Definitions, Theorems, Axioms, Postulates, and Properties Flashcards
theorem
properties/facts that have been established with proof
postulate/property/axiom
properties/facts accepted without proof
Line Axiom
There are infinitely many points A, B, C,…, and infinitely many lines l, m, n, … .
Two points determine a line:
(1) Given any two distinct points, exactly one line contains them both.
(2) Each line contains at least two points.
Given a line, there exists a point not on the line.
The Distance Assignment Postulate
(i) To every pair of distinct points there corresponds a unique positive number. The number is called the distance between the two points.
(ii) The distance between two points is zero if and only if the two points are not distinct.
The Segment Existence Postulate
Given ray XY and line segment AB, there exists exactly one point P on ray XY such that line segment XP is congruent to line segment AB.
Line Segment Extension Postulate
Given any 2 distinct points A and B, there exists a point C such that A, B, and C are collinear and C is not between A and B.
Angle Measure Assignment Postulate
To every angle there corresponds a unique real number between 0 and 180. This number is called its measure.
Angle Existence Postulate
Given ray XY, a point P on one side of line XY, and a real number k between 0 and 190, there exists exactly one ray, XP, such that the measurement of angle PXY is equal to k.
The Partition Postulate
A whole is equal to the sum of its parts.
(1) If point B is between points A and C, then AB + BC = AC.
(2) If P lies in the interior of angle ABC, then the measurement of angle ABP plus the measurement of angle PBC is equal to the measurement of angle ABC.
Some More Existence Postulates
a) . Each line segment has a unique midpoint.
b) . Each angle has a unique angle bisector.
The Addition Postulate of Algebra
If equals are added to equals then their sums are equal.
Substitution Postulate
We may replace an expression with an equivalent expression in any statement.
The Addition Theorem of Congruence
If congruent segments (angles) are added to congruent segments (angles) then their sums are congruent.
The Subtraction Postulate of Algebra
If equals are subtracted from equals then their differences are equal.
The Subtraction Theorem of Congruence
If congruent segments (angles) are subtracted from congruent segments (angles) then their differences are congruent.
The Transitive Property for Congruence
If two line segments (angles) are congruent to the same line segment (angle) then they are congruent to each other.
The Reflexive Property for Congruence
A line segment (angle) is congruent to itself.
The Symmetric Property for Congruence
A congruence is reversible.
Note: The symmetric property differs from Reversibility of Notation, which tells us that line segment SW and line segment WS name the same segment.
The Multiplication Postulate of Algebra
If a = b then ca = cb.
The Division Postulate of Algebra
If a = b then a / c = b / c (provided that c is not equal to zero).
The Division Theorem for Geometry
Quotients of congruences are congruent.
Special Cases:
1) . Halves of congruent line segments (angles) are congruent.
2) . Thirds of congruent line segments (angles) are congruent.
The Multiplication Theorem for Geometry
Multiples of congruence are congruent.
Special Cases:
1) . Doubles of congruent line segments (angles) are congruent.
2) . Triples of congruent line segments (angles) are congruent.
“First Theorem”
Two distinct lines intersect n at most one point.
“The Theorems”
- If two angles are right angles then they are congruent.
- If two angles are complementary to the same angle then they are congruent.
- If two angles are supplementary to the same angle then they are congruent.
- If two angles are complementary to congruent angles then they are congruent.
- If two angles are supplementary to congruent angles then they are congruent.
Adjacent Angles
Adjacent angles are two angles in the same plane that share a common vertex and a common ray but share no other points.
Linear Pair
Angles that form a linear pair are two adjacent angles such that their non-common rays are mutually opposite.
“Another Postulate”
If 2 angles form a linear pair then they are supplementary.
Vertical Pair Theorem
If two angles form a vertical pair, then they are congruent.
Definition Reversed: Two angles form a vertical pair if the rays forming one angle are the opposite rays of the other.
Note: This does not mean that the angles form a linear pair.
First Theorem Based on the Transitive Property
If two angles are congruent to congruent angles then they are congruent to each other.
Second Theorem Based on the Transitive Property
If two line segments are congruent to congruent line segments then they re congruent to each other.
Undefined Terms:
point, line, plane, half-plane, between, distance, measure and area
open sentence
a sentence with at least one undetermined pronoun and has no truth value that can be specified
closed sentence
a sentence that has exactly one truth value
collinear
If three or more points all belong to the same line they are said to be collinear.
concurrent
If three or more lines contain the same point they are said to be concurrent.
same side
Points B and C are said to be on the same side of point A if A, B, and C are distinct collinear points and point A is not between points B and C.
line segment
A line segment is a set of two points of a line and all of the points between them.
ray
A ray is a set of points consisting of a fixed point of a line and all the points of the line on the same side of the fixed point.
opposite rays
Opposite rays are 2 rays of the same line that have a common endpoint and no other point in common.
angle
An angle is a set of points consisting of the union of 2 distinct rays with a common endpoint.
interior
A point P lies in the interior of an angle if there exist two points, one on each ray, neither at the endpoint such that the point P lies not the line segment joining the two points.
congruent
same shape, same size
similar
same shape, different size
congruent line segments
Congruent line segments are line segments that are equal in measure.
congruent angles
Congruent angles are angles that are equal in measure
midpoint
The midpoint of a line segment is the point of the line segment that forms two congruent segments.
line segment bisector
A line segment bisector is a line, line segment, or ray, that intersects the line segment at its midpoint. There are infinitely many.
intersect
Two lines are said to intersect if they have a point in common.
perpendicular lines
Perpendicular lines are two lines that intersect and form a right angle.
angle bisector
An angle bisector is a ray whose endpoint is the vertex of the angle and that forms two congruent angles
