1. Foundations of Geometry1
CHAPTER
Foundations of Geometry
Copyright © 2014 Pearson Education, Inc.

2. Copyright © 2014 Pearson Education, Inc.
Definitions
The word geometry comes from the Greek language and means to measure the earth. Before we talk more about geometry and mathematical systems, let’s check our reasoning skills, or logic. We need to learn how to use reason and logic in order to recognize patterns, make conjectures, and develop the laws of this system.
Copyright © 2014 Pearson Education, Inc.

3. Using Logic to Recognize Patterns
Sketch the next figure in this pattern. Solution From left to right, each figure is a square, with a triangle shaded in a different corner. From left to right, the shading is moving clockwise. Thus, we predict that the next figure is
Copyright © 2014 Pearson Education, Inc.

4. Using Logic to Recognize Patterns
Look for a pattern and predict the next number. a. 2, 10, 50, 250, … b. −3, 0, 3, 6, 9, … Solution a. Each number is 5 times the previous number. We predict the next number to be 1250 because 250(5) = 1250
Copyright © 2014 Pearson Education, Inc.

5. Using Logic to Recognize Patterns
Look for a pattern and predict the next number. a. 2, 10, 50, 250, … b. −3, 0, 3, 6, 9, … Solution b. Each number is 3 more than the previous number. We predict the next number to be 12 because = 12
Copyright © 2014 Pearson Education, Inc.

6. Mathematical Structure
Undefined terms are the most basic of terms that we do not formally define, but have a meaning or description that we agree upon. To form definitions, we use undefined terms or already defined terms to formally define them. Postulates (or Axioms) are statements that we accept as true and do not try to prove. Theorems are statements that we prove using logic, the foundation above, and other previously proved theorems.
Copyright © 2014 Pearson Education, Inc.

7. Mathematical Structure
Now we can use our logic and reasoning skills to develop the mathematical system of geometry. Begin with undefined terms, which we first describe. Use these undefined terms to formally define terms. Use our defined terms to write statements that we do not prove, but instead agree and accept them to be true. These statements are called postulates or axioms. Our mathematical system grows by using terms, postulates, and axioms to prove theorems.
Copyright © 2014 Pearson Education, Inc.

8. Copyright © 2014 Pearson Education, Inc.
Euclid’s Postulates
1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
Copyright © 2014 Pearson Education, Inc.

9. Euclid's fifth postulate
Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)
Copyright © 2014 Pearson Education, Inc.

10. Inductive and Deductive Reasoning
Inductive Reasoning— Observing patterns or examples, and then using them to form a general conclusion that may or may not be true. Deductive Reasoning— Using known information to write a logical argument that arrives at a specific conclusion that is true. (This is how we prove theorems.)
Copyright © 2014 Pearson Education, Inc.

11. Copyright © 2014 Pearson Education, Inc.
Deductive Reasoning
Deductive reasoning is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved true by deductive reasoning is called a theorem.
Copyright © 2014 Pearson Education, Inc.

12. Copyright © 2014 Pearson Education, Inc.
Deductive Reasoning
Axiomatic System Examples
Determine if the conclusion follows logically from the premises and state whether the reasoning is inductive or deductive. Premise: Adrian Peterson is a football player for the MN Vikings. Premise: Adam Theilen is a football player for the MN Vikings. Premise: Teddy Bridgewater is a football player for the MN Vikings. Conclusion: All football players play for the MN Vikings.
Because we are reasoning from three specific examples and drawing a general conclusion, the process involves inductive reasoning. The conclusion does not follow from the premises.
Copyright © 2014 Pearson Education, Inc.

13. Copyright © 2014 Pearson Education, Inc.
Deductive Reasoning
Axiomatic System Examples
Determine if the conclusion follows logically from the premises and state whether the reasoning is inductive or deductive. Premise: Amanda’s only dog is a pug. Premise: Scout is Amanda’s dog. Conclusion: Scout is a pug.
This is an example of deductive reasoning. We do not know if Scout is a pug but the conclusion follows from the premises.
Copyright © 2014 Pearson Education, Inc.

14. Copyright © 2014 Pearson Education, Inc.
Fallacy
A Fallacy is a conclusion the does not necessarily follow from the premises.
Copyright © 2014 Pearson Education, Inc.

15. Copyright © 2014 Pearson Education, Inc.
Fallacy Example
Premise: If you are smart and work hard, you will get good grades. Premise: Beth gets good grades. Conclusion: Beth is smart and works hard.
Solution: Only if you are smart and work hard will you earn good grades. The premise did not say anything about other ways to get good grades (lenient teacher, etc, taken the class before, etc). All we can logically conclude from the given premises is that Beth gets good grades.
Copyright © 2014 Pearson Education, Inc.

16. Copyright © 2014 Pearson Education, Inc.
Given the figure, can you conclude the statement? This is a square? Solution No, we cannot assume that the figure is a square. A square has 4 right angles and 4 sides of equal length. Correctly marked, this is a square.
Copyright © 2014 Pearson Education, Inc.

17. Copyright © 2014 Pearson Education, Inc.
Sheep in Scotland
A mathematician, a physicist, and an engineer are riding a train through Scotland… The engineer looks out the window, sees a black sheep, and exclaims, "Hey! They've got black sheep in Scotland!" The physicist looks out the window and corrects the engineer, "Strictly speaking, all we know is that there's at least one black sheep in Scotland." The mathematician looks out the window and corrects the physicist, " Strictly speaking, all we know is that is that at least one side of one sheep is black in Scotland."
Copyright © 2014 Pearson Education, Inc.