A Framework with Definitions Lesson 1.4 A Framework with Definitions pp. 16-20

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

Theorems Postulates Definitions Undefined terms

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.

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.

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.

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

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

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.”

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.

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

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.

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.

Homework pp. 19-20

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

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

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

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.

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.

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

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

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

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

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

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

■ Cumulative Review 21. Define space.

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

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

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

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