Math 7 geometry 02 postulates and theorems on points, lines, and planes
Download as PPTX, PDF18 likes10,727 views
This document covers basic concepts in geometry including: 1. Definitions, undefined terms, postulates, and theorems related to points, lines, and planes. Undefined terms include points, lines, and planes. Definitions clearly define concepts like line segments. 2. Postulates are statements accepted as true without proof, including the ruler postulate, segment addition postulate, and plane postulate. 3. Theorems are important statements that can be proven, such as the intersection of lines theorem and the theorem regarding a line and point determining a unique plane.
Math 7 geometry 02 postulates and theorems on points, lines, and planes
- 1. The session shall begin shortly…
- 3. Basics of Geometry 2 Postulates and Theorems Related to Points, Lines and Planes
- 4. Foundations of Geometry Definitions Undefined Terms Postulates and Theorems on Points, Lines, and Planes Postulates Theorems Euclid Father of Geometry
- 5. Definitions DEFINITIONS are words that can be defined by category and characteristics that are clear, concise, and reversible. Postulates and Theorems on Points, Lines, and Planes Example: Definition of Line Segment B A LINE SEGMENT (or segment) is a set of points consisting of two points on a line, and all the points on the line between the two points C
- 6. Postulates POSTULATES statements accepted as true without proof. They are accepted on faith alone. They are considered self-evident statements. Postulates and Theorems on Points, Lines, and Planes They are also called AXIOMS.
- 7. Ruler Postulate PART 1: There is a one-to-one correspondence between the points of a line and the set of real numbers. This means that every point on the number line corresponds to a UNIQUE real number Postulates and Theorems on Points, Lines, and Planes
- 8. Ruler Postulate PART 2: the distance between any two points equals the absolute value of the difference of their coordinates. a b a bDistance = Postulates and Theorems on Points, Lines, and Planes
- 9. Segment Addition Postulate If B is a point between A and C, then AB + BC = AC A B C Note that B must be on AC. Postulates and Theorems on Points, Lines, and Planes
- 10. Definition of “Betweenness” If A, B, and C are points such that AB + BC = AC, then B is between A and C. A B C Postulates and Theorems on Points, Lines, and Planes
- 11. Segment Addition Postulate Examples Postulates and Theorems on Points, Lines, and Planes 1. has length 10 cm and has length 8 cm. If A is between P and K, find the length of PA AK PK
- 12. Segment Addition Postulate Examples Postulates and Theorems on Points, Lines, and Planes 2. Is on a number line and O is between B and X. If the coordinates of B and O are 3 and 8, respectively, and BX = 12, what is the coordinate of X? BX
- 13. The Midpoint of a Line Segment Postulates and Theorems on Points, Lines, and Planes The MIDPOINT of a line segment is a point that divides the segment into two equal segments. A M B M is the midpoint ofAB 1 2 AM MB AB
- 14. Examples Postulates and Theorems on Points, Lines, and Planes 3. has length 10 cm. If J is the midpoint of , what are the lengths of the following? KL KL a. KJ b. JL The Midpoint of a Line Segment
- 15. Examples Postulates and Theorems on Points, Lines, and Planes 4. Find the coordinate of the midpoint of on the number line if the coordinates of L and N are –3 and 7, respectively. LN The Midpoint of a Line Segment
- 16. Line Postulate Through any two points there is exactly one line. Restated: 2 points determine a unique line. Postulates and Theorems on Points, Lines, and Planes
- 17. Plane Postulate Part 1: Through any three points there is at least one plane. Part 2: Through any three non- collinear points there is exactly one plane. Postulates and Theorems on Points, Lines, and Planes
- 18. Three collinear points can lie on multiple planes. M While three non- collinear points can lie on exactly one plane. (Three noncollinear points determine a unique plane) Postulates and Theorems on Points, Lines, and Planes Plane Postulate
- 19. With 3 non-collinear points, there is only one plane – the plane of the triangle. B A C Postulates and Theorems on Points, Lines, and Planes Plane Postulate
- 20. Flat Plane Postulate If two points of a line are in a plane, then the line containing those points in that plane. M A B Postulates and Theorems on Points, Lines, and Planes
- 21. Intersection of Planes Postulate If two planes intersect, then their intersection is a line. Remember, intersection means points in common or in both sets. Postulates and Theorems on Points, Lines, and Planes
- 22. Intersection of Planes Postulate If two planes intersect, then their intersection is a line. H G F E D CB A Remember, intersection means points in common or in both sets. Postulates and Theorems on Points, Lines, and Planes
- 23. H G F E D CB A Intersection of Planes Postulate If two planes intersect, then their intersection is a line. Remember, intersection means points in common or in both sets. Postulates and Theorems on Points, Lines, and Planes
- 24. Final Thoughts on Postulates Postulates are accepted as true on faith alone. They are not proved. Postulates need not be memorized. Those obvious simple self-evident statements are postulates. It is only important to recognize postulates and apply them occasionally. Postulates and Theorems on Points, Lines, and Planes
- 25. Theorems Theorems are important statements that are proved true. Postulates and Theorems on Points, Lines, and Planes These are statements that needs to be proven using logical valid steps. The principles and ideas used in proving theorems will be discussed in Grade 8
- 26. Intersection of Lines Theorem If two lines intersect, then they intersect in exactly one point. This is very obvious. To be more than one the line would have to curve. But in geometry, all lines are straight. Postulates and Theorems on Points, Lines, and Planes
- 27. Theorem Through a line and a point not on the line there is exactly one plane that contains them. A Postulates and Theorems on Points, Lines, and Planes Restatement: A line and a point not on the line determine a unique plane.
- 28. Theorem Through a line and a point not on the line there is exactly one plane that contains them. WHY? A B C Postulates and Theorems on Points, Lines, and Planes If you take any two points on the line plus the point off the line, then… The 3 non-collinear points mean there exists a exactly plane that contain them. If two points of a line are in the plane, then line is in the plane as well.
- 29. If two lines intersect, there is exactly one plane that contains them. Theorem Postulates and Theorems on Points, Lines, and Planes Restatement: Two intersecting lines determine a unique plane.
- 30. If you add an additional point from each line, the 3 points are noncollinear. Through any three noncollinear points there is exactly one plane that contains them. If two lines intersect, there is exactly one plane that contains them. WHY? Theorem Postulates and Theorems on Points, Lines, and Planes
- 31. Postulates and Theorems on Points, Lines, and Planes
- 32. Foundations of Geometry: 1 Undefined terms: Point, Line & Plane 2 Definitions 3 Postulates 4 Theorems Statements accepted without proof. Statements that can be proven true. Primitive terms that defy definition due to circular definitions. Words that can be defined by category and characteristics that are clear, concise, and reversible. Summing it up! Postulates and Theorems on Points, Lines, and Planes