2. Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 2-1Inductive Reasoning and Conjecture Lesson 2-2Logic Lesson 2-3Conditional Statements Lesson 2-4Deductive Reasoning Lesson 2-5Postulates and Paragraph Proofs Lesson 2-6Algebraic Proof Lesson 2-7Proving Segment Relationships Lesson 2-8Proving Angle Relationships

5. Lesson 5 Contents Example 1Points and Lines Example 2Use Postulates OBJECTIVE: To identify and use basic postulates about points, line, and planes (2.8.8E) (M8.D.1.1) KEY CONCEPT: Five essential parts of a good proof: 1.State the theorem or conjecture to be proven. 2.List the given information 3.If possible, draw a diagram to illustrate given info. 4.State what is to be proved. 5.Develop a system of deductive reasoning

6. Example 5-1a 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.

7. Example 5-1b Solve Label 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.

8. Example 5-1b Answer: 15 Examine In the figure, are all segments that connect the vertices of the snow crystal.

9. Example 5-1c 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

10. Example 5-2a Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. 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.

11. Example 5-2b Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes. For, if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Determine whether the following statement is always, sometimes, or never true. Explain.

12. Example 5-2c Answer: Never; noncollinear points do not lie on the same line by definition. contains three noncollinear points. Determine whether the following statement is always, sometimes, or never true. Explain.

13. Example 5-2d 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 2.7 states that if two planes intersect, then their intersection is a line. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line.

14. Example 5-2e c. Two planes will always intersect a line. Answer: Sometimes; Postulate 2.7 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. Determine whether each statement is always, sometimes, or never true. Explain.

15. Review 1.Determine the number of segments that can be drawn connecting 4 noncollinear points. 2. Determine the number of segments that can be drawn connecting 6 noncollinear points. 6 15

16. Line BD and Ray BR are in plane P, and W is on Line BD. State the postulate or definition that can be used to show each statement is true 3.B, D, and W are collinear. 4. E, B, and R are coplanar. 5. R and W are collinear. Definition of collinear. Through any 3 points not on the same line, there is exactly one plane. Through any two points, there is exactly one line.

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