DEFINITIONS, POSTULATES, AND THEOREMS
Definition – tells the meaning of a term. Definitions are always biconditional (reversible).
Postulate – rules that are accepted as being true.
Theorems – rules which must be proven true.
Points Postulate – a line contains at least 2 points; a plane contains at least 3 non-collinear points;
space contains at least 4 non-collinear, non-coplanar points.
5) Line Postulate – 2 points are contained in one and only one line.
6) Plane Postulate – 3 non-collinear points are contained in one and only one plane.
7) Flat Plane Postulate – If 2 points are contained in a plane, then the line through them is contained in
the same plane.
8) Line Intersection Postulate – If 2 lines intersect, then they intersect in exactly one point.
9) Plane Intersection Postulate – If 2 planes intersect, then they intersect at a line.
10) Congruent – same size and shape.
11) Segment (def) – a straight path from one point to another.
12) Ray (def) – an endless straight path from one starting point.
13) Opposite Rays (def) – 2 rays that share a common endpoint to form a line.
14) Space (def) – the set of all points.
15) Collinear Points (def) – points that are contained in one line.
16) Non-collinear Points (def) – points NOT contained in the same line.
17) Coplanar Points (def) – points that are contained in the same plane.
18) Non-coplanar Points (def) – points NOT contained in the same plane.
19) Coplanar Lines (def) – lines that are contained in the same plane.
20) Non-coplanar Lines (def) - lines NOT contained in the same plane.
21) Angle (def) – Is the union of two non-collinear rays which have the same endpoint.
22) Interior of an Angle (def) – inside of the angle.
23) Exterior of an Angle (def) – outside of the angle.
24) Right Angle (def) – an  is right  its measure is 90º.
25) Acute Angle (def) – an angle whose measure is < 90º.
26) Obtuse Angle (def) – an angle whose measure is > 90º.
27) Straight Angles (def) – an angle whose measure = 180 º.
28) Segment Addition Postulate – If point P is between points A and B, then AP + PB = AB.
Sum of parts equal whole.
29) Angle Addition Postulate- If B is in the interior of  APC, then m  APB + m  BPC = m  APC.
Sum of parts equal whole.
30) Adjacent Angles (def) – two coplanar angles with a common side and no common interior points.
31) Perpendicular Lines (def) – lines that intersect to form a right angle. (have negative reciprocal slopes)
2 lines   they form 90º  s.
32) Midpoint of a Segment (def) – a point is a midpoint  it divides a segment into 2  segments.
33) Bisector of a Segment (def) – a set of points whose intersection with the segment is the midpoint of
the segment.
34) Perpendicular Bisector of a Segment (def) – a line is a perpendicular bisector  it is perpendicular
to the segment and goes through the segment’s midpoint.
35) Angle Bisector (def) – a ray is an  bisector  it divides an  into 2   s.
36) Vertical Angles (def) – 2  s are vertical  they are nonadjacent  s formed by intersecting lines.
37) Linear Pair of Angles (def) – 2  s are a linear pair  they are adjacent  s whose noncommon sides
are opposite rays.
38) Complementary Angles (def) – 2  s are complementary  the sum of their measures is 90º.
39) Supplementary Angles (def) – 2  s are supplementary  the sum of their measures is 180º.
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2)
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40) Linear Pair Theorem – If 2  s form a linear pair  they are supplementary.
41) Vertical Angles Theorem – If 2  s are vertical  they are  .
42) Def. of Congruence – If two angles or segments are  , then they have = measures or lengths.
43) All right  s are  .
44) If 2   s are supplementary  they are right  s.
PROPERTIES FROM ALGEBRA:
Let a, b, and c be real numbers.
Addition Property – If a = b, then a + c = b + c. (add same thing to both sides of an equation)
Subtraction Property – If a = b, then a – c = b – c. (subtract same thing from both sides of an equation)
Multiplication Property – If a = b, then ac = bc. (multiply both sides by same thing)
a b
Division Property – If a = b and c ≠ 0, then  . (divide both sides by the same thing)
c c
Reflexive Property – For any real number a, a = a.
Transitive Property – If a = b and b = c, then a = c.
Substitution Property – If a = b, then a may be substituted for b in any equation or expression.
Distributive Property – If a(b + c), then ab + ac.
Formulas to Know:
Distance
d  ( x2  x1 )2  ( y2  y1 )2
Midpoint
 x1  x 2 y1  y 2 
,


2 
 2