2.4 Use Postulates & Diagrams

1. 2.4 Use Postulates & Diagrams

2. Objectives • Identify and use basic postulates about points, lines, and planes. • Write paragraph proofs.

3. Postulates • In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry. Postulates are also sometimes called axioms. • Postulates are always accepted as true.

4. Postulates • In Chapter 1 you already learned 4 postulates. • Postulate 1 – Ruler Postulate • Postulate 2 – Segment Addition Postulate • Postulate 3 – Protractor Postulate • Postulate 4 – Angle Addition Postulate

5. Postulates • Postulate 5 – Through any two points, there is exactly one line. • Postulate 6 – A line contains at least 2 points. • Postulate 7 – If 2 lines intersect, then their intersection is exactly 1 point. • Postulate 8 – Through any three points not on the same line, there is exactly one plane.

6. More Postulates • Postulate 9 – A plane contains at least three points not on the same line. • Postulate 10 – If two points lie in a plane, then the entire line containing those points lies in that plane. • Postulate 11 – If two planes intersect, then their intersection is a line.

7. Example 1: SNOW CRYSTALS Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal? Explore The snow crystal has six vertices since a regular hexagon has six vertices. Plan Draw a diagram of a hexagon to illustrate the solution.

8. Example 1: SolveLabel the vertices of the hexagon A, B, C, D, E, and F. Connect each point with every other point. Then, count the number of segments. Between every two points there is exactly one segment. Be sure to include the sides of the hexagon. For the six points, fifteen segments can be drawn.

9. Examine In the figure, are all segments that connect the vertices of the snow crystal. Example 1: Answer: 15

10. Your Turn: ART Jodi is making a string art design. She has positioned ten nails, similar to the vertices of a decagon, onto a board. How many strings will she need to interconnect all vertices of the design? Answer: 45

11. Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. Example 2a: Answer: Always; Postulate 10 states that if two points lie in a plane, then the entire line containing those points lies in the plane.

12. Determine whether the following statement is always, sometimes, or never true. Explain. For , if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes. Example 2b:

13. contains three noncollinear points. Example 2c: Determine whether the following statement is always, sometimes, or never true. Explain. Answer: Never; noncollinear points do not lie on the same line by definition.

14. Your Turn: Determine whether each statement is always, sometimes, or never true. Explain. a. Plane A and plane B intersect in one point. b. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. Answer: Never; Postulate 11 states that if two planes intersect, then their intersection is a line. Answer: Always; Postulate 5 states that through any two points, there is exactly one line.

15. Your Turn: Determine whether each statement is always, sometimes, or never true. Explain. c. Two planes will always intersect a line. Answer: Sometimes; Postulate 11 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect.

16. Theorems • We use undefined terms, definitions, postulates, and algebraic properties of equality to prove that other statements or conjectures are true. Once a statement or conjecture has been shown to be true, it is called a theorem. • Once proven true, a theorem can be used like a definition or postulate to justify other statements or conjectures.

17. Assignment • Geometry: Pg. 99 – 102 #3 – 4, 6 – 24, 39