Welcome to Proofs!

The Basics

Structure:

Given information (sometimes assumption)

Step-by-step reasoningConclusion

Types of reasoning:

Inductive-use a number of specific examples to arrive at conclusion (called a “conjecture”)

Deductive- use facts, rules, definitions, or properties to reach a conclusion

Example of inductive reasoning

Write a conjecture that describes the pattern.

Movie show times: 8:30am, 9:45am, 11:00am, 12:15am…

The show time is 1 hour and 15 minutes greater than the previous show. The next show time will be 12:15am + 1:15= 1:30pm.

The examples of movie times showed us a pattern, and we were able to come to a conclusion based on that pattern.

Example of deductive reasoning

At Fumio’s school if you are late 5 times, you will receive a detention. Fumio has been late to school 5 times; therefore he will receive a detention.

Are the following examples of inductive or deductive reasoning? Discuss with your partner.

1.) Every Wednesday Lucy’s mother calls. Today is Wednesday, so Lucy concludes that her mother

will call.

2.) A person must have a membership to work out at a gym. Jesse is working out at a gym. I conclude that Jesse has a membership to the gym.

3.) A dental assistant notices a patient has never been on time for an appt. She concludes the patient will be late for her next appt.

4.) If Eduardo decides to go to a concert tonight, he will miss football practice. Tonight, Eduardo went to a concert, so he missed football practice.

Conditional Statements (“If-Then” statements)

Uses a hypothesis-conclusion format

Example 1: If the forecast is rain (hypothesis), then I will take an umbrella (conclusion)

Example 2: A number is divisible by 10 (conclusion) if its last digit is 0 (hypothesis)

If p q

If q p (converse)

If –p -q (inverse)

If –q -p (contrapositive)

If-Then StatementsWith your partner, determine what the hypothesis and conclusion are of each statement. Also, determine if the converse is true.

If a polygon has 6 sides, then it is a hexagon.

Tamika will advance to the next level of play if she completes the maze in her computer game.

A five-sided polygon is a pentagon.

An angle that measures 45° is an acute angle.

Who remembers the book If You Give a Mouse a Cookie?

2.5 POSTULATES AND PARAGRAPH PROOFS

Objective:

I will be able to…

-Prove relationships between points, lines, and planes using deductive reasoning and paragraph proofs

2.5 POSTULATES AND PARAGRAPH PROOFS

2.5 POSTULATES AND PARAGRAPH PROOFS

The proof process

Example #1

Given that M is the midpoint of write a paragraph proof to show that .

Step 1- Given: M is the midpoint of .Step 2- Prove:

Steps 3 & 4- If M is the midpoint of , then from the definition of midpoint of a segment, we know that XM = MY. This means that and have the same measure. By the definition of congruence, if two segments have the same measure, then they are congruent.

Step 5- Thus, . QED

We just proved this!

Example #2

In the figure at the right, and C is the midpoint of and . Write a paragraph proof to show that .

Since C is the midpoint of and , and by definition of midpoint. We are given so by the definition of congruent segments. By the multiplication property, . So, by substitution, .

Try one with your partner!

2.6 Algebraic proof

A proof that is made up of a series of algebraic statements.

Reasoning uses properties of real numbers (addition, subtraction, multiplication, division, reflexive, distributive, substitution, etc.)

Two-Column Proofs

Left column contains statements, right column contains reasons

Let’s go back to a problem we did earlier and complete a two-column proof

Given that M is the midpoint of write a paragraph proof to show that .

Statement ReasonM is the midpoint of . Given XM = MY Definition of midpoint of a

segmentDefinition of congruence

QED

Solve algebraically to find x.

Now, complete an two-column algebraic proof to show that if then .

What was your first step?

Try one on your own!

Show that if then using a two-column proof.

2.7 Proving segment relationships

Ruler Postulate: The points on any line or line segment can be put into one-to-one correspondence with real numbers (think of a ruler).

Segment Postulate Postulate: If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.

Example: Prove that if and using a two-column proof.

C

F

GE

D

Properties of Segment Congruence

Reflexive Property of Congruence:

Symmetric Property of Congruence: If , then

Transitive Property of Congruence: If and , then