1. A Framework with Definitions
Lesson 1.4
A Framework with Definitions pp

2. Objectives:
1. To identify the structure of a postulate system.
2. To define key geometric concepts.
3. To define postulate and theorem.

3. Theorems
Postulates
Definitions
Undefined terms

4. Definitions Collinear points are points that lie on the same line. k F
Points A, C, and F are collinear points because they lie on line k.

5. Definitions
Noncollinear points are points that do not lie on the same line. X Y m W
Points W, X, and Y are noncollinear points because no line could contain all of them.

6. Definitions
Concurrent Lines are lines that intersect at a single point. P a b c
Lines a, b, and c are concurrent because they intersect at point P.

7. Definitions Coplanar points are points that lie in the same plane. A Bq C B
Points A, B, and C are coplanar because they all lie in plane q.

8. Definitions Coplanar lines are lines that lie in the same plane. n m k
Lines m and n are coplanar lines in plane k.

9. Definitions Parallel lines are coplanar lines that do not intersect.
If lines l and m are parallel, we write l || m. The symbol “||” is read “is parallel to.”

10. Definitions Skew lines are lines that are not coplanar. l m k
Lines m and l are skew lines. No plane could contain both lines.

11. Definitions Parallel planes are planes that do not intersect. k s
Planes k and s are parallel: k||s.

12. A statement that can be shown to be true by a logical progression of previous terms and statements is a theorem. The process of justifying a theorem is called proving a theorem.

13. Postulates (sometimes called axioms) are assumed to be true
Postulates (sometimes called axioms) are assumed to be true. Theorems are proven from definitions, postulates and previous theorems.

14. Homework pp

15. 1. Name the lines that contain point K.
►A. Exercises
1. Name the lines that contain point K. K L M N

16. 3. Name all the lines shown.
►A. Exercises
3. Name all the lines shown. K L M N

17. 5. Name three sets of collinear points.
►A. Exercises
5. Name three sets of collinear points. K L M N

18. 7. Name three concurrent lines that intersect at point H.B D G F E
7. Name three concurrent lines that intersect at point H.

19. 9. Name the intersection of HC and CB.p G D F E n B C A H m
9. Name the intersection of HC and CB.

20. 11. Give four noncoplanar points.D F E n B C A H m
11. Give four noncoplanar points.

21. 11. Give four noncoplanar points.D E H p G D F E n B C A H m
11. Give four noncoplanar points.

22. 13. Name three coplanar lines.G D F E n B C A H m
13. Name three coplanar lines.

23. ►B. Exercises
15. Do you ever have to prove a postulate?
No, postulates are assumed to be true without proof.

24. ►B. Exercises
17. Do skew lines ever intersect?
No, skew lines are lines that are not coplanar.

25. ►B. Exercises
19. What is logic?
Logic is valid reasoning; it is step-by-step, principle-upon-principle thinking.

26. ■ Cumulative Review
21. Define space.

27. ■ Cumulative Review 22. Define subset (assume set and
element as undefined terms).

28. ■ Cumulative Review
23. If A  B and B  A, what can you
conclude?

29. ■ Cumulative Review 24. If A  B, what can you conclude
about sets A and B?

30. ■ Cumulative Review
25. Draw a picture to illustrate AB  CD
= {P}.