Vocab. Flashcards
axiomatic system parts
1) Undefined terms
2) Definitions
3) Axioms
4) Theorems
5) Proofs
undefined terms (In an Axiomatic System)
A minimal number of technical words that will be used in the subject that we will not explicitly define.
definitions (In an Axiomatic System)
Used to allow statements to be made concisely and to identify and highlight key structures and concepts.
axioms (In an Axiomatic System)
Used interchangeably with postulate, they are statements that are accepted without proof. They are where the subject begins and everything else in the system should be logically deduced from them.
theorem/proof
Can be used synonymously, they are statements that are proved using the axioms. The 1st proof uses only the axioms, subsequent proofs use the axioms and previous proofs.
interpretation (In an Axiomatic System)
Also called a model, is a particular way of giving meaning to the undefined terms in the system.
independent (In an Axiomatic System)
A statement that is impossible to either prove or disprove as a logical consequence of the axioms. A good way to show that a statement is independent of the axioms is to exhibit one model for the system in which the statement is true and another model in which it is false.
consistent (In an Axiomatic System)
No logical contradiction can be derived from the axioms in the axiomatic system.
incident
Used to mean that same thing as “lie on”. As in, “P lies on l” or “P is incident with l”.
interpretation (In an Axiomatic System)
Also called a model - A particular way of giving meaning to the undefined terms in that system.
collinear
Three points that lie on the same line.
finite geometry
A geometry that contains only a finite number of points.
parallel lines
The lines do not intersect.
statement
An assertion that can be classified as true or false.
propositional function
A function whose domain consists of x numbers and whose range consist of the values “True” and “False”.
conditional statement
A compound statement of the form “If……, then…..” in which the first set of dots represents a statement called the hypothesis (or antecedent) and the second set of dots represents a statement called the conclusion (or consequent).
theorem
A conditional statement that has been proved true.
Not (P then Q)
P is true while Q is false.
converse of P then Q
Q then P
contrapositive of P then Q
Not Q then not P.
biconditional statement
“if and only if” - The statement and its converse are true.
existential quantifier
Something exists.
universal quantifier
Holds for all objects in a certain class.
negation of a quantifier
Negation interchanges the two quantifiers.
unique
Often used in connection with the existential quantifier.
Justifications for proof steps
1) by hypothesis
2) by axiom
3) by previous theorem
4) by definition
5) by an earlier step in this proof
6) by one of the rules of logic
(The reasons should be written in parenthesis after the statement)
indirect proof
Also called proof by contradiction - For proofs in the form P then Q, you assume P is true and Q is false and then show that this leads to a logical contradiction. This allows you more information to work with because you assume both P and not Q.
contrapositive proof
not Q then not P
quantifiers
One of the distinctions that must be made clear is whether you are asserting that every object of a certain type satisfies a condition or whether you are simply asserting that there is one that does.
existential quantifier
Asserts that something exists
universal quantifier
Asserts that some property holds for all objects in a certain class.
unique
Used to indicate that there is exactly one object the satisfies a particular condition.
neutral geometry
Five undefined terms (point, line, distance, half-plane, angle measure. Six Axioms: 1) The Existence Postulate 2) The Incidence Postulate 3) The Ruler Postulate 4) The Plane Separation Postulate 5) The Protractor Postulate 6) The Side-Angle-Side Postulate Does not make any assumptions regarding which of the competing Parallel Postulates are true.
plane geometry
two-dimensional geometry
Axiom 3.1.1 (The Existence Postulate)
The collection or all points forms a nonempty set. There is more than one point in that set.
The Side-Angle-Side Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
congruent triangles
The assertion that two triangles are congruent is really an assertion that there are six congruencies, three angle congruencies and three segment congruencies.
superposition
Euclid’s proof of Proposition 4 relied on the ability to move one object so that is lies on top of another object without distorting the object you are moving. This does not hold true in all geometries.
isosceles triangle
A triangle with a pair of congruent sides.
base angles
The two angles of an isosceles triangle not included between the congruent sides.
Postulate I
Given any two points, you can draw a line.
Postulate II
A line can be extended indefinitely.
Postulate III
To draw a circle, you need only know the center and the radius.
Postulate IV
Right angles are congruent
Postulate V
The parallel postulate - Lines are parallel if a straight line draw through both form right angles with both. Additionally, lines are parallel if when extended the lines do not intersect, or there does not exist a point P such that P lies on both lines.
Incidence Axiom 1
For every pair of distinct points P and Q there exists exactly one line t such that both P and Q lien on t.
Incidence Axiom 2
For every line t there exist at least two distinct points P and Q such that both P and Q lie on t.
Incidence Axiom 3
There exist three points that do not lie on any one line.
rhombus
A quadrilateral in which all four sides have equal lengths.
rectangle
A quadrilateral in which all for angles have equal measures (90 degrees).
translation
A transformation that moves all of the points of a figure the same distance in the same direction.
reflection
A transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane.
rotation
A figure can be turned around a point.
dilation
A figure can be enlarged or reduced.
isometry
A congruent transformation.
line of symmetry
Some figures can be folded so that the two halves match exactly. The fold is a line called the line of symmetry.
point of symmetry
A common point of reflection.
grid reflection
A composition of a translation and a reflection in a line parallel to the direction of translation.
µ
Used to indicate the measure of an angle.
exterior angles
The angle which forms a linear pair with and interior angle at one vertex of a triangle.
remote interior angles
The two interior angles at the other two vertices from the exterior angle in question.
foot
The point at which a perpendicular intersects a line.
linear pair
In order to be a linear pair, two angles must share a side. If two angles for a linear pair, then they are supplements.
transformational perspective
The view that the essence of any geometry is captured by the transformations that preserve the structures of the geometry.
isometry
A transformation that preserves distances.
transformation
A function T: P →P that is both one-to-one and onto.
identity function
Is an isometry.
dilation
The dilation with center O and constant k (OP’ = k*OP):
Each point is either moved toward O or away from O, depending on whether k is less than or greater than 1.
composite transformation
When one transformation is combined with another.
fixed point of a transformation
A point than when the transformation is applied, results in the same point. T(P) = P.
quadrilateral
The union of four segments AB, BC, CD, and DA. The four segments are called sides and the points are called vertices of the quadrilateral.
congruent quadrilateral
There is a correspondence between their vertices so that all four corresponding sides are congruent and all four corresponding angles are congruent.
diagonals of the quadrilateral
Segments AC and BD
convex quadrilateral
Each vertex of the quadrilateral is contained in the interior of the angle formed by the other three vertices.
defect of a triangle
defect = 180 - (the sum of all the angles of the triangle)
By the Saccheri-Legendre Theorem, the defect of every triangle is nonnegative.
glide reflection
- A translation maps P onto P’.
- A reflection in a line k parallel to the
direction of the translation maps P’ onto P’’.
Inversion in a circle
A method to convert geometric figures into other geometric figures. It is similar to reflection across a line:
1) Any figure can be reflected across a line or inverted in a circle.
2) Reflecting a figure across the same line twice returns it to its original form. The same is true for inversion in a circle.
3) Reflection takes points to the other side of the line; inversion takes points to the “other side” of the circle. In other words, points inside are inverted to the outside and vice-versa.
4) There is a fairly easy mathematical relationship between a figure and its reflection or between a figure and its inversion.
5) Sometimes it is much easier to work with the reflected version or the inverted version of a figure.
The basic definition of inversion of a point in a circle
If k is a circle with center O and radius r, and P is any point other than O, then the point P’ is the inversion of P if:
1) P’ lies on the ray OP.
2) |OP|*|OP’|=r^2.
