Geometric Postulate
For example, a postulate is a statement accepted without proof. Postulates are the next logical step up
from definitions. Every mathematical system must have postulates. From postulates, defined terms, and
undefined terms, it is possib
Defined Terms
(Defintions)
Every set of 3 points is copanlar
Point B is between A and C if A, B, and
C are collinear and the equation AB +
betweenness of BC = AC is true, where AB, BC, and
points
AC are the distances between points A
and B, B and C, and A
Operations of sets
complement
The elements in the universe that are not in the set.
of a set
Sets that have no
common
disjoint sets
elements, whose
intersection is cfw_.
intersection of sets A set containing elements in common for all sets being considere
Glossary and Credits
betweenness
A condition of one point being between two other points on a line.
A line or segment that intersects a segment at its
bisector
midpoint.
collinear
Occurring on the same line.
coplanar
Occurring on the same plane.
geometry
Using Deductive Reasoning
Model 1:
Specific
Example
2x + 2 = 6
If x = y, then x - a = y - a
2x + 2 - 2 = 6 - 2
a + (-a) = 0, additive inverse 2x + 0 = 6 - 2
a + 0 = a, addition of zero
2x = 6 - 2
Substitution, 4 for 6 - 2
2x = 4
If x = b, then ax = ab
( )
EXTERIOR AND REMOTE INTERIOR ANGLES OF A
TRIANGLE
We will now use some of the statements about parallels to prove some useful theorems
concerning triangles and other polygons.
Objectives
Define exterior and remote interior angles of a triangle
Find the me
Conjunctions
conjunction
A statement formed by combining two statements with the word and
In English, certain words have a special function when they are used to create a logical structure.
Note the role that each of the italicized words plays when combin
CONVERSE, INVERSE, CONTRAPOSITIVE
CONVERSE, INVERSE, CONTRAPOSITIVE
We can do three things to a conditional statement to get new statements
converse of a
conditional
A statement formed by interchanging the hypothesis and the conclusion
in a conditional st
GEOMTRIC THEOREMS
A theorem is a statement that is proved by deductive logic
A theorem is the product of mathematics. If you remember our discussion from the beginning of this
unit, people arrive at these products by "thinking mathematically." They used o
SET THEORY
Near the end of the 19th century, Georg Cantor (1845-1918) was instrumental in working out the
mathematical theory of abstract sets. It is his basic ideas that we briefly studied in 7th and 8th level
math and algebra. The ideas refine the preci
CONDITIONAL OR IMPLICATION STATEMENTS
Where p and q represent statements, the compound statement written in the symbols p
conditional of implication. It means if p , then q and p implies q
conditional or
implication
q is called a
Two statements connected
OTHER POLYGONS
We will now use some of the statements about parallels to prove some useful theorems
concerning triangles and other polygons.
Working with polygons will be easier now that you have studied the properties of triangles and
the theorems relate
Properties of Algebra
a=a
a = b then b = a
a = b and b = c then a= c
Reflexive property
Symmetric property
Transitive property
REMEMBER: Properties of equality make these operations possible:
Addition:
If a = b, then a + c = b + c and c + a = c + b
Subtra
DISJUNCTIONS
In the previous lesson, we listed several ways that statements can be combined. In this lesson, we
will explore some of those ways in some depth.
We can combine two statements in another way by using the word or.
For example:
3 + 2 = 5 or 7 7
Green cement may set CO2 fate in concrete, Carrie Sturrock, S.F.
Chronicle
Stanford Professor Brent Constantz (59) prior medical patents for bone
cement
Waste CO2 + Seawater + Mg = Cement
Calcining Limestone produces CO2
Worldwide, 2.5 billion tons of ce
Structure: Recrystallization = new fabrics
Tectonics: Dehydration reactions release fluids & trigger quakes
Geochronology: Datable minerals hold both parent and daughter isotopes
Petrology: Partition or order/disorder of elements serves as
geothermometer
What Minerologist do:
Crystallography: Forms, symmetry, XRD
Crystal Chemistry: Inorganic, substitution, kinetics of formation
Classification: Composition & Structure, ~50 new minerals a year, ~4000
total
Paragenesis: Geological occurrence, assemblage, set
A mineral is a wondrous thing. At least it is to me,
For in its ordered structure lies a world of mystery.
The secrets that it has withheld for countless ages past,
And clung to most tenaciously and being learned at last.
Each year using new techniques o
Euhedral: good, well formed faces, taking its characteristic crystal form.
E.g. hexagonal quartz prisms or cubic pyrite
Subhedral: Some good faces - some curved
Anhedral: Mineral lacking crystal faces, curved, rounded, embayed,
irregular
Compact: too fin
Physical Properties of Minerals
(Interplay with light)
Luminescence
Lustre
Magnetism
Parting
Phosphorescence
Piezo-, Pyroelectricity
Play of colours
Radioactivity
Tenacity
Streak
Asterism
Crystal form
Crystal Habit
Chatoyancy
Cleavage
Colour
Density (S.G.