Portal:Euclidean geometry/Chapter 1

This chapter is the introduction to geometry, and introduces some basic, essential, concepts of geometry, which include the concepts of undefinable terms, postulates, and theorems. To see the textbook related to this chapter, see the Wikibook chapter.

Undefinable terms

Points

In Euclid's Elements in Book 1 and Definition 1, Euclid addresses the definition of Point. "The Elements is the prime example of an axiomatic system from the ancient world. Its form has shaped centuries of mathematics. An axiomatic system should begin with a list of the terms that it will use. This definition says that one term that will be used is that of point." As stated by David E. Joyce Professor of Mathematics and Computer Science at Clark University.

Euclid writes: A point is that which has no part

This is a difficult concept to grasp to be sure but a definition is needed and this is what the father of geometry gives us.

Every point is represented with the dot, and the letter that labels it. All labels are uppercase for points. This is important because the opposite is true for lines.

Lines

Euclid, in Definition 2 states:

A line is breadthless length.

"Line" is the second primitive term in the Elements. The description, "breadthless length," says that a line will have one dimension, length, but it won't have breadth or depth.

Although lines have no width, we do draw them that way; it's hard to draw a widthless line!

This line has two parts, like the point. It has a figure that represents the line, and a label for the line, in this case b. Notice that the label is lowercase. It is key to be able to identify lines from points.

To represent the fact that lines go on forever, we have a pair of arrows on the line. These show the way the line continues on forever in.

Euclid, in Definition 3 states:

The ends of a line are points.

This statement can be taken as indicating that between certain lines and points a relation holds, that a point can be an end of a line. It doesn't say what ends are. It also doesn't indicate how many ends a line can have. For instance, the circumference of a circle has no ends, but a finite line has its two end points.

Euclid, in Definition 4 states:

A straight line is a line which lies evenly with the points on itself.

This statement indicates, at least, that the term "straight line" refers to a kind of line. It is hard to tell what else it means, if anything. Various commentators have interpreted in a variety of ways. There are a some postulates that come a little later in Book I and give meaning to straight lines. Postulate 1 says that a straight line can be drawn between any two points.

Axioms/Postulates

The definition of an axiom is that it is a statement which is taken to be self-evident, and cannot be proved.

Quizzes

Lesson 1.1

What was the name of the first Geometry book?

Elementary Geometry Incorrect

The Elements Correct

Euclidean Geometry Incorrect

How to draw geometric figures Incorrect

What is the defintion of geometry?

The measure of all things geometric Incorrect

The branch of maths that has to do with proofs Incorrect

The measure of the earth This was the old defintion, but it is accepted

The branch of maths that has to do with spatial relationships Correct

Lesson 1.2

What is the definition of inductive reasoning?

Process of reaching a conclusion based on previous observations Correct

Process of reaching a conclusion by hearing it from somewhere else that you trust Incorrect

Process of reaching a conclusion by combining known truths to create a new truth Incorrect

Process of reaching a conclusion in which the argument supports the conclusion based upon a rule Incorrect

What is the definition of deductive reasoning?

Process of reaching a conclusion based on previous observations Incorrect

Process of reaching a conclusion by hearing it from somewhere else that you trust Incorrect

Process of reaching a conclusion by combining known truths to create a new truth Correct

Process of reaching a conclusion in which the assumption of an argument supports the conclusion, but does not ensure it Incorrect

Lesson 1.3

What are the three undefinable terms?

Points, circles, and planes Incorrect

Points, lines, and planes Correct

Points, line segments, and planes Incorrect

Points, lines, and circles Incorrect

What is the defintion of a point?

A dot Incorrect

A point in space that has a location but no dimensions Incorrect

An infinitely small point that exists on a line Incorrect

This is a trick question Correct

Lesson 1.4

What is the defintion of an axiom/postulate?

A statement that proves itself Incorrect

A statement which is taken to be self-evident, and cannot be proved Correct

A statement which is proven based on other statements Incorrect

A proposition that has been or is to be proved on the basis of explicit assumptions Incorrect

Which of these is not an axiom/postulate?

A straight line segment can be drawn joining any two points. Incorrect

All right angles are congruent. Incorrect

Given any straight line segment, a circle can be drawn having the segment as its radius and one endpoint as its center. Incorrect

All line segments are congruent. Correct

Geometry Main Page

• Topic:Euclidean Geometry/Chapter 1 Definitions and Reasoning (Introduction)
• Topic:Euclidean Geometry/Chapter 2 Proofs
• Topic:Euclidean Geometry/Chapter 3 Logical Arguments
• Topic:Euclidean Geometry/Chapter 4 Congruence and Similarity
• Topic:Euclidean Geometry/Chapter 5 Triangle: Congruence and Similiarity
• Topic:Euclidean Geometry/Chapter 6 Triangle: Inequality Theorem
• Topic:Euclidean Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
• Topic:Euclidean Geometry/Chapter 8 Perimeters, Areas, Volumes
• Topic:Euclidean Geometry/Chapter 9 Prisms, Pyramids, Spheres
• Topic:Euclidean Geometry/Chapter 10 Polygons
• Topic:Euclidean Geometry/Chapter 11 $R,$ $R^{2},$ $R^{3}$
• Topic:Euclidean Geometry/Chapter 12 Angles: Interior and Exterior
• Topic:Euclidean Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
• Topic:Euclidean Geometry/Chapter 14 Pythagorean Theorem: Proof
• Topic:Euclidean Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
• Topic:Euclidean Geometry/Chapter 16 Constructions
• Topic:Euclidean Geometry/Chapter 17 Coordinate Geometry
• Topic:Euclidean Geometry/Chapter 18 Trigonometry
• Topic:Euclidean Geometry/Chapter 19 Trigonometry: Solving Triangles
• Topic:Euclidean Geometry/Chapter 20 Special Right Triangles
• Topic:Euclidean Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
• Topic:Euclidean Geometry/Chapter 22 Rigid Motion