Lesson 1.6 Incidence Theorems pp. 25-29

Objectives: 1. To prove some incidence theorems from the Incidence Postulates.

Theorem 1.1 If two distinct lines intersect, they intersect in one and only one point.

If two lines intersect, how many points of intersection are there?

Consider the diagram. A B C How many lines pass through both A and B?

Consider the diagram. A B C How many planes pass through AB?

Consider the diagram. A B C How many planes can pass through A, B, and C?

Consider the diagram. A B C How many planes pass through any three given noncollinear points?

Consider the diagram. A B C How many planes pass through point C and AB?

Theorem 1.2 A line and a point not on that line are contained in one and only one plane.

Theorem 1.3 Two intersecting lines are contained in one and only one plane.

Consider these lines:

Theorem 1.4 Two parallel lines are contained in one and only one plane.

Consider these lines:

Homework pp. 28-29

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 1. Does AB lie in plane m? m A C D B E

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 3. How many planes pass through points A, E, and D? m A C D B E

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 5. How many planes pass through CD and point E? m A C D B E

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 7. What is the intersection of ED and DB? m A C D B E

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 7. Is there more than one point of intersection? m A C D B E

►A. Exercises Answer each question, explain your answer, and state a definition, postulate, or theorem that supports your answer. 9. If AD is parallel to CB, will these two lines ever intersect? m A C D B E

11. A line and a point not on that line ►B. Exercises 11. A line and a point not on that line are contained in one and only one plane. B A C D H E G F

13. Two parallel lines are contained in one and only one plane. ►B. Exercises 13. Two parallel lines are contained in one and only one plane. B A C D H E G F

15. If two lines intersect, then the lines ►B. Exercises 15. If two lines intersect, then the lines are contained in one and only one plane. B A C D H E G F

►B. Exercises 17. Are BC and EF skew lines? B A C D H E G F

►B. Exercises 19. Give the reasoning behind this statement: “There are at least three lines in any plane.”

■ Cumulative Review 27. Name three undefined terms.

■ Cumulative Review 28. Name three defined terms.

■ Cumulative Review 29. Name a postulate.

■ Cumulative Review 30. State a theorem.

■ Cumulative Review 31. According to the postulates and theorems so far, must there be an infinite number of points in space? Explain.