Lesson 1.1 Sets, Statements, and Reasons
Statement: A statement is a set of words or symbols that make a claim that can be classified as
true or false.
Negation: The negation of a statement P makes a claim opposite of the original statement
(negation of the statement). In symbols this would be represented ~P.
Conjunction: P and Q (P is true and Q is true). A conjunction is true only when both statements
Disjunction: P or Q (P is true or Q is true). A disjunction is true when either of the statements is
Conditional statement (implication): If P, then Q. P Q“P implies Q.” A conditional
statement consists of two clauses, one of which begins with the word “if” or “when” or some
Hypothesis: In the conditional statement P, then Q,” P is the hypothesis. The hypothesis is
the “if” clause.
Conclusion: In the conditional statement “If P, then Q,” Q is the conclusion. The conclusion is
the “then” clause.
Inductive reasoning: Inductive reasoning uses experiments and observations to draw
conclusions. Experiments and observations form and confirm hypotheses. Inductive reasoning is
the method for drawing conclusions from a limited set of observations.
Deductive reasoning: Deductive reasoning uses formal argument to prove a tested theory.
Deductive reasoning uses logic to draw conclusions from statements accepted as true.
Intuition: Intuition involves statements made without applying a formal reasoning.
Deduction: Deduction is the type of reasoning in which the knowledge and acceptance of
selected assumptions guarantee the truth of a particular conclusion.
Premises: Premises are statements that are treated as facts in a valid argument (direct proof).
Conclusion: A conclusion is what must follow on the basis of the premises.
Law of detachment (chains of reason—syllogisms) On the basis of premises, a particular
conclusion must follow. A syllogism is an argument of the form,
If you accept the first two premises, you must accept the conclusion.
Counterexample: a counterexample disproves the validity of an argument
Example of a valid argument:
1) If P, then Q
conclusion) therefore () Q
Example of an invalid argument:
1) If P, then Q
conclusion) therefore () P
This is not valid because the premises tell us nothing about what happens if Q is true.
Venn (Euler) diagrams: To make a Venn diagram for a conditional statement, draw two circles,
one inside the other. The interior of the smaller circle represents the hypothesis; the interior of
the larger circle represents the conclusion.
Intersection: The intersection of two sets P and Q is exactly those elements common to both P
and Q. This is represented P Q.
Union: The union of two sets P and Q is the set of elements that are in either P or Q. This is
represented P Q.
1.2 Informal Geometry and Measurement
Point: A point has location, but no size. It is represented by a dot and labeled with upper case
Line: A line is an infinite collection of points that has length but no width. A line extends
infinitely in both directions and is labeled with a lower case letter (usually italicized) or with two
points on the line with arrows on either end.
Plane: A plane is two-dimensional. It has length and width but no thickness.
Collinear: If three points lie on the same line, they are said to be collinear. Points are collinear if
there is a line that contains all of them.
Noncollinear: Points are noncollinear if no single line contains them all.
Concurrent: Lines are concurrent if they contain the same point. Three or more lines that
intersect at one point are concurrent.
Betweeness of points: If one point lies between two others, then the points are collinear. This is
symbolized A-X-B, or B-X-A. This tells us that X is between A and B on AB.
Vertex: The point at which sides meet is called a vertex.
Line segment: A line segment consists of two points and every point between them. A line
segment is part of a line bounded by two endpoints.
Midpoint: If A-B-C and AB BC , then B is the midpoint of AC. The midpoint of a line
segment divides a line segment into two equal segments.
Segment congruence: Segments can be congruent if the lengths of the segments (distances
between endpoints) are equal.
Parallel lines: Parallel lines lie in the same plane and never intersect.
Empty set: The empty set is the set of no elements.
Bisect: To bisect something means to separate it into two equal parts.
Straight angles: Straight angles are angles having a measure of 180
Right angles: Right angles are angles having a measure of 90 .
Perpendicular: Two lines are perpendicular if the intersect to form congruent adjacent angles.
Constructions: Constructions are drawings made
using only a compass and straightedge.
Circle: A circle is the set of all points a given distance (radius) from a given point (center).
Arc: An arc is two points on a circle and all points on the circle between them. An arc is part of
Radius: A radius is a line segment joining the center of a circle to a point on the circle.
Addition and subtraction property of equality: If a b, then a c b c. If
a b, then a c b c.
Copying a segment, Perpendicular bisector construction (midpoint)
1.3 Early definitions and Postulates
Mathematical system: A mathematical system is a system of undefined terms (starting points),
definitions, assumptions, and principles that follow from logic.
Axioms/postulates (assumptions): Properties of a system that are apparent—they do not need to
be proved. A postulate is a statement that is assumed to be true without proof.
Theorems: Theorems follow by logic from the definitions and postulates. Theorems must be
proved. A theorem is a statement that is proved by reasoning deductively from already accepted
Definition: A good definition 1) names the term being defined, 2) places the term into a set or
category, 3) distinguishes it from other terms, 4) it is reversible. Both a b, and b a . This
can be written as a b which means “a, if and only if b.” When we define a word in geometry,
the word and its definition are understood to have exactly the same meaning.
Definition of an isosceles triangle: An isosceles triangle is a
triangle that has two congruent
Definition of a line segment: A line segment is part of a line segment that consists of two points
known as endpoints and all the points between them. A line segment is part of a line bounded
by two endpoints.
Postulate 1) (two points determine a line): Through two distinct points there is exactly one line.
Postulate 2) (ruler postulate): the measure of any line segment is a unique positive number.
Distance between two points: The distance between two points A and B is the length of the line
segment AB that joins the two points.
AB represents line AB
AB represents line segment AB
AB represents the length of segment AB (this distance between A and B).
Postulate 3) (segment addition postulate): If X is a point of AB and A-X-B, then AX XB AB
Congruent segments: Congruent ( ) segments are two segments that have the same length.
Midpoint: The midpoint of a line segment is the point that separates the line segment into two
congruent parts. The midpoint is equidistant from the endpoints of a segment.
Bisector: A bisector is any line, ray, or segment that separates a segment into two congruent
parts. A bisector passes through the midpoint of a segment.
Ray: Ray AB, symbolized by AB, is the union of AB and all points X on AB such that B is
between A and X (A-B-X). A ray is part of a line that extends endlessly in one direction.
Opposite rays: Opposite rays are two rays that share a common endpoint and form a line.
Intersection: The intersection of two figures is the set of all points that the two figures have in
Postulate 4) (two points determine a line): If two lines intersect, they intersect at one point.
Parallel lines: Parallel lines are lines that lie in the same plane and never intersect.
Coplanar: Points are coplanar if there is a plane that contains all of them. Coplanar points are
points that lie in the same plane.
Postulate 5) (three points determine a plane): Through three noncollinear points there is exactly
Space: Space is the set of all possible points.
Postulate 6) (the intersection of two planes is a line): If two distinct planes intersect, then their
intersection is a line.
Postulate 7): Given two distinct points in a plane, the line containing them also lies in the plane.
Therorem 1.3.1: The midpoint of a line segment is unique. (This will be proved later).
1.4 Angles and their relationships
Angle: an angle is the union of two rays that share a common endpoint. The rays are called the
sides of the angle and their common endpoint is called the vertex of the angle.
Postulate 8) (protractor postulate): The measure of an angle is a unique positive number.
Acute angle: An acute angle is an angle whose measure is less than 90 .
Right angle: A right angle is an angle whose measure is exactly 90 .
Obtuse angle: An obtuse angle is an angle whose measure is greater than 90 and less than 180 .
Straight angle: A straight angle is an angle whose sides for opposite rays (a straight line). A
straight angle has a measure of exactly 180 .
Reflex angle: A reflex angle is an angle whose measure is greater than 180 and less than 360 .
Postulate 9) (angle addition postulate): If a point D lies in the interior of ABC , then
mABD mDBC mABC .
Congruent angles: Congruent angles ( ' s ) are two angles with the same measure. Congruent
angles are angles that coincide exactly when superimposed.
Bisector of an angle: The bisector of an angle is the ray that separates the given angle into two
For example, if NP is the angle bisector of MNQ, then mMNP mPNQ. It also
follows that mMNP mMNQ.
Complementary angles: Two angles are complementary if the sum of their measures is 90 .
Each angle in the pair is said to be the complement of the other angle.
Supplementary angles: Two angles are supplementary if the sum of their measures is 180 .
Each angle in the pair is said to be the supplement of the other angle.
Vertical angles: When two lines intersect, the pairs of nonadjacent angles formed are known as
Theorem 1.4.1 (the angle bisector of an angle is unique): There is one, and only one, angle
bisector for a given angle. (This theorem will be proved later).
Copying an angle, Construction of an angle bisector