Geometry Postulates and Proofs: Intersections of Lines and Planes

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2. Five-Minute Check (over Lesson 2–4) CCSS Then/Now New Vocabulary Postulates: Points, Lines, and Planes Key Concept: Intersections of Lines and Planes Example 1: Real-World Example: Identifying Postulates Example 2: Analyze Statements Using Postulates Key Concept: The Proof Process Example 3: Write a Paragraph Proof Theorem 2.1: Midpoint Theorem Lesson Menu

3. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.Given: A and B are supplementary.Conclusion: mA + mB = 180 A. valid B. invalid 5-Minute Check 1

4. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.Given: Polygon RSTU has 4 sides.Conclusion: Polygon RSTU is a square. A. valid B. invalid 5-Minute Check 2

5. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.Given: A andB are congruent.Conclusion: ΔABC exists. A. valid B. invalid 5-Minute Check 3

6. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.Given: A and B are congruent.Conclusion: A and B are vertical angles. A. valid B. invalid 5-Minute Check 4

7. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.Given: mY in ΔWXY = 90.Conclusion: ΔWXY is a right triangle. A. valid B. invalid 5-Minute Check 5

8. How many noncollinear points define a plane? A. 1 B. 2 C. 3 D. 4 5-Minute Check 6

9. Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. CCSS

10. You used deductive reasoning by applying the Law of Detachment and the Law of Syllogism. • Identify and use basic postulates about points, lines, and planes. • Write paragraph proofs. Then/Now

11. postulate • axiom • proof • theorem • deductive argument • paragraph proof • informal proof Vocabulary

12. Concept

13. Concept

14. Identifying Postulates ARCHITECTUREExplain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q. Answer: Points F and G lie on line m, and the line lies in plane Q. Postulate 2.5, which states that if two points lie in a plane, the entire line containing the points lies in that plane, shows that this is true. Example 1

15. Identifying Postulates ARCHITECTUREExplain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. B. Points A and C determine a line. Answer: Points A and C lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true. Example 1

16. ARCHITECTURERefer to the picture. State the postulate that can be used to show the statement is true.A. Plane P contains points E, B, and G. A. Through any two points there is exactly one line. B. A line contains at least two points. C. A plane contains at least three noncollinear points. D. A plane contains at least two noncollinear points. Example 1

17. ARCHITECTURERefer to the picture. State the postulate that can be used to show the statement is true.B. Line AB and line BC intersect at point B. A. Through any two points there is exactly one line. B. A line contains at least two points. C. If two lines intersect, then their intersection is exactly one point. D. If two planes intersect, then their intersection is a line. Example 1

18. A. 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. Analyze Statements Using Postulates 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. Example 2

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

20. A. Determine whether the statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point. A. always B. sometimes C. never Example 2

21. B. Determine whether the statement is always, sometimes, or never true. 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. A. always B. sometimes C. never Example 2

22. Concept

23. Given: Proof: and must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on . Points A, C, and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line. Write a Paragraph Proof Prove: ACD is a plane. Example 3

24. Example 3

25. Proof: ? Example 3

26. A. Definition of midpoint B. Segment Addition Postulate C. Definition of congruent segments D. Substitution Example 3

27. Concept

28. End of the Lesson