CHAPTER 1
3 undefined terms:
point – a location that has neither shape nor size
line – made up of points and has no thickness or width
plane – a flat surface made up of points that extends infinitely in all directions
Collinear points – points that lie on the same line
Coplanar points – points that lie in the same plane
Intersection of two or more geometric figures – the set of points they have in
common
Space – a boundless, three-dimensional set of all points [space can contain lines and
planes]
Dimension
- a points has no dimension
- a line has one dimension
- a plane, circle, square is two dimensional
- a pyramid, prism [box] is three dimensional
locus – a set of points that satisfy a particular condition
line segment (segment) – a measurable part of a line that consists of two points,
called endpoints, and all of the points between them
Betweenness of Points
Point M is between points P and Q if and only if
P, Q, and M are collinear and
PM + MQ = PQ
Congruent Segments – congruent segments have the same measure
Midpoint of a segment – the point halfway between the endpoints of a segment. If X
is the midpoint of AB, then AX = XB.
Midpoint Formula (in Coordinate Plane)
Segment bisector – any segment, line, or plane that intersects a segment at its
midpoint
Ray – a part of a line that has one endpoint and extends indefinitely in one direction
Opposite rays – two rays that share a common endpoint and form a line
Angle – formed by two noncollinear rays that have a common endpoint
[the rays are the sides of the angle and the common endpoint is the vertex of the
angle]
acute angle – an angle with a measure less than 900
right angle – an angle with a measure of 900
obtuse angle – an angle with a measure greater than 900 and less than 1800
straight angle – an angle with a measure of 1800
angle bisector – a ray that divides an angle into two congruent angles
adjacent angles – two angles that lie in the same plane and have a common vertex
and a common side, but no common interior points [they do not overlap]
linear pair – a pair of adjacent angles with noncommon sides that are opposite rays
[linear pair form a straight angle]
vertical angles – two nonadjacent angles formed by two intersecting lines
complementary angles – two angles with measures that have a sum of 900
[an angle and its complement equal to 900]
supplementary angles – two angles with measures that have a sum of 1800
[an angle and its supplement equal to 1800]
perpendicular - lines, segments or rays that form a right angle
CHAPTER 2
Inductive reasoning – reasoning that uses a number of specific examples to arrive at
a conclusion.
Conjecture – a concluding statement reached using inductive reasoning
Counterexample – false example, used to prove a conjecture is false
Statement – a sentence that is either true or false
Compound statement – two or more statements joined by the word “and” or “or”
-
conjunction – compound statement using the word “and” [notation: p q ]
-
disjunction – compound statement using the word “or” [notation: p q ]
negation – a statement that has the opposite meaning and truth value of an original
statement [ notation: ~p, read “not p”]
conditional statement – a statement that can be written in if-then form.
Related conditionals
Converse – formed by exchanging the hypothesis and conclusion of the conditional
statement [notation: q p ]
Inverse – formed by negating both the hypothesis and conclusion of the conditional
statement [notation: ~ p ~q ]
Contrapositive – formed by negating both the hypothesis and the conclusion of the
converse of the conditional [notation: ~ q ~p ]
Biconditional – the conjunction of a conditional and its converse
Deductive reasoning – uses facts, rules, definitions, or properties to reach logical
conclusions from given statements.
Valid methods for proving a conjecture
LAW OF DETACHMENT
If p q is a true statement and p is true, then q is true
LAW OF SYLLOGISM
If p q and q r are true statements, then p r is a true statement
Postulate or axiom – a statement that is accepted as true without proof
Proof – a logical argument in which each statement you make is supported by a
statement that is accepted as true
Algebraic proof – a proof that is made up of a series of algebraic statements
Two-column proof – contains statements and reasons organized in two columns
0
