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:

 

 

 

 

 

 

 

 

 

 

 

2.3 Conditional Statements

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, then <A is a right angle.

 

 

 

 

 

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

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

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

 

 

 

 

 

 

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 MidpointThe point halfway between the endpoints of a segment.      

So...if M is the midpoint of segment AB, then AM =MB

 

 

 

Postulate 2.9 (Segment Addition Postulate If B is between A and C, then AB + BC = AC

 

 

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 Theoremif 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 TheoremIf 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 TheoremIf 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 TheoremIf two angles are supplementary to the same angle (or to congruent angles), then they are congruent. 

Theorem 2.7 (Congruent Complements TheoremIf 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.