An Ideal Euclidean Model Lesson 1.5 An Ideal Euclidean Model pp. 20-24

Objectives: 1. To describe the characteristics of an ideal geometric system: consistent, independent, and complete. 2. To identify and use the incidence postulates.

Logic - (1) interrelation or sequence of facts or events when seen as inevitable or predictable, (2) something that forces a decision apart from or in opposition to reason

A geometry is a system of definitions, postulates, and theorems that is built in a logical progression.

The key to a good geometry is its set of postulates or basic assumptions. The system should be... 1. consistent 2. independent 3. complete

Consistent A postulate in the system does not contradict any of the others.

Independent The system has no postulates that can be proved from the other postulates in the system.

Complete The postulates leave no unanswered questions about the system.

Euclidean Model Based on 5 incidence postulates. These are the foundation to the system of geometry that we are studying.

What is an incidence postulate What is an incidence postulate? Incidence is the partial overlapping of two figures or of a figure and a line.

A postulate (also known as an axiom) is something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument. A fundamental element; a basic principle.

Therefore, an incidence postulate is a self-evident truth about the intersection of two figures or of a figure and a line.

Postulate 1.1 Expansion Postulate. A line contains at least two points. A plane contains at least three noncollinear points. Space contains at least four noncoplanar points.

Postulate 1.2 Line Postulate. Any two points in space lie in exactly one line.

Postulate 1.3 Plane Postulate. Three distinct noncollinear points lie in exactly one plane.

Postulate 1.4 Flat Plane Postulate. If two points lie in a plane, then the line containing these two points lies in the same plane.

Postulate 1.5 Plane Intersection Postulate. If two planes intersect, then their intersection is exactly one line.

Plane Intersection Postulate

False, they could be parallel. True/False based on the postulates. 1. Two planes always intersect. False, they could be parallel.

True, by the Expansion Postulate. True/False based on the postulates. 2. Four noncoplanar points determine space. True, by the Expansion Postulate.

True, by the Expansion Postulate and Line Postulate. True/False based on the postulates. 3. A plane always contains at least two lines. True, by the Expansion Postulate and Line Postulate.

False, not guaranteed by the given postulates. True/False based on the postulates. 4. A plane must contain infinitely many points. False, not guaranteed by the given postulates.

False, not guaranteed by the given postulates. True/False based on the postulates. 5. A plane must contain at least 5 points. False, not guaranteed by the given postulates.

Homework pp. 23-24

►A. Exercises 1. Name the three characteristics of an ideal system of postulates and give a brief explanation of each.

►A. Exercises 3. Observing the Line Postulate, draw as many lines as possible through the following points. E F D

5. Three noncollinear __________ lie in a plane. ►A. Exercises Place the word plane(s), line(s), point(s), or space in the blank to complete each sentence. 5. Three noncollinear __________ lie in a plane. points

7. An infinite number of lines can intersect in one __________ . ►A. Exercises Place the word plane(s), line(s), point(s), or space in the blank to complete each sentence. 7. An infinite number of lines can intersect in one __________ . point

9. If a line lies in a certain plane, then there are at least two ►A. Exercises Place the word plane(s), line(s), point(s), or space in the blank to complete each sentence. 9. If a line lies in a certain plane, then there are at least two __________ that also lie in that plane. points

17. Plane k contains points A and C; AC also lies in k. ►B. Exercises Give the postulates (by name) that would verify the following statements. 17. Plane k contains points A and C; AC also lies in k. Flat Plane Postulate

21. Any three points lie in exactly one plane. ►B. Exercises Identify each statement as true or false based on the five incidence Postulates. If the answer is false, draw a diagram to illustrate. 21. Any three points lie in exactly one plane. False, they could be collinear

■ Cumulative Review Identify each statement as true or false. 26. If 5  A, and A  B, then 5  B.

■ Cumulative Review Identify each statement as true or false. 27. If A  B, and B  C, then A  C.

■ Cumulative Review Identify each statement as true or false. 28. If 8  A  B, then 8  A and 8  B.

■ Cumulative Review Identify each statement as true or false. 29. If 7  A  B, then 7  A and 7  B.

■ Cumulative Review Identify each statement as true or false. 30. If C  A  B, then C  A or C  B.