Chapter 2Connecting Reasoning and Proof

In this chapter, you will:

Make conjecturesUse the laws of logic to make

conclusionsSolve problems by looking for a patternWrite algebraic proofsWrite proofs

Why It’s Important

Law – The job of a lawyer is to present the client’s for guilt or innocence so that jurors can use logical reasoning to determine whether the client is guilty or not. In this chapter, you will learn about two basic types of logic that can be used to help a person determine whether something is true or false.

Inductive reasoning relies on patterns in past occurrences to reach a conclusion.

Deductive reasoning uses a rule to reach a conclusion.

Lawyers may use both types of logic as they present their cases to juries.

2-1 Inductive Reasoning and Conjecturing

Inductive Reasoning – Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction. When you observe the same thing happening again and again and form a conclusion from those observations, you are using inductive reasoning.

Deductive Reasoning – A system of reasoning used to reach conclusions that must be true whenever the assumptions on which the reasoning is based are true.

When you use laws of logic and statements that are known to be true to reach a conclusion, you are using deductive reasoning.

Conjecture – An educated guess. Looking at several specific situations to arrive at a conjecture is called inductive reasoning.

Conjectures are based on observations of a particular situation.

Conjecture based on several observations may be true or false.

Counterexample – An example used to show that a given general statement is not always true.

Example 1 – Page 70

Some conjectures are:The ball will strike the long side of the

table at its midpoint.The ball will then bounce off the rail at

the same angle.The ball will continue on a path and

touch the opposite corner.

Example 2

For points A, B, and C, AB = 10, BC = 8, and AC = 5. Make a conjecture and draw a figure to illustrate your conjecture.

Conjecture: A, B, and C are noncollinear

Conjectures are made based on observations of a particular situation.

A conjecture based on several observations may be true or false.

Example 3

Eric Pham was driving his friends to school when his car suddenly stopped two blocks away from school. Make a list of conjectures that Eric can make and investigate as to why his car stopped.

Some conjectures are:

The car ran out of gas.The battery cable lost its contact with the

battery.

Example 4

Given that points P, Q, and R are collinear, Joel made a conjecture that Q is between P and R. Determine if his conjecture is true or false.

Page 72

Explain the meaning of conjecture. Why three points on a circle could never be

collinear. Determine if the conjecture is true or false. Given: <1 and <2 are supplementary angles. <1 and <3 are supplementary angles. Conjecture: <2 = <3

Give a conjecture

Lines l and m are perpendicular.If l and m are perpendicular, then they

form a right angle.Points H, I, and J are each located on

different sides of a triangle, make a conjecture about points H, I, and J.

Determine if the conjecture is true or false. Explain your answer and give a counterexample if the conjecture is false

Given : FG = GHConjecture: G is the midpoint of FH.