Geometry/Chapter 1

From testwiki
Jump to navigation Jump to search

"Let's start at the very beginning. A very good place to start" - do-re-mi

Objective

Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Section 1.1 - Introduction

Geometry roughly translates to measuring the earth. Geometry is also defined as the branch of math that has to do with spatial relationships. What this all means is that Geometry is used to measure things. Some of those things might be impossible to measure with instruments so we'll have to use math to figure out how big they are. To date no one has taken a tape measure around earth. And yet we are pretty confident that the distance around the earth at the equator is 24,901.473 miles . How do we know that? The first known case of calculating the distance around the earth was done by Eratosthenes around 240 BC. What tools do you think current scientists might use to measure the size of the earth? There is more to Geometry than measuring the earth. Geometry is useful for figuring out how big (or small) a lot of things are. This study of Geometry is more than that.

If you were to ask someone who had taken Geometry in High School what it is that they remember most likely they will say, "Proofs". If you were to ask them what it is that they liked the least they would probably say, "Proofs." A study of Geometry does not have to include proofs. Proofs are not unique to Geometry. Proofs could have been done in Algebra or delayed until Calculus. The reason that High School Geometry almost always spends a lot of time with proofs is that the first great Geometry textbook, "The Elements", was written exclusively with proofs.

This textbook is based on Elementary Geometry. Elementary doesn't mean that it is for elementary school or that it is beginning Geometry. Elementary refers to a book written over a 2,000 years ago called The Elements. The book was written by a man named Euclid. In the book Euclid started with some basic concepts. He built upon those concepts creating more and more complex concepts which he used to create even more complex concepts. His structure and method influence the way Geometry is taught today. Euclid's book or interpretations of it were used as High School curriculum even until the beginning of the 20th century. This textbook is not a re-interpretation of Euclid's The Elements. However it will include more than just facts about geometric objects. The ability to "prove" that a particular answer is correct is part of the course.

Section 1.2 - Reasoning

There are two general ways of reaching conclusions, inductive reasoning and deductive reasoning.

Inductive Reasoning

Inductive reasoning is what we use most often, without even realizing that we are doing so. Inductive reasoning is essentially reaching a conclusion based on previous observations. For example if I notice that every day the sun rises in the East, through inductive reasoning I may conclude that it will do the same tomorrow. In math we may notice a pattern from which we draw conclusions. Look at the following pattern.

12=1 1≤1
22=4 2≤4
32=9 3≤9
(−1)2=1 −1≤1
(−2)2=4 −2≤4

Through inductive reasoning it may be concluded that whenever a number is squared the result is a number which is greater than or equal to the original number. Based on observations this appears to be true. Inductive logic is not certain. Looking at the example above you may have already surmised that for the equation:

(12)2=14

12>14

the conclusion does not hold true.

The same can be applied to problems outside of Math. A beginning observer of American baseball may conclude, after watching several games, that the game is over after 9 innings. He will only realize that this observation is false after observing a game which is tied after 9 innings. Inductive reasoning is useful but not certain. There will always be a chance that there is an observation that will show the reasoning to be false. Only one observation is needed to prove the conclusion to be false.

Much of the reasoning in geometry is like this, consisting of three simple stages (see example A):

1. Look for commonalities
A pattern.
2. Make a conjecture-
An unproven statement that you will prove.
3. Prove/Disprove
The conjecture.

Deductive Reasoning

Deductive reasoning is reaching a conclusion by combining known truths to create a new truth. Unlike inductive reasoning, deductive reasoning is certain, provided that the previously known truths are in fact true themselves. In order to use deductive reasoning there must be a starting point. The starting point or points are truths that cannot be called into question. These truths are true because we said so. These truths are true in the sphere in which they are created. Let's start with a simple example. The truth I will start with is that all High School students love math. It cannot be argued whether this is true or not. It is true because in the world in which this problem exists we said it is true. Another truth that we will assert is that Mike is a High School student. Based on these two truths I can conclude with absolute certainty that Mike loves Math.

Vocabulary

conjecture: An unproven statement that is based on observations.

Examples

^ is to the power of

Example A: Making a Conjecture

Complete this conjecture:
The sum of the first x odd positive integers is _?_

Solution:
sum of the first 1 odd positive integers: 1 = 1 = 1^2
sum of the first 2 odd positive integers: 1 + 3 = 4 = 2^2
sum of the first 3 odd positive integers: 1 + 3 + 5 = 9 = 3^2
sum of the first 4 odd positive integers: 1 + 3 + 5 + 7 = 16 = 4^2
sum of the first 5 odd positive integers: 1 + 3 + 5 + 7 + 9 = 25 = 5^2
sum of the first 6 odd positive integers: 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6^2

And so on...

The sum of the first x odd positive integers is x^2.

Exercises

1) All vegetables are healthy. Broccoli is a vegetable. Therefore, broccoli is healthy. This is an example of what type of reasoning?

2) Broccoli is a vegetable. Broccoli is green. Therefore, all vegetables are green. Why is this conclusion invalid?

Section 1.3 - Undefined Terms

In Geometry, there are 3 undefined terms: points, lines, and planes. We can give descriptions of these three terms. We also use these terms to help us write definitions of other terms such as segment, or ray. The reason these are called undefined terms is that they can only be defined circularly which means in terms of each other or in terms of themselves. For example, if I were to try to define point I might say "a point is a point in space that has a location but no dimensions." The trouble is, I had to use the word point to define the word point, which is no real help at all. Because of this, Geometry starts with three concepts that have meanings and characteristics that everyone agrees on, but no one knows how to define: point, line, and plane. This may seem like a strange place to start, but it actually works pretty well. Since I have already failed to define point, let me continue to fail to define line and plane. A line is the set of all points that are arranged such that they continue infinitely in two directions and have exactly one dimension: length. Notice I failed to define line because I failed to define point and I used point in the definition. A plane is the set of all points that are arranged such that they continue infinitely in any direction that permits the set to have only two dimensions: length and width. Again, I failed to define plane because I used the term point which I cannot define. There are probably more definitive failures to define these undefined terms, but these should suffice for our purposes.

With the non-definitions out of the way, let's look at how these things work. A point is usually represented by a dot on a piece of paper. It is, in reality, infinitely small (so an infinite number of points fit inside the dot on the piece of paper). It is so small that it can never be measured because it has no dimensions at all. Having said that, a point is useful because it tells us exactly where something is, and we can then build observations, conjectures, and rules from that information. For example, we can say that two points determine a line. What this means is that once you know where two points are, you know where the line that contains both of the points must be. Notice that if you only know where one point is, there are an infinite number of lines that can contain that one point, and if you know where three points are, there is a pretty good chance that there isn't any single line that would contain all three points.

Okay, we have talked about lines, now, so how do they behave? A line is always straight and travels forever in two directions. We usually represent a line by drawing it on a piece of paper, but, like the point, since the line has no width dimension, we could actually fit an infinite number of lines in that drawing of a line. A line has only its length (which is infinite). We can take pieces of a line and call them line segments (more on that later in the book) and we can cross two lines and get both a point (where they intersect) and some angles (more on those later too). We can also choose to ignore half of a line by cutting it off at a point and calling what we have left a ray (again, more later).

A plane has two dimensions: width and length. Both of these dimensions are infinite, and, because there are only two dimensions, a plane is perfectly flat and infinitely thin, meaning it has no thickness dimension. Because of this, a plane doesn't really have a top or a bottom because whatever is on the top is also on the bottom. If you take two planes and make them intersect, you get a line (more on that later) and if you take three points that are not all in the same line, there is only one plane that can contain all three (more on that later too). Planes are useful because a plane can hold all of the two dimensional (flat) shapes that geometry uses. We usually think of one side of a piece of paper (or a computer screen) as part of a plane. While this is not exactly correct, like the representations of a point and a line, this is useful.

Exercises

Section 1.4 - Axioms/Postulates

A postulate or axiom is a statement which is taken to be self-evident, and cannot be proved. They are the starting point from which any system in Mathematics (such as geometry) is built up. In Euclidean geometry, there are five such statements:

  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 its radius and one endpoint as its center.
  4. All right angles are congruent.
  5. Given a line and a point off the line, exactly one new line can be drawn through the point that is parallel to the given line.

From these postulates we can deduce all the theorems of Euclidean geometry.

Section 1.5 - Theorems

Postulates:

1) Between any two points, there exists one and only one line.

2) If two lines intersect, then their intersection is a point.

3) Given any 3 non-collinear points, there is exactly one plane that can be constructed which will include all of them.

4) If two planes intersect, then their intersection is a line.

Exercises

1) Draw a point on a piece of paper. How many lines can you draw through that point?

2) Draw two points on a piece of paper. How many lines can you draw through both points?

3) Draw three points on a piece of paper. How many lines can you draw through all three points? Why? What undefinable object could connect all three points?

Chapter Review

Vocabulary

  • Inductive Reasoning - process of reasoning in which the assumption of an argument supports the conclusion, but does not ensure it
  • Deductive Reasoning - process of reasoning in which the argument supports the conclusion based upon a rule
  • Conjecture - a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove
  • Theorem - a proposition that has been or is to be proved on the basis of explicit assumptions
  • Hypothesis - a proposed explanation which can be a proposition ("A causes B")
  • Postulate - a mathematic statement which is used but cannot be proven
  • Axiom - a formal logical expression used in a deduction to yield further results

Navigation

Geometry Main Page

Retrieved from "https://en-labs.wikimedia.beta.math.wmflabs.org/w/index.php?title=Geometry/Chapter_1&oldid=390"