extraction 183

Understanding the exterior angle theorem

Equal alternate interior angles

the exterior angle at one vertex of a triangle is greater

in value than that of an interior angle at either of the

remaining two vertices.

See also

CONCAVE

/

CONVEX

.

exterior-angle theorem In his famous work T

HE

E

LEMENTS

, the geometer E

UCLID

(ca. 300–260

B

.

C

.

E

.)

established the following result, which he called the

exterior-angle theorem:

For a given triangle ABC with interior angles x

and yand exterior angle zas shown, we have z

> xand z> y.

The result is proved as follows:

Let Mbe the midpoint of side BC, and extend

a line from point Athrough Mto a new point

Dso that AM and DM are the same length.

Consider triangles AMB and DMC. They share

two sides of the same lengths and a common

angle at M. By the SAS principle for similarity,

the two triangles are congruent figures. Conse-

quently, angle MCD matches angle MBA,

which is y. Since zis clearly larger than angle

MCD, we have that z> y.

An analogous argument based on a line

drawn through the midpoint of side AC estab-

lishes that zis also greater than x.

This theorem has one very important consequence.

If two lines cut by a transversal produce equal

alternate interior angles, then the two lines

are parallel.

In the diagram above, if the two angles labeled xare

indeed equal, then the lines cannot meet to the right to

form a triangle: the exterior angle xcannot be greater

than the interior angle x. Similarly, the lines cannot

meet to the left to form a triangle by the same reason.

(Work with the angle 180 – x.) It must be the case then

that the lines are parallel.

This result is the

CONVERSE

of Euclid’s famous, and

controversial,

PARALLEL POSTULATE

.

It is important to note that Euclid proved the

exterior-angle theorem and its consequence without

assuming that the parallel postulate holds. If one is

willing to assume that the three angles in a triangle

always sum to 180°(a statement equivalent to the par-

allel postulate), then the proof of the exterior angle

theorem is

TRIVIAL

.

extraction The process of finding the

ROOT

of a

number or the solution to an algebraic equation is

sometimes called extraction. For example, extracting

the square root of 3 is the process of finding its square

root. (One might use H

ERON

’

S METHOD

, for example,

to compute √

–

3 = 1.7320508…)

The term digit extraction is often used to describe

any method that allows one to compute a specific digit

of a number without computing earlier digits. For

example, in 1995 mathematician Simon Plouffe discov-

ered the following remarkable formulae for π:

It has led mathematicians to a technique that com-

putes the Nth digit of πin base 16 without having to

calculate the preceding N– 1 digits. (In base 16, the

number πappears as 3.243F6A8885A308D … where