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Axiom 1

Given two points, there exists a unique line containing these two points. For every line there are points which belong to this line and points which do not belong to this line.

Axiom 2

Given three points on a line, only one of them is between the other two points

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Axiom 3

To each segment AB, we can associate a positive real number AB called length of this segment. If B is a point between the points A and C then we assume that AB + BC = AC

Axiom 4

Given a ray →OE and a positive real number α, there is a unique point A on →OE such that OA = α

Axiom 5

Every line in a plane will divide the plane into two half-planes. Any segment with its endpoints in separate half-places will have a point in common with the given line.

Axiom 6

Each angle can be measured with a protractor and will be assigned a degree measure. This will be a unique number between 0° and 180°. Straight angle is 180°. The degree measure of the angle ∠AOB will be denoted by m∠AOB.

Axiom 7

Given ray →AB and one of the half-planes π' determined by the line ↔AB, for any positive number α such that 0<α<180, there is a unique ray →AW, the point A being in π' such that m∠WAB = α°

Axiom 8

Given ΔABC, ray →A'D' and half-plane π' determined by this ray, there is unique triangle ΔA'B'C' congruent to ΔABC such that C' belongs to π' and ray →A'B' equals →A'D'

Median

Line segment joining a vertex to the midpoint of the opposite side

Triangle

Given 3 points not on the same line, a triangle is a figure made of 3 segments having these 3 points as its end points.

Figure

Every subset of points on a plane

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Interior angle

The set of all points between the sides of an angle

Lines & Planes

(nonempty) set of points.

A point may belong to a line or not belong to a line

Segment

Given two points on a line ↔AB, a segment is the set of all points between A and B including A and B

Initial Point

Starting point of a ray

Half Plane

Any line that lies in a plane separates the plane into two half-planes.

Angle

A figure formed by two rays that have the same endpoint.

Common endpoint that is shared between two rays; usually at the point of origin.

Congruence

denoted with symbol = (squiggle on top)

reflexive, symmetric, and transitive relation

Two figures have the same shape/size

midpoint

divides a segment into 2 congruent segments

Vertices and sides of a triangle

Points are called vertices of a triangle

Segments are called sides

Parallel

If two lines do not have a point in common (do not intersect)

Axiom 9

Given line a and point B not on the line, there exists a unique line b passing through point B, and parallel to line a

Conditional Statment

statements in the form if...then...

Continuation rays or Opposite rays

2 rays that share the same end point and extend in opposite directions

supplementary

Two angles that add up to 180°

Acute/Obtuse

0<m∠A<90°

90°<m∠A<180°

Vertical Angles

2 non-adjacent angles formed by 2 intersecting lines.

Perpendicular bisector of a segment

A line, ray, or segment that is perpendicular to a segment at its midpoint creating two equal segments and 90 degree angles

SAS

Given two triangles such that two sides and the angle formed by these two sides of the first triangle are congruent to with the corresponding two sides and angle formed by these two sides of the second triangle, then they are congruent.

ASA

If two given triangles are such that a side and two angles on this side in the first triangle are congruent with the corresponding side and two angles on that side in the second triangle, then they are congruent.

Isosceles

A triangle with 2 sides that are congruent

The non-congruent side is usually called the base

Equilateral

All sides of a given triangle are congruent

Altitude

The perpendicular, dropped from a given vertex to the line determined by the side opposite to this vertex

angle bisector

the ray that divides an angle into two, congruent, adjacent parts

SSS

If 3 sides of one triangle are congruent with corresponding 3 sides of another triangle, then the triangles are congruent

Proof by Contradiction

Proof of a conditional statement P => Q by verifying that P & (not Q) is always a false statement

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