Download Axiom A-1: To every angle there corresponds a unique, real number

Axiom A-1: To every angle there corresponds a
unique, real number 2, 0 < 2 < 180.
We denote the measure of pABC by mpABC.
(Temporary Definition): A point D lies in the
interior of pABC iff there exists a segment
with E-D-F, and such that neither E nor F
are the point B, and
.
Axiom A-2: (Angle Addition Postulate) If D lies
in the interior of angle pABC, then
mpABC = mpABD + mpDBC
The book takes as part of this axiom the
converse of this statement, but we can actually
prove it after the next section.
Definition:
Suppose that
,
,and
are concurrent
rays, all having the same endpoint O. Then
ray
is said to be between rays
and
, written
if and only if
these rays are distinct, and if
mpAOB + mpBOC = mpAOC,
Note: This tells you that the ability to add
angle measures is equivalent to
“betweenness” for rays.
In practice, what this says is that before you add
angle measures, be sure the appropriate
betweenness relationships are in place.
Axiom A-3: Protractor Postulate.
The set of rays having a common origin O and
lying on one side of line l containing ray
,
may be assigned to the real numbers 2 for
which 0 # 2< 180, called coordinates, in such a
manner that:
(1) each ray is assigned a unique coordinate 2.
(2) each coordinate 2 is assigned to a unique
ray.
(3) the coordinate of
is 0.
(4) if rays
and
have coordinates 2 and
N, then mpPOQ = *2 - N*.
Note: This connects angle measures to real
numbers, so there are now as many rays coming
off a point in one half plane as there are real
numbers between 0 and 180.
Using the protractor postulate, we are able to
create angles with various properties. In
particular, we can:
“Copy” angles.
Bisect angles.
“Double” angles.
We prove a particular theorem of this kind
below, using a little lemma first:
Lemma: Let rays
have
coordinates a, b, and c respectively. Then
iff a < b < c or c < b < a.
Proof: Just like Theorem 3 in section 2.4.
Theorem 1. Angle Construction Theorem:
Given any two angles pABC and pDEF such
that mpABC < mpDEF, there is a unique ray
such that mpABC = mpGEF and
.
Sketch of proof: Let a = mpABC and b =
mpDEF, so 0 < a < b < 180. Using the
protractor postulate, assign coordinates to rays
on the D-side of
with 0 assigned to
.
The coordinate of
must then be b. Make
the unique ray
with coordinate a. By
lemma above,
. Then
mpGEF is |a-0| = a = mpABC .
Basic Facts About Angles:
Using the ruler postulate, it is easy to show that
If A-B-C,
c
is the line containing A, B,
and C. We call such rays opposing or opposite
rays. We note that given any ray, we can
always find a unique opposing ray. (How?)
Definition: If the sides of one angle are opposite
rays to the respective sides of another angle, the
angles are said to form a vertical pair.
Definition: Two angles are said to form a linear
pair iff they have one side in common and the
other two sides are opposite rays. We call any
two angles whose angle measures sum to 180 a
supplementary pair, or more simply,
supplementary, and two angles whose angle
measures sum to 90 a complementary pair, or
complementary.
Theorem 2: Two angles which are
supplementary (or complementary) to the same
angle have equal angle measures.
Outline of proof: Suppose angles " and $
are both supplementary to angle (. Then
mp" + mp( = 180 = mp$ + mp( .
Canceling mp( from both sides gives the
result. Proof for complementary case is
similar.
Now another Axiom that we need to make our
geometry work:
Axiom A-4. A linear pair of angles is a
supplementary pair.
Definition: An angle having measure 90 is
called a right angle. Angles having measure
less than 90 are called acute angles, and those
with measure greater than 90, obtuse angles.
Definition: If line l intersects another line m at
some point A and contain the sides of a right
angle, then l is said to be perpendicular to m,
and we write l z m.
Lemma: One line is perpendicular to another
line iff the two lines form four right angles at
their point of intersection.
Outline of proof: Use the linear pair axiom.
Corollary: The relation of perpendicularity is
symmetric, that is, if l z m, then m z l.
Theorem 3: If line
meets segment
at an
interior point B on that segment, then
iff the adjacent angles at B have
equal measures.
Proof: Easy.
Theorem 4 (Existence and Uniqueness of
Perpendiculars): Suppose that in some plane
line m is given and an arbitrary point A on line
m is located. Then there exists a unique line l
that is perpendicular to m at A.
Outline of proof: Locate a point B … A on
m. Use the protractor postulate to establish
as having coordinate 0. Find the
unique ray
having coordinate 90. Then
line
and line m contain the sides of a
right angle and are perpendicular by
definition. Uniqueness follows from the
protractor postulate.
Theorem 5 (Vertical Pair Theorem) Vertical
angles have equal measures.
Outline of proof: Referring to the diagram,
angles 1 and 3 form a linear pair, as do
angles 2 and 3. By axiom A-4, angles 1 and
2 are each supplementary to a common
angle 3, and by Theorem 2, must have equal
measures.