1. §2.1 Introductory Material The student will learn about: and the beginning postulates to be used in this course. definitions of basic terms, 1

2. Assumptions We are assuming knowledge of all of the arithmetic and algebra properties that are known. 2 Basic terms such as point, line and plane will not be defined. The postulate system we will be using is attributed to David Hilbert. 1862 –1943

3. The Distance Postulate Postulate 1. The Distance Postulate. To every pair of different points there corresponds a unique positive number. 3

4. The Ruler Postulate Postulate 2. The Ruler Postulate. The points of a line can be placed in correspondence with the real numbers in such a way that 4 1)To every point of the line there corresponds exactly one real number; 2)To every real number there corresponds exactly one point of the line; and 1)To every point of the line there corresponds exactly one real number; 2)To every real number there corresponds exactly one point of the line; and 3)The distance between any two points is the absolute value of the difference of the corresponding numbers.

5. The Ruler Placement Postulate Postulate 3. The Ruler Placement Postulate. Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. 5

6. Betweenness Definition If A, B and C are points on the same line, then B is between A and C iff AB + BC = AC And we write A – B – C or C – B – A. 6

7. Theorem 2 - 1 If A, B, and C are three different points of the same line, then exactly one of them is between the other two. Proof assigned for homework. Hint: show that exactly one of the following must hold: AB + BC = AC or AC + CB = AB or BA + AC = BC 7

8. The Line Postulate Postulate 4. The Line Postulate. For every two points there is exactly one line that contains both points. 8

9. Some Definitions Definition. For any two points A and B, the segment is the set whose points are A and B, together with all the points between A and B. the points A and B are called the endpoints of Definition. The number AB is called the length of segment. 9

10. Some Definitions Definition. Let A and B be points on line L. The ray is the set which is the union of the segment and the set of all points C for which it is true that A – B – C. the point A is called the end point of. Definition. If A is between B and C, then and are opposite rays. 10

11. Segments, Rays, Lines & Angles. 11 Segment AB: Ray AB: Line AB: Angle ABC:

12. Some Definitions Definition. A point B is called a midpoint of a segment if A – B – C and AB = BC. The midpoint is said to bisect the segment. 12

13. Theorem Every segment has exactly one midpoint. Given: B is the midpoint of segment AC. Prove B is unique. 13 A B D C Assume that D is also a midpoint of segment AC. Arrive at a contradiction. What is our strategy?

14. Theorem Every segment has exactly one midpoint. StatementReason 1. AB + BC = AC and AB = BCGiven, B is midpoint. Given: B is the midpoint of segment AC. Prove B is unique. Assume that D is also a midpoint of segment AC. Arrive at a contradiction. 2. AB = ½ ACArithmetic 3. AD + DC = AC and AD = DCAssumed, D is midpoint 4. AD = ½ ACArithmetic →← of the ruler postulate. Hence there is only one midpoint. 14 A B D C

15. Some Definitions Definition. The set of all points is called space. 15 Definition. A set of points is collinear if there is a line which contains all the points of the set. Definition. A set of points is coplanar if there is a plane which contains all the points of the set.

16. Postulate Postulate 5. (a) Every plane contains at least three noncollinear points. (b) Space contains at least four nonplaner points. 16

17. Theorem If two distinct lines l and m meet, their intersection is a single point. Given: Distinct lines l and m with l ∩ m = A Prove: A is unique. l m A 17 Proof for homework. Strategy?

18. Postulate Postulate 6. If two points of a line lie in a plane, then the line lies in the same plane. 18 Theorem If a line intersects a plane not containing it, then the intersection contains only one point. This theorem is a direct result of the postulate.

19. Postulate Postulate 7. The plane postulate. any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane. 19

20. Theorems Theorem. given a line and a point not on the line, there is exactly one plane containing both. This theorem is a direct result of the postulate. 20 Theorem Given two intersecting lines, there is exactly one plane containing both. This theorem is a direct result of the postulate.

21. Some Definitions Definition. The interior of  BAC is the set of all points P in the plane of  BAC such that P and B are on the same side of and P and C are on the same side of. The exterior of  BAC is the set of all points of the plane of  BAC that lie neither on the angle nor in its interior. 21 P A C B

22. Postulate Postulate 8. The Angle Measurement Postulate. To every angle  BAC there corresponds a real number between 0 and 180. 22

23. Postulate 23 Postulate 9. The Angle Construction Postulate. For any two angles  ABC and  DEF such that m  ABC < m  DEF, there is a unique ray such that m  ABC = m  GEF and B A C D G FE

24. Postulate 10 Postulate 10. The Angle Addition Postulate. If D is in the interior of  BAC, then m  BAC = m  BAD + m  DAC 24

25. Some Definitions Definition. If and are opposite rays, and is any other ray, then  BAC and  CAD form a linear pair. 25 Definition. If the sum of the measures of two angles is 180, then the angles are called supplementary, and each is called a supplement of the other.

26. Postulate 11 Postulate 11. The Supplement Postulate If two angles form a linear pair, then they are supplementary. 26

27. Some Definitions 27 Definition. If two angles in a linear pair have the same measure, then each of them is a right angle. Definition. If, then the lines and are called perpendicular. Definition. If and form a right angle, then they are called perpendicular rays, and we write

28. Some Definitions Definition. If the sum of the measures of two angles is 90, then they are called complementary, and each of them is called a complement of the other. An angle with measure less than 90 is called acute. And angle with measure greater than 90 is called obtuse. 28 Definition. Two angles with the same measure are called congruent.

29. Some Definitions Definition. Two angles are vertical angles if their sides form two pairs of opposite rays. 29 There are many theorems involved with supplementary, complementary, vertical angles and perpendicularity. Some will be assigned for homework.

30. Assignment: §2.1