then the lines, if extended on the

other side of the given line segment, will form a triangle

on that side. (The sum of the supplementary interior

angles will be less than 180°.) Furthermore, if the lines

never meet, that is, if they are

PARALLEL

, then the sum

of the two angles xand yshown cannot be less than nor

greater than 180°, and so must equal 180°. We have:

For a pair of parallel lines with a

TRANSVERSAL

,

as shown in the diagram below right, the angles

labeled xand ymust sum exactly to 180°. Con-

sequently, the alternate interior angles of the

transversal are equal.

This is often taken as the statement of Euclid’s fifth

postulate. For this reason, it is often called the parallel

postulate.

As the parallel postulate is intimately connected

with the formation of triangles, it is not surprising that

one can use it to prove:

The sum of the interior angles of any triangle

equals 180°.

As we show below, it is possible to alternatively take

this statement as a basic postulate and use it to estab-

lish the parallel postulate. (Thus the parallel postulate

and this claim about triangles are logically equivalent

starting assumptions.)

In his 1795 text Elements of Geometry, Scottish

mathematician John Playfair offered a third, logically

equivalent version of the parallel postulate:

Given a line Lin the plane and a point Pnot

on that line, then there exists one, and only

one, line through Pparallel to L.

This alternative version of the postulate is today

called P

LAYFAIR

’

S AXIOM

. In 1829 it provided Russian

mathematician N

IKOLAI

I

VANOVICH

L

OBACHEVSKY

(1792–1856) the inspiration to consider, and discover,

a consistent theory of a

NON

-E

UCLIDEAN GEOMETRY

.

We now have three assertions:

1. Euclid’s parallel postulate

2. Angles in a triangle sum to 180°

3. Playfair’s axiom

These three statements are logically equivalent, that is,

any one implies the other two. To see this, it suffices to

show that (3) implies (2), (2) implies (1), and that (1)

implies (3). We will need to make use of the converse of

Euclid’s parallel postulate:

If a transversal across a pair of lines yields

equal alternate interior angles, then the lines

are parallel.

As the

EXTERIOR

-

ANGLE THEOREM

shows, this state-

ment can be proved without making use of any of the

three statements above.

Consider now the three implications:

(3) implies (2):

Given a triangle with angles a, b, and cas shown in

the diagram on page 378, draw two lines at the apex

that copy angles aand b. Given that both yield equal

alternate interior angles across a transversal, both lines

are parallel to the base of the triangle. By Playfair’s

axiom there is only one such line. Thus angles a, b, and

call lie on the same straight line and so sum to 180°.

(2) implies (1):

In the middle diagram on page 378, suppose that

lines Land Mare parallel, and so never meet. Could it

parallel postulate 377

Understanding the parallel postulate