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reflexive property

a=a (any number is equal to itself)

substitution property

if a=b, then a can be substituted for b in any expression

addition property

if a=b, then a+c = b+c

subtraction property

if a=b, then a-c = b-c

multiplication property

if a=b, then ac = bc

division property

if a=b, and c doesnt equal 0, then a/c = b/c


the points on an infinite number line that are numbered so that to every point there corresponds exactly one real number, and to every real number there corresponds exactly one point.


a real number that illustrates betweenness of points

ruler postulate

the points on a line can be numbered so that positive number differences measure distances.

betweenness of points (definition)

a point is between two other points on the same line iff its coordinate is between their coordinates

the betweenness of points theorem

if A-B-C, then AB + BC = AC


the unit of measuring angles (created by the Babylonians); each is 1/360 of a circle

rotation of rays

all of the positions in which a ray can be of a circle


all of the rays that correspond to a semicircular protractor (aka 180 degrees)

measure of the angle

the positive difference between the coordinates of the rays

protractor postulate

the rays in a half-rotation can be numbered from 0 to 180 so that positive number distances measure angles


iff the angle is less than 90 degrees


iff the angle = 90 degrees


iff the angle is more than 90 degrees but less than 180 degrees


iff the angle = 180 degrees

betweenness of rays (definition)

a ray is between two others in the same half-rotation iff its cordinate is between their coordinates

the betweenness of rays theorem

if OA-OB-OC, then angle AOB + angle BOC = angle AOC

midpoint of a line segment (definition)

a point that divides the line segment into 2 equal segments

bisection of an angle (definition)

a line that divides the angle into two equal parts


"coinciding exactly when superimposed"; divided into two equal parts (which are called this term) e.g. line segments are... when they have equal lengths, or angles are.. when they have equal measures


a theorem that can be easily proved as a consequence of a postulate or another theorem

corollary to the ruler postulate

a line segment has exactly one midpoint

corollary to the ruler postulate

an angle has exactly one ray that bisects it

complimentary angles

iff two angles' sum = 90 degrees


each angle of a complimentary angle to the other, found by subracting the other angle from 90


iff two angles' sum = 180 degrees


each angle of a supplimentary angle to the other, found by subtracting the other angle from 180

theorem of compliments of the same angle

compliments of the same angle are equal

theorem of supplements of the same angle

suppliments of the same angle are equal

opposite rays

when 2 rays point in opposite directions

linear pair

two angles are this iff they have a common side and their other sides are opposite rays

vertical angles

two angles are this iff the sides of one angle are opposite rays of the other

theorem of the angles in a linear pair

the angles in a linear pair are supplimentary

theorem of vertical angles

vertical angles are equal

perpendicular lines (definition)

two lines are this iff they form a right angle

theorem of perpendicular lines

perpendicular lines form four right angles

corollary to the definition of a right angle

all right angles are equal

theorem of the angles in a linear pair being equal

if the angles in a linear pair are equal, then their sides are perpendicular

parallel (definition)

two lines are this iff they lie in the same plane and do not intersect

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