CMA Geometry Unit 1 Introduction Week 2 Notes

Transcription

1 CMA Geometry Unit 1 Introduction Week 2 Notes Assignment: 9. Defined Terms: Definitions betweenness of points collinear points coplanar points space bisector of a segment length of a segment line segment midpoint of a segment opposite rays ray Point B is between A and C if A, B, and C are collinear and the equation AB + BC = AC is true, where AB, BC, and AC are the distances between points A and B, B and C, and A and C, respectively. A set of two or more points all on the same line. A set of two or more points all on the same plane. The set of all possible points. A line or segment that intersects the segment at its midpoint. The distance between the endpoints of a segment. The set of two different endpoints and all points between them. The point on a segment that divides the segment into two equal segments. Ray BA and ray BC are opposite rays if A, B, and C are collinear and B (the endpoint of both rays) is between A and C. A ray, AB, is the set of points beginning at point A and going infinitely in the direction of point B. The three undefined terms of geometry are: point, line and plane. Think about the three dimensional world in which we live. We can move up and down, forward or backward, and left or right. Think of all the locations that we can move into. The set of all points in our three-dimensional world is referred to as space.

2 Using the concepts of line and point, we can define points on the same line as collinear. With this figure, we can also introduce the concept of betweenness. Point C is between points A and E. It is also between points B and D, but not between points A and B, as points A, B and C are not collinear. This figure depicts two intersecting planes. Points A, B, C and D are on plane P and are referred to as coplanar.

3 Points E, F and G are coplanar on plane Q. Because it takes three non-collinear points to define a plane, at least four points must be named in a set that is not coplanar. From this figure, we can see that points E, F, G and C make up a set of points that is not coplanar. A line segment can be thought of as a part of a line and is named according to its endpoints. A line segment is written with the two endpoints and a bar above them. Unlike a line, a segment has a finite length; the length is a positive real number and is designated by writing AB with no bar above. In this figure point M would be the midpoint of AB if the length of AM equals the length of MB. As we are discussing lengths of segments, notice that there is no bar above AM or MB.

4 Associated with midpoint is the concept of bisecting. To bisect means to cut into two equal parts. Notice that line l passes through the midpoint M. This means that line l bisects segment AB into two congruent segments of AM and MB. In other words, line l is the bisector of segment AB. A ray begins at a point called its endpoint and then extends infinitely far in one direction. To name a ray, a single pointed arrow is place over the two letters. The endpoint is always listed first followed by any point on the ray to show direction. [Ray CD is written by placing an arrow pointing right above the letters C and D.] [Line l is shown, with points R, S, and T. Segment SU intersects l at point S] In this figure we have a line, four segments and three rays. Rays SR and ST are considered opposite rays because they are collinear and share a common endpoint. Rays SR and SU share a common endpoint, but are not opposite rays, as they are not collinear.

5 Assignment: 11. Geometric Postulates Postulate 1: Postulate 1a: A line contains at least two points. A plane contains at least three points not all on one line. Postulate 1b: Space contains at least four points not all in one plane. Postulate 2: Postulate 3: Postulate 4: Postulate 5: Through any two different points, exactly one line exists. Through any three points that are not on one line, exactly one plane exists. If two points lie in a plane, the line containing them lies in that plane. If two planes intersect, then their intersection is a line. Did you know the foundation of mathematics involves undefined terms and assumptions? For example, a postulate is a statement accepted without proof. Postulates are the next logical step up from definitions. Every mathematical system must have postulates. From postulates, defined terms, and undefined terms, it is possible to use logic and arrive at a theorem--a statement that can be proved. A postulate, sometimes called an axiom, is an accepted statement of fact. We don't have to prove its truthfulness. We simply accept it as is. These are often things that seem like common sense.

6 A postulate is a statement that is accepted without proof. With the use of defined terms, undefined terms, and logic, a postulate can be proven. A proven postulate or statement is a theorem. The selection of postulates for a mathematical system is a difficult task. Proper selection is important because a theorem can only be proven true if the set of postulates the theorem is based on is true. For postulates to have any important significance, they must be independent, complete, and consistent. For each postulate to be independent, they must able to stand alone. That is, they cannot be proven or deduced using another postulate. For a set of postulates to be complete, there must be enough of them to sufficiently describe the system. Imagine the game of basketball with only one rule; the ball must go into the hoop. The game would probably be much different than one we play and watch today. For a set of postulates to be consistent, no postulate within the system can contradict another postulate in that set. For this course, we will use five postulates about points, lines and planes. These postulates are known as Hilbert s Axioms of Inci1dence and Order. [Postulates: Adequate number, consistent.] Postulate 1 gives us more information about lines, planes, and space. It tells the minimum number of points that each of them has. This postulate states that a line cannot have just one point, a plane cannot have just two points, and space cannot have less than four points. Postulate 2 tells us that for any two points, only one line can be drawn containing them. We also can say that two points determine a line. Exactly one means at least one but no more than one. Postulate 3 tells us that if we have three noncollinear points, then only one plane contains them. We also can say that three points determine a plane. It reminds us that the plane in question contains,, and, and that the plane is flat. Postulate 4 tells us that if we take two points of the plane and determine a line with them, then the line is entirely in that plane.

7 Postulate 5 tells us that two planes cannot intersect in a point, but in a line. Assignment: 12. Review of Algebraic Postulates (Please check the note of Properties of the Real Numbers Notes in this website.) Assignment: 13. Geometric Theorems A theorem is a statement that is proved by deductive logic. A theorem is the product of mathematics. If you remember our discussion from the beginning of this unit, people arrive at these products by "thinking mathematically." They used other products of the mathematical system such as undefined terms, defined terms, postulates, and other theorems as well as thinking processes including deductive reasoning, conjecture, calculating, testing, hypothesizing, assuming, abstracting, comparing, and others to arrive at a statement. In algebra, one way to solve a system of two equations was to graph both equations. When you graph y = 2x + 4 and y = negative x minus 2, notice that they intersect at a single point (negative 2, 0). In math, a statement that has been proven true is called a theorem. The lines intersect at (negative 2, 0)] This means that the solution to the system of equations is (negative 2, 0). This also illustrates Theorem 1-1 which states that if two lines intersect then they intersect at

8 exactly one point. The power of this theorem is beyond just the one example but that it says that all lines that intersect will intersect at one point. Mathematical theorems are very important things that describe a large number of situations. Proving these theorems will be an important preoccupation of this course. Theorems Since we have a set of undefined terms, definitions, and postulates, we now have the capability to prove theorems. Furthermore, once we have proven a theorem we may use it in our repertoire. Let's begin by looking at three theorems and strategies for their proofs. Theorem 1-1: If two lines intersect, then their intersection is exactly one point. Theorem 1-2: Exactly one plane contains a given line and a given point not on the line. Theorem 1-3: If two lines intersect, then exactly one plane contains both lines. THEOREM 1-1: The Plan: Let a and b be the two given lines and P, their given point of intersection. Suppose they also intersect in some other point Q. However, for two lines to intersect in P and Q is impossible because Postulate 2 tells us that only one line contains two given points. In other words, if lines a and b intersect in points P and Q (see Graphic), then line PQ is one line not two. However, we were given two distinct lines ( a and b ). So the statement that the two given lines intersect also in Point Q is an impossible condition, and is therefore false. This form of proof is not the more common one; in most cases, a statement is proved true, instead of false: this latter form is called an indirect proof. Geometric Theorems 2 Theorem one-two states that exactly one plane contains a given line and a given point not on the line. As this is a theorem, we should be able to prove this is true with the use

9 of postulates, defined and undefined terms Postulate 1 tells us that at least two points are on line a. We will label these as points R and S. Since point P is not on line a, we have three non-collinear points. Postulate 3 tells us that exactly one plane contains three points. Finally, Postulate 4 justifies line a being in the plane since the points R and S of line a are in the plane. The progression from postulate 1 to 3 to 4 takes us from having two points on a line to exactly one plane containing a given line and a point not on that line. [On a plane, lines A and B are shown intersecting at point P.] Let s take a look at the proof for Theorem 1-3, which states that if two lines intersect, then exactly one plane contains both lines. Theorem 1-1 tells us the intersection of lines a and b is one point; call it P. Postulate 1 tells us each line has at least two points, so another point must be on a and another point on b. Call them R and S. [Point R is shown on line A. Point S is shown on line B.] Postulate 3 tells us that exactly one plane contains those three points. Lines a and b will have to lie in this plane, because Postulate 4 says so. Good proofs follow a well thought out plan.

10 Assignment: 14. Review of Properties of Algebra a = a a = b then b = a a = b and b = c then a= c Reflexive property Symmetric property Transitive property REMEMBER: Properties of inequality: Subtraction: If a < b, then a - c < b - c Division: If a < b and c > 0 then < If a < b and c < 0 then > Substitution: If a = b, a may be replaced by b and b by a in any equation or inequality. Zero Product Property: If ab = 0, then a = 0 or b = 0, or a and b = 0 Modeling Mathematical Practices Click here to learn how an important mathematical practice can be applied to this topic. These postulates were used in your algebra course to solve equations. Example: x 2 + 4x + 3 = 0 First, we would use factoring to simplify the quadratic equation: (x+1)(x + 3) = 0 Now, we use the Zero Product Property and set each of the individual factors equal to zero: x + 1 = 0 or x + 3 = 0 Finally, solve each equation for the variable by using the Subtraction Postulate of Equality: x = -1 or x = -3 Modeling Mathematical Practices Click here to learn how an important mathematical practice can be applied to this topic.