1. Two Column Proofs
PROOF Geometry

2. Finding & Describing Patterns
Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns.
Inductive reasoning is used to find and describe patterns.

3. Using Inductive Reasoning
Inductive reasoning was used to discover many of the theorems and postulates we use today
You can use inductive reasoning to make your own conjectures when you are given a situation and you need to find a conclusion

4. Steps involved in using Inductive Reasoning
Look for a Pattern: Look at several examples. Use diagrams and tables to help discover a pattern.
Make a Conjecture. Use the example to make a general conjecture.
A conjecture is an unproven statement that is based on observations.
Discuss the conjecture with others. Modify the conjecture, if necessary.

5. EXAMPLE: Making a Conjecture
Conjecture: The sum of the two positive even integers is _____________.
How to proceed: List some specific examples and look for a pattern.

6. ExAMPLE: Making a Conjecture
Some evens added: 2+2 = = = = 14 EVEN

7. To show that a conjecture is True
We need to prove it for all cases.
A proof like this will typically involve algebra, geometry definitions, postulate and theorems and deductive reasoning.
Let the first integer be 2n and the second 2m
Sum = 2n + 2m
=2 (n + m) always even

8. To show that a conjecture is false
Show the conjecture is false by finding a counterexample.
Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

9. Finding a counterexample - Solution
Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.
The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to In fact, any number between 0 and 1 is a counterexample.

10. Conditional Statement
We often write conjectures as conditional statements
Definition:
A conditional statement is a statement that can be written in if-then form.
“If _____________, then ______________.”
Example:
If your feet smell and your nose runs, then you're built upside down.

11. Conditional Statements have two parts:
The hypothesis is the part of a conditional statement that follows “if” (when written in if-then form.)
-The hypothesis is the given information, or the condition.
The conclusion is the part of an if-then statement that follows “then” (when written in if-then form.)
-The conclusion is the result of the given information.
Every theorem can be written as a conditional statement.

12. Using the Laws of Logic Definition:
Deductive reasoning uses postulates, definitions, and theorems in a logical order to write a logical argument.
The logical argument is called a proof
This differs from inductive reasoning, in which previous examples and patterns are used to form a conjecture.

13. Comparison of Inductive and Deductive Reasoning

14. Comparison of Inductive and Deductive Reasoning

15. Types of proof A paragraph proof
Most commonly used in upper-level mathematics

16. Types of proof A two column proof: Most commonly used in high school.
Has numbered statements and reasons that show the logical order of an argument.

17. Types of proof
A flow chart proof

18. Justification in Proofs using Properties of equality

19. Definition of …
Used when using a definition of defined term to justify statements
Examples:
Congruent to equal: AB ≅ CD Given AB = CD Def. of congruent segments
Equal to congruent:
AB = CD Reason
AB ≅ CD Def. of congruent segments

20. Definition of … Congruent to equal angles Equal to congruent angles
Given
Definition of congruent angles
Equal to congruent angles
Given
Definition of congruent angles

21. Definition of … Example
1 and 2 are complementary Given m1 + m2 = 90 Def. of complementary angles

22. Two Column Proofs We will start with simple, short, easy proofs.
With the short proofs I will show you how to do a Formal Proof which will include writing down each logical step – even if it seems obvious.
Once the proofs get more complex we will write more Compact Proofs so we don’t get bogged down in notation.
Although the proofs are called compact I will still require accurate use of logic, definitions, postulates, theorems and the two-column format.

23. Two Column Proofs We will use the following method:
Create two columns, label the left Statements and the right Reasons.
Start with a given and if possible make some conclusions before using another given.
Number each line and providing a reason for each statement.
Sometimes your reasons will include references to previous line numbers.
When you have finished the proof, write Q.E.D. (quod erat demonstrandum, which is Latin for "which was to be shown") or make a filled-in square (a "bullet") at the end of the proof.

24. Two Column Proof: Algebra Example
If 5x – 18 = 3x +2, prove that x= 10
5x – 18 = 3x + 2
2x – 18 = 2
2x = 20
x = 10
Q.E.D.
Given
Subtraction prop. of eq.
Addition prop. Of eq.
Division prop. Of eq.

25. Homework
Beginning Proofs Worksheets