Chapter 2 Reasoning and Proof

__
2.1 Inductive
Reasoning and Conjecture__

**Conjecture--**an educated guess based on known
information.

**Inductive Reasoning**--reasoning that uses a number of
specific examples to make a prediction.

__Examples:__ Predict
the next number in the pattern

1 3 6 10 15.....

-8 -5 -2 1 4.....

-4 12 -36 108...

**Counterexample**--an individual example that shows that a
conjecture is false.

__Example:__

Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: Points A, B and C

Conjecture: Points A, B and C form a triangle

__Example:__

Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: <1 and <2 are complementary. <1 and <3 are also complementary.

Conjecture: m<2 = m<3

__Example:__

Make a conjecture based on the given information.

Given: ray BD is an angle bisector of <ABC

Conjecture:

**Conditional Statement**has
two parts, **hypothesis** and **
conclusion**.

Written in * if-then* form

**Hypothesis** follows the * if*

**Conclusion** follows the * then*

Example: *
If *m<A = 90,

**Converse of a Conditional Statement**is written by
switching the hypothesis and conclusion.

Example statement: * If*
m<A = 90,

Example Converse: * If*
<A is a right
angle,

**Negation of a Statement**is written by writing the
**negative** of the statement.

Example statement <A is right..

Example Negation
is: <A is **not** right.

**Inverse of a Statement**is written by negating the
hypothesis and conclusion of a conditional statement.

Example statement: If m<A = 90, then <A is a right angle

Example Inverse: If m<A
**does not**
= 90, then <A **is not** a right angle.

**Contrapositive of a Statement**is written by
negating the hypothesis and conclusion of the converse of a conditional
statement.

Example Statement: If m<A = 90, then <A is a right angle.

Example converse: If <A is a right
angle, then m<A = 90.

Example contrapositive: If <A is not a
right angle, then m<A is not = 90.

**Any of the statements above can be either true or false.**

**Biconditional Statement**is a statement that
contains the phrase if and only if.

It is equivalent to writing a conditional statement *and* its
converse.

Example: <A is a right angle if and only if m<A = 90.

__2.5
Postulates and Paragraph Proofs__

__Point, Line, and Plane Postulates__

**Postulate 2.1**: Through
any two points there exists exactly one line.

**Postulate 2.2**: Through
any three noncollinear points there exists exactly one plane.

**Postulate 2.3**:
A
line contains at least two points.

**Postulate 2.4**: A
plane contains at least three noncollinear points.

**Postulate 2.5**: If
two points lie in a plane, then the line containing them lies in the plane.

**Postulate 2.6**: If
two lines intersect, then their intersection is exactly one point.

**Postulate 2.7**: If
two planes intersect, then their intersection is a line.

__
2.6
Algebraic Proof
__

**Postulate
or Axiom**--a statement describing a relationship between basic geometric
terms.

**Theorem**--a statement or conjecture
that has been show to be true.

**Proof**--an organized logical argument
in which each step is supported by an accepted statement of truth.

__
__

**Algebraic Properties of Equality**

**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 does not = 0, then a/c = b/c

**Reflexive Property:**
For any real number a, a = a

**Symmetric Property**:
If a = b, then b = a

**Transitive Property:**
If a = b and b = c, then a = c

**Substitution Property:** If a = b, then a can be substituted for b in any expression
or equation.

**Distributive Property:** a(b + c) = ab + ac

*These properties can be applied to geometric measures
of segments or angles.
*

__2.7
Proving Segment Relationships__*
*

**Theorem 2.1: (Midpoint Theorem)**--If M
is the midpoint of segment AB, then segment AM is congruent to segment MB

__ Definition of a
Midpoint__: The point halfway between the
endpoints of a segment.

**Postulate
2.9 (Segment Addition Postulate)
**

**Theorem
2.2 (Segment Congruence)** Congruence of segments is reflexive,
symmetric and transitive.

**Definition
of Congruent Segments:** Two Segments are congruent if and only
if their measures are equal.

__2.8
Proving Angle Relationships__

**Definition of
Congruent Angles**--If two angles are congruent, then their measures
are equal. (the converse is also true)

**Theorem 2.5**--(**Angle
Congruence**)--Congruence of angles is
reflexive, symmetric, and transitive

**Postulate
2.11 (Angle Addition Postulate)--**
If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS.

**Definition of an Angle
Bisector**--If a ray bisects an angle, then the ray
divides the angle into two congruent angles. (the converse is also true)

**Theorem 2.3 (Supplement
Theorem)**if
two angles form a linear pair, then they are supplementary.

**Definition of
Supplementary Angles**--If two angles have a sum of 180, then they are
supplementary. (the converse is also true)

**Theorem 2.4 (Complement
Theorem)**If
the non-common sides of two adjacent angles form a right angle, then the angles
are complementary.

**Definition of
Complementary Angles**--If two angles have a sum of 90, then they are
complementary. (the converse is also true)

**Theorem
2.8 (Vertical Angles Theorem)**If two angles are vertical angles, then they are congruent.

**Theorem
2.9** (**Perpendicular Lines Theorem**)---Perpendicular lines intersect to form four right angles.

**Theorem 2.6 (Congruent Supplements
Theorem)**If two
angles are supplementary to the same angle (or to congruent angles), then they
are congruent.

**Theorem 2.7 (Congruent Complements
Theorem)**If two
angles are complementary to the same angle (or to congruent angles), then they
are congruent.

**Theorem
2.10 ** All right angles are congruent.

**Theorem
2.11** Perpendicular lines form four congruent adjacent angles.

**Theorem
2.12 ** If two angles are congruent and supplementary, then each angle is a
right angle.

**Theorem
2.13** If two congruent angles form a linear pair, then they are right
angles.