Download Angles An angle ∠AOB is the union of two noncollinear rays O r A

Angles
An angle
r ∠AOBr is the union of two noncollinear
rays O A and O B that share a common endpoint.
In the same way that we can measure lengths by
attaching
numbers (distances) to line segments, we
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can measure “rotational distance” spanned by
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angles. This is accomplished by means of angle
axioms:
[A-1] To every angle ∠AOB = ∠BOA we associate a
real number strictly between 0 and 180
called the measure of the angle; this
measure is labeled m ∠AOB = m∠BOA.
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The use of the interval 0 < m∠AOB < 180 for angle
measure is entirely a matter of convenience. We
€
could just as easily use the interval 0 < m∠AOB < π
or even 0 < m∠AOB < 1. In particular, note the
€
absence of reference to degrees above; the intention
here is to make use of the familiarity of degree
€
measure without needing to provide a special
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definition for degree.
The interior of ∠AOB is the set of all points in the
plane of A,s O
on the
r and B that lie simultaneously
sr
A-side of OB and the B-side of OA , that is,
sr
sr
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Interior ∠AOB = H(A, OB) ∩ H(B, OA ).
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sr
Theorem If A-X-B and O does not lie on AB , then
X is interior to ∠AOB. //
r
Corollary If A-X-B, then every point
on O X ,
€
except €
for O, is interior to ∠AOB. //
Recall that we used the additivity
€ of distances to
define the betweenness of points (A-B-C means
€
AB + BC = AC); we would like to do the same for
angles. For this, we need the Angle Addition
Postulate:
[A-2] If C is an interior point of ∠AOB, then
m∠AOC + m∠COB = m∠AOB.
In fact, the converse is also true:
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Theorem
If m∠AOC + m∠COB = m∠AOB, then
r
ray O C must pass through an interior
point of
r
∠AOB. Indeed, every point on O C is interior to
∠AOB.
€ //
From this raxiom,
define betweenness for
r we can
€r
rays: if O A , O B and O C are
r three rays sharing
r a
common
endpoint, then O C lies
O A and
r
r between
r
r
O B , expressed by writing O A -O C -O B , if
+ m∠COB
= m∠AOB. Betweenness for
€m∠AOC
€
€
rays behaves in€a manner entirely€similarly to
betweenness for points. The connection between
€ € €
these relations can be seen in the following
Theorem Suppose
r
r A, rB, C lie on line l and O is not
on l. Then O A -O C -O B if and only if A-C-B. //
Furthermore, in the same way that we were able to
formulate a Ruler Postulate that gives coordinates
€ € €
to points on a line, we can formulate a Protractor
Postulate that gives coordinates to rays within a
half plane:
r
[A-3] Given any ray O A , there exists a one-to-one
correspondence
between thes set
r
r of all rays
O X on oner side of the line OA , together with
the ray€O A itself, and the interval of real
numbers 0 ≤ θ < 180 (the number
r θ is called
ther coordinate of€the ray O X rand we write
€
this) sorthat O A [0], and
€O X [θ ] to signify
r
whenever
O B [θ ] and O C€[ φ ], then
€
m∠BOC =|θ − φ |. €
€ €
€
Of course,
theorems relating
€ the
€ corresponding
€ €
coordinates and betweenness of rays are true (and
€ their proofs handled similarly as well):
Theorem Howeversone
the rays
r coordinatizes
r
r on
oner side of therline OA (where O
r A [0]),
r ifr O B [θ ],
O C [ φ ] and O D [ ω ], we have O B -O C -O D if and
only if either θ < φ < ω or ω < φ < θ . //
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Theorem [Angle Construction] If ∠ABC, ∠DEF
are two angles for whichr m ∠ABC < m ∠DEF
r
r , then
r
there is a unique ray E G such that E D - E G - E F
and m ∠ABC = m ∠GEF . // €
€
€
€
Corollary rEvery angle has a unique bisector.
€
€ € €
O C is the bisector of ∠AOB if
€ (The ray €
m∠AOC = m∠COB .) //
r
r
Rays
O A and O B are called
opposite rays if
€
€
A-O-B, that is, the union of the two rays is a line.
It is easy to show the
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r
€
Theorem
Every ray O A has a unique opposing
r
ray O B . //
Two angles ∠AOC
and ∠COB are rsaid to form a
€
linear pair if theyrshare a side
O C and their
r
remaining sides O A and O B are opposite rays.
Note
Postulate makes no
€ that the Protractor
€
provision for assigning coordinates
to all three of
€
these rays simultaneously since the coordinates of
€
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all rays must be less than 180. For this, we need
another axiom. Calling any two angles whose
measures sum to 180 supplementary angles
allows us to state the Linear Pair Axiom:
[A-4] A linear pair of angles is supplementary.
An angle is right if its measure is 90, and any two
angles whose measures sum to 90 are called
complementary. An angle is acute if its measure
is less than 90 and obtuse if its measure is greater
than 90.
Two distinct lines l and m are perpendicular if
they contain sides of a right angle and we write
l ⊥ m to denote this. (We can also extend the
defintion of perpendicularity to segments and rays
if the lines that contain them are perpendicular.)
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Theorem Perpendicular lines make four right
angles at their point of intersection. //
r
If angles ∠AOC and ∠COB share side O C and
have no interior points in common, we call the
angles adjacent.
sr €
€
€s r
Theorem If BD
s rmeets
s r AC at the point B between
A and C, then BD ⊥ AC if and only if the adjacent
angles ∠ABD and ∠DBC have equal measure. //
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Theorem There is a unique perpendicular to a
€
given
line at any point on that line. //
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Two angles form a vertical pair if the sides of one
are the opposing rays of the sides of the other.
Theorem Vertical angles have equal measure. //
Moritz Pasch, one of the nineteenth century
geometers who, like Hilbert, worked on developing
the axiomatization of geometry, posed the following
statement as an axiom and used it to prove the
Plane Separation Postulate as a theorem.
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A triangle ΔABC is the union of three segments
A B , B C and C A , called sides of the triangle; the
points A, B and C, called the vertices of the
triangle, must be noncollinear.
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Theorem [Postulate of Pasch] Let A, B, C be
noncollinear points in a plane. If l is any line in
this plane that passes through a point D interior to
side A B , then l must pass through some point E on
the triangle not on A B ; that is, E satisfies exactly
one of the following three properties:
• E is an interior point on side C A ;
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• E = C; or
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• E is an interior point on side C B . //
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Theorem If A-X-B, then X is an interior point of
∠AOC. //
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Theorem [Crossbar
r Theorem] If D is interior to
∠ABC, then ray B D must meet segment A C . //
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