DEFINITIONS, POSTULATES, AND THEOREMS
A definition uses known words to describe a new word. Point, Line, and Plane must be commonly
understood without being defined.
Undefined terms:
Point- understood to be a dot that represents a location in a plane or in space.
Line-understood to be straight, contains infinitely many points, extends infinitely in 2 directions, and has
no thickness
Plane-understood to be a flat surface that extends infinitely in all directions.
Definition – tells meaning of term. Definitions are always biconditional.
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) Plane Intersection Postulate- If 2 planes intersect, then they intersect at a line.
9) Congruent- same size and shape
10) Similar- same shape does not have to be same size
11) Segment (def)- a straight path from one point to another.
12) Ray (def)- an endless straight path from a 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)- Acute angle < 90
26) Obtuse Angle (def)- Obtuse angle > 90
27) Straight Angles (def)- Straight angle = 180.
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) Intersecting Lines (def)- are coplanar and have exactly one point in common. If intersecting lines do
not meet at right <s they are oblique.
32) Parallel Lines (def)- are lines that are coplanar and do not intersect. (same slope)
33) Skew lines (def)- two lines that do not lie in the same plane. Noncoplanar lines that do not intersect.
34) Perpendicular Lines (def)- lines that intersect to form a right angle. (negative reciprocal slopes)
2 lines   they form 90º  ’s.
1)
2)
3)
4)
35) Midpoint of a Segment (def)- A point is a midpoint  it divides a segment into 2  segments.
36) Bisector of a Segment (def)- is a set of points whose intersection with the segment is the midpoint of
the segment.
37) Perpendicular Bisector of a Segment (def)- A line is a perpendicular bisector  it is perpendicular
to the segment and goes through the segments midpoint.
38) Angle Bisector (def)- A ray is an  bisector  it divides an  into 2   ’s.
39) Vertical Angles (def)- 2  ’s are vertical  they are nonadjacent  ’s formed by intersecting lines.
40) Linear Pair of Angles (def)- 2  ’s are a linear pair  they are adjacent  ’s whose noncommon
sides are opposite rays.
41) Complementary Angles (def)- 2  ’s. are complementary  their sum is 90º.
42) Supplementary Angles (def)- 2  ’s. are supplementary  their sum is 180º.
43) Linear Pair Theorem- If 2  ’s that form a linear pair  they are supplementary.
44) Congruent Supplements Theorem- If 2  ’s are supplementary to the same  or   's  they are
.
45) Congruent Complements Theorem- If 2  ’s are complementary to the same  or   's  they are
.
46) Vertical Angles Theorem- If 2  ’s are vertical  they are  .
47) Def. of congruent angles or segments – If two angles or segments are congruent, then they have equal
measure.
48) All right  ’s are  .
49) Congruent Complements Theorem - If 2  ’s are complementary to the same  or   's  they are
.
50) Congruent Supplements Theorem - If 2  ’s are supplementary to the same  or   's  they are
.
51) If 2   's are supplementary  they are right  ’s.
52) Common Segments Theorem - If 2 segments are formed by a pair of  segments and a shared
segment  the resulting segments are  .
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)
Division Property If a = b and c = 0, then a / c = b / c. (divide both sides by the same thing)
Reflexive Property For and real number a, a = a.
Symmetric Property If a = b, then b = 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  x2 y1  y2
,
)
2
2