Geometry Chapter 2

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Reasoning and Proof

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.

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