Introduction to Neutral Geometry
( Sections 3.1, and 3.2 )
Neutral Geometry is the axiom system with the same undefined terms as those of Euclidean Geometry but using the SMSG Postulates 1 15 as the only axioms. One can think of Ne utral Geometry as the g
THE CONVERSE OF THE A I A THEOREM IS FALSE IN THE POINCARE UPPER HALF-PLANE AND FALSE IN HYPERBOLIC GEOMETRY!
Lines and m are parallel and line t is a transversal.
The fact that these pairs of alternate interior angles are NOT CONGRUENT shows that the CON
Proofs of Some EPP Equivalences (in Neutral Geometry)
Recall the correct wording of the EPP (Euclidean Parallel Postulate): For every line l and every point P not on line l , there is one and only one line m that contains P and is parallel to l . Note tha
More on Neutral Geometry II (Including Sections 3.4 and 3.5) Section 3.4: The Place of Parallels Theorem 3.4.1 The "Alternate Interior Angles (AIA)" Theorem or also the "Congruent Alternate Interior Angles Make Parallel Lines" Theorem: Two lines may or ma
More on Neutral Geometry I (Including Section 3.3) ( "NIB" means "NOT IN BOOK" ) Theorem (NIB), The "The Adjacent Supplementary Angles" Theorem (Converse of Postulate 14) : If two adjacent angles are supplementary, then they form a linear pair. Proof: Sup
Theorem 4.4.8 (The Pythagorean Theorem): In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Proof: Given: b
C a [ NTS: c 2 = a 2 + b 2 ]
A
c
B
By the (NIB) "Drop a Perp
Triangle Similarity Problems
Problem 1: Given Q G: J Q 15 5 7 G P 4 R GHJ are similar triangles, 21 H ? PQR and GHJ as shown such that
Show that PQR and and determine JH . Solution: JG = PQ PQR
15 GJ = =3; 5 PQ GH QR = 3 , and
GH 21 = =3. QR 7 PQR JGH (g
The Basic Proportionality Theorem Initial Results:
All triangles with a common base and common altitude have EQUAL AREAS . Common base =
AB = 6.60 inches CH = 2.93 inches Area ABC = 9.65 inches 2 Area ABD = 9.65 inches 2 1 2 AB CH = 9.65 inches 2
Common a
The Area Concept , Similarity in Triangles, and Applications of the Basic Proportionality Theorem Section 4.3: The Area Concept (SMSG Postulates 17 20)
The three properties which are necessary for all area measure assignments: 1) Every polygon is assigned
Notes on Median Concurrence
Recall that we are working within Euclidean Geometry. Therefore, the following statements are all true: 1. The Euclidean Parallel Postulate is true: Given any line l and any point P not on line l, there is one and only one line
Introduction to Euclidean Geometry (Section 4.2) We now include the Euclidean Parallel Postulate as an axiom along with SMSG Postulates 1 15: SMSG Postulate 16: (The Euclidean Parallel Postulate) Through a given external point, there is at most one line p
The "Common Perpendicular" Theorem In the axiom system being developed, the "Common Perpendicular" Theorem follows the "Drop a Perpendicular" Theorem immediately. Theorem (NIB), The "Common Perpendicular" Theorem: Given any line l and any point P not on l
BUS030: Personal Finance | Assignment
Assignment 2
Instructions
Save the file in your course folder, and name it with Assignment, the section number, and your first initial and last
name. For example, Jessie Robinson's assignment 1R for Section 1 would be
The Three Alternatives for a Parallel Postulate (P.P.)
Given a line l and a point P not on line l, . . . Angle Sum Alternative Postulate Completion Alt 1) there is no line through P and parallel to l . Type Elliptic P.P. in a Triangle Greater than 180 A M
THREE FLAWS
in the Axiom System in Euclid's "Elements"
1) Euclid did not begin with a set of undefined terms.
2) Some statements, not listed postulates, were assumed to be true without proof.
3) Euclid sometimes used diagrams or drawings to substitute for
Congruences between Two Triangles
"Congruence" is a relationship that two geometric objects of the same type may or may not have. Two segments and are congruent segments ( have the same measure (that is, the same length). Two angles and are congruent angl