segment
a straight set of points that has a beginning and an end
rays
a straight set of points that begins at and endpoint and continues forever in one direction
line
a set of straight points that extend forever in both direction
angle
the union of 2 rays with a common endpoint
union (u)
this OR that
intersection (upside down u)
this AND that
collinear
points that lie on the same line
noncollinear
points that do not lie on the same line (3 or more points)
betweeness
the center point in a set of 3 collinear points
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Things to assume
straight lines/angles, collinearity, betweenness, relative position of points
things not to assume
right angle, congruent segments/angles, relative sizes of segments/angles
True or False: Saying B is in between A and C is the same thing as saying B is in the middle of A and C
False, hypothesis not satisfied
Theorem 1
If two angles are right angles, then they are congruent.
Theorem 2
If two angles are straight angles, then they are congruent.
What are the labels for the two columns of a proof?
Statement and reason
bi/tri-sected definition
A (point, segment, ray, or line) that divides a (segment, angle) into (2,3) congruent parts (bisects, trisects) the (segment, angle)
Midpoint definition
A point that divides a segment into two congruent segments is called the midpoint of a segment
Counter-example
an example that shows that something cannot be proven
deductive structure
a system of thought in which conclusions are justified by means of previously assumed or proven statements.
4 parts of deductive structure
undefined terms, postulates, definitions, theorems
Undefined terms
concepts that we agree on but can't define (God, a point)
Postulate
an unproved assumption (The Bible is true, 2 points determine a unique line)
Definitions
states the meaning of a term (right angles, salvation)
Theorems
mathematical statements that can be proved (if, then statements) (if the Bible is true, then man is inherently sinful)
syllogisms
deductive conclusions drawn from two premises
If a player is the MVP, then he scored a goal. Messi was the MVP.
Messi scored a goal.
Conditional statements
If hypothesis, then conclusion
Negation
the opposite of the original statement (~)
Converse
not logically equivalent, switching the order of the if, then statements
Logically equivalent
words are different, meaning is the same
Inverse
If not p, then not q, negating both sides, not LE
Contrapositive
logically equivalent, the converse of the inverse
a ->b, b->c
a->c
Probability
total successes/ total outcomes; good space/winning space/ total space