Post on 31-Dec-2015
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§2.1 Introductory Material
The student will learn about:
and the beginning postulates to be used in this course.
definitions of basic terms,
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Assumptions
We are assuming knowledge of all of the arithmetic and algebra properties that are known.
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Basic terms such as point, line and plane will not be defined.
The postulate system we will be using is attributed to David Hilbert.
1862 –1943
The Distance Postulate
Postulate 1. The Distance Postulate.
To every pair of different points there corresponds a unique positive number.
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The Ruler Postulate
Postulate 2. The Ruler Postulate.The points of a line can be placed in correspondence with the real numbers in such a way that
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1) To every point of the line there corresponds exactly one real number;
1) To every point of the line there corresponds exactly one real number;
2) To every real number there corresponds exactly one point of the line; and
1) To every point of the line there corresponds exactly one real number;
2) To every real number there corresponds exactly one point of the line; and
3) The distance between any two points is the absolute value of the difference of the corresponding numbers.
The Ruler Placement Postulate
Postulate 3. The Ruler Placement Postulate.
Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.
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Betweenness
Definition
If A, B and C are points on the same line, then B is between A and C iff
AB + BC = AC
And we write A – B – C or C – B – A.
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Theorem 2 - 1
If A, B, and C are three different points of the same line, then exactly one of them is between the other two.
Proof assigned for homework.
Hint: show that exactly one of the following must hold:AB + BC = AC or AC + CB = AB or
BA + AC = BC
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The Line Postulate
Postulate 4. The Line Postulate.
For every two points there is exactly one line that contains both points.
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Some DefinitionsDefinition.
For any two points A and B, the segment is the set whose points are A and B, together with all the points between A and B. the points A and B are called the endpoints of
AB
AB
Definition.
The number AB is called the length of segment .AB
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Some DefinitionsDefinition.
Let A and B be points on line L. The ray is the set which is the union of the segment and the set of all points C for which it is true that A – B – C. the point A is called the end point of .
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AB��������������
Definition.
If A is between B and C , then and are opposite rays.
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AC��������������
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Segments, Rays, Lines & Angles.
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Segment AB: AB X : A X B, X A, or X B
Ray AB: AB X : A X B, A B X, X A, or X B
Line AB: AB X : X A B, A X B, A B X, X A, or X B �������������� �
Angle ABC: ABC BA BC
(A, B and C noncollinear)
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Some DefinitionsDefinition.
A point B is called a midpoint of a segment if A – B – C and AB = BC. The midpoint is said to bisect the segment.
AC
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Theorem
Every segment has exactly one midpoint.
Given: B is the midpoint of segment AC. Prove B is unique.
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A B D C
Assume that D is also a midpoint of segment AC. Arrive at a contradiction.
What is our strategy?
Theorem
Every segment has exactly one midpoint.
Statement Reason
1. AB + BC = AC and AB = BC Given, B is midpoint.
Given: B is the midpoint of segment AC. Prove B is unique. Assume that D is also a midpoint of segment AC. Arrive at a contradiction.
2. AB = ½ AC Arithmetic
3. AD + DC = AC and AD = DC Assumed, D is midpoint
4. AD = ½ AC Arithmetic
→← of the ruler postulate. Hence there is only one midpoint.
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A B D C
Some DefinitionsDefinition.
The set of all points is called space.
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Definition.
A set of points is collinear if there is a line which contains all the points of the set.
Definition.
A set of points is coplanar if there is a plane which contains all the points of the set.
Postulate
Postulate 5.
(a) Every plane contains at least three noncollinear points.
(b) Space contains at least four nonplaner points.
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TheoremIf two distinct lines l and m meet,
their intersection is a single point.
Given: Distinct lines l and m with l ∩ m = A
Prove: A is unique.
l m
A
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Proof for homework.
Strategy?
Postulate
Postulate 6.
If two points of a line lie in a plane, then the line lies in the same plane.
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Theorem
If a line intersects a plane not containing it, then the intersection contains only one point.
This theorem is a direct result of the postulate.
Postulate
Postulate 7. The plane postulate.
any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane.
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Theorems
Theorem. given a line and a point not on the line, there is exactly one plane containing both.
This theorem is a direct result of the postulate.
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Theorem
Given two intersecting lines, there is exactly one plane containing both.
This theorem is a direct result of the postulate.
Some Definitions
Definition.
The interior of BAC is the set of all points P in the plane of BAC such that P and B are on the same side of and P and C are on the same side of . The exterior of BAC is the set of all points of the plane of BAC that lie neither on the angle nor in its interior.
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AB�������������� � AC
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P
A
C
B
Postulate
Postulate 8. The Angle Measurement Postulate.
To every angle BAC there corresponds a real number between 0 and 180.
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Postulate
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Postulate 9. The Angle Construction Postulate.
For any two angles ABC and DEF such that m ABC < m DEF, there is a unique ray such that m ABC = m GEF and EG
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B
A
C
D
G
FE
Postulate 10
Postulate 10. The Angle Addition Postulate.
If D is in the interior of BAC, then
m BAC = m BAD + m DAC
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Some Definitions
Definition.
If and are opposite rays, and is any other ray, then BAC and CAD form a linear pair.
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Definition.
If the sum of the measures of two angles is 180, then the angles are called supplementary, and each is called a supplement of the other.
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Postulate 11
Postulate 11. The Supplement Postulate
If two angles form a linear pair, then they are supplementary.
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Some Definitions
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Definition.
If two angles in a linear pair have the same measure, then each of them is a right angle.
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Definition.
If , then the lines and are called perpendicular.
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Definition.
If and form a right angle, then they are called perpendicular rays, and we write
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Some DefinitionsDefinition.
If the sum of the measures of two angles is 90, then they are called complementary, and each of them is called a complement of the other. An angle with measure less than 90 is called acute. And angle with measure greater than 90 is called obtuse.
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Definition.
Two angles with the same measure are called congruent.
Some Definitions
Definition.
Two angles are vertical angles if their sides form two pairs of opposite rays.
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There are many theorems involved with supplementary, complementary, vertical angles and perpendicularity. Some will be assigned for homework.
Assignment: §2.1