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51 terms

Geometry Chapter 2

Reasoning and Proof
STUDY
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Points
Through any two _______ there exists exactly one line.
line
A _____ contains at least two points.
exactly one
If two lines intersect, then their intersection is ______ _______ point.
three
Through any _______ noncollinear points there exists exactly one plane.
non collinear
A plane contains at least three ___________ points.
plane
If two points lie in a plane, then the line containing them lies in the ________.
planes
If two _________ intersect, then their intersection is a line.
Addition Property
If a=b, then a+c=b+c.
Subtraction Property
If a=b, then a-c=b-c.
Multiplication Property
If a=b, then ac=bc.
Division Property
If a=b and c≠0, then a/c=b/c.
Substitution Property
If a=b, then a can be substituted for b in any equation or expression.
Distributive Property
a(b+c)=ab+ac, where a, b, and c are real numbers.
Reflexive Property
--Real Numbers--For any real number a, a=a.
--Segment Lengths--For any segment, line segment AB, AB=AB
--Angle Measure-- For any angle <A, m<A=m<A
Symmetric Property
--Real Numbers-- For any real numbers a and b, if a=b, then b=a
--Segment Length-- For any segments _-AB-_ and _-CD-_, if AB=CD, then CD=AB
--Angle Measure-- For any angles <A and <B, if m<A=m<B, then m<B=m<A
Transitive Property
--Real Numbers-- For any real numbers a, b, and c,if a=b and b=c, then a=c
--Segment Length-- For any segments _-AB-_, _-CD-_, and _-EF-_, if AB=CD, and CD=EF, then AB=EF.
--Angle Measure-- For any angles <A, <B, and <C, if m<A=m<B and m<B=m<C, then m<A=m<C
Conjecture
An unproven statement that is based on observations.

EXAMPLE: Conjecture: All prime numbers are odd.
Educated guess
Inductive Reasoning
The process of arriving at a conclusion based on a set of observations or examples

EXAMPLE: Given the number pattern 1,5,9,13,..., you can use inductive reasoning to determine that the next number in the pattern in 17.
Counterexample
A specific case that shows a conjecture is false.

EXAMPLE: All prime numbers are odd.
Counterexample: 2, a prime number that is not odd.
Conditional Statement
A logical statement that has two parts, if/then
Hypothesis
It's what logically comes first after the "if". The "p" of a conditional statement.
Conclusion
It is what logically comes second after the "then". The "q" of a conditional statement.
Negation
Opposite ( ~ )
Converse
If q, then p.
Inverse
If ~p, then ~q.
Contrapositive
If ~q, then ~p.
Logically Equivalent Statements
Two statements that are both True or both False. They both have the same truth value.
Perpendicular Lines
Lines that intersect at 90 degrees
Angle Congruence Postulate
Two angles are congruent if and only if their measures are equal.
Segment Congruence Postulate
Two segments are congruent if and only if they have the same length.
Supplement Theorem
If two angles form a linear pair, then they are supplementary.
Ray
One endpoint and all the points of the line on the one side of the endpoint. EXAMPLE: Ray AB (The endpoint is always listed first.)
Opposite Rays
Two collinear rays with the same endpoint. EXAMPLE: Ray AB and Ray AC (The endpoints are the same, ray go in opposite directions.)
Definition of Right Angles
If Angle 1 is a right angle
then m Angle 1= 90 degrees
Definition of Perpendicular Lines
If Line l is perpendicular to line k,
then m<1=m<2=m<3=m<4=90
Definition of Supplementary Angles
If m<1+m<2 = 180
then <1 is supp. to <2
Definition of Complementary Angles
If m<1+m<2 = 90
then, <1 is comp. to <2
Definition of Congruent Angles
If <1 is congruent to <2
then m<1=m<2
Definition of Segment Bisector
If AB bisects XY at A
then XA=AY
Definition of Angle Bisector
If BD bisects <ABC
then m<ABD=m<DBC
Definition of a Midpoint
If M is the midpoint of AB
then AM=MB
Two angles supplementary to the same angle or to congruent angles are congruent
If <1 supp. <2 and <2 supp. <3
then <1 is congruent <3
Two angles complementary to the same angle or to congruent angles are congruent
If <1 comp. <2 and <2 comp. <3
then <1 is congruent <3
All right angles are congruent
If <1 and <2 are right angles
then <1 is congruent <2
Vertical Angles are Congruent
If Angle 1 is opposite to angle 2
then <1 is congruent <2
Angle Addition Postulate
m<ABD+m<DBC= m<ABC
Segment Addition Postulate
XY+YZ=XZ
Linear Pair Postulate
If <1 and <2 are adjacent and supplementary
then m<1+m<2=180
Theorem
A statement that must be proved using undefined terms, definitions, postulates or other proven theorems
Postulate
A statement that is widely accepted without proof.
Deductive Reasoning
The process by which a person makes conclusions based on previously known facts, definitions, or rules.