Chapter 6 Lines and Planes in Space

1. Advanced Geometry Chapter 6Lines and Planes in Space

2. Ch 6: Lines and Planes in Space Objectives After studying this chapter, you will be able to: • 6.1 Relating Lines to Planes • Understand basic concepts relating to planes • Identify four methods for determining a plane • Apply two postulates concerning lines and planes • 6.2 Perpendicularity of a Line and a Plane • Recognize when a line is perpendicular to a plane • Apply the basic theorem concerning perpendicularity of a line and plane • 6.3 Basic Facts about Parallel Planes • Recognize lines parallel to planes, parallel planes, and skew lines • Use properties relating parallel lines and planes

3. Ch 6: Lines and Planes in Space New Vocabulary 6.1 Foot [of a line] - The point of intersection of a line and a plane. (p 270) 6.3 Skew [lines] – Two lines that are not coplanar. (p 283)

4. 6.1 Relating Lines to Planes Important Related Vocabulary from previous chapters: Collinear – lying on the same LINE (1.3, P 18) Plane – SURFACE such that if any two points on the surface are connected by a line, all points of the line are also on the surface (4.5, P 192) Coplanar – Lying on the same PLANE (4.5, P 192) Noncoplanar - points, lines, segments (etc.) that DO NOT lie on the same PLANE (4.5, P 192) Intersect - to OVERLAP a figure or figures geometrically so as to have a point or set of points in COMMON (1.1, Pp 5 - 6) Parallel [lines] - COPLANAR lines that DO NOT intersect (4.5, P 195)

5. 6.1 Relating Lines to Planes In Chapter 3, you learned that two points determine a line . . . A AB, or line “m” B m

6. 6.1 Relating Lines to Planes Did you notice how all three Theorems stem from the postulate: “Three Noncollinear points determine PLANE?” In 6.1, you will learn the Four Ways to Determine a Plane (you must memorize and know these!!!) • Postulate: Three non-collinear points • determine a plane. • Thm 46: Two intersecting lines • determine a plane. m A B n C • Thm 47: Two parallel lines • determine a plane. 2. Thm 45: A line and a point not on the line determine a plane. A m m n

7. 6.1 Two Postulates Concerning Lines and Planes that may be assumed : Postulate: If a line intersects a plane not containing it, then the intersection is exactly one point. a line intersects a plane S one POINT m P Postulate: If two planes intersect , their intersection is exactly one line. two planes intersect m one LINE B n A

8. 6.2 Perpendicularity of a Line and a Plane Two types of Perpendicularity: Definition: TWO LINES are perpendicular if they intersect at right angles. intersect EVERY ONE of the lines Definition: A line is perpendicular TO A PLANE if it is perpendicular to EVERY ONE of the lines in the plane that pass through its foot. FOOT A Points E, C, and D determine plane m. Given: AB  m C E foot Then: AB  BC, and B m D AB  BD, and AB  BE

9. 6.2 Perpendicularity of a Line and a Plane Theorem 48: If a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane. Given: BF and CF lie in planem A AF  FB AF  FC m Prove: AF  m B F C Hint: See Theorem 48!

10. 6.2 Perpendicularity of a Line and a Plane Sample Problem #1: If ∡STR is a right angle, can we conclude that ST  m ? S ∡ STR is Rt ∡ Given ST  TR Rt ∡  segs m ST  m ? Can’t be done with only one  line on m! T R n See Theorem 48 again! To be perpendicular to plane m, ST must be perpendicular to at least TWO lines that lie in m, AND pass through T, the FOOT of ST!

11. 6.3 Basic Facts About Parallel Planes Theorem 49: If a plane intersects two PARALLEL PLANES, the lines of intersection are parallel.

12. 6.3 Basic Facts About Parallel Planes Definition: A line and a plane are PARALLEL if they do not intersect Definition: Two planes are PARALLEL if they do not intersect Definition: Two lines are SKEW if they are NOT coplanar

13. 6.3 Properties Relating Parallel Lines and Planes • Parallelism of Lines and Planes – • If two planes are perpendicular to the same line, they are parallel to each other.

14. 6.3 Properties Relating Parallel Lines and Planes • Parallelism of Lines and Planes – • If a line is perpendicular to one of two parallel planes, • it is perpendicular to the other plane as well. n n || m m

15. 6.3 Properties Relating Parallel Lines and Planes • Parallelism of Lines and Planes – • If two planes are parallel to the same plane, • they are parallel to each other. m|| n m m||p n p || n p

16. 6.3 Properties Relating Parallel Lines and Planes • Parallelism of Lines and Planes – • If two lines are perpendicular to the same plane, • they are parallel to each other. m

17. 6.3 Properties Relating Parallel Lines and Planes • Parallelism of Lines and Planes – • If a plane is perpendicular to one of two parallel lines, • the plane is perpendicular to the other line as well. m

18. Ch 6 Sample Problems 6.1 Relating Lines to Planes n AB R m ∩ n = ___?___ A m W V P B S

19. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P A, B & V determine plane ___?___ B S m

20. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P Name the foot of RS in m: ___?___ B S P

21. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P AB & RS determine plane: ___?___ B S n

22. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P B R or S AB & point ___?___ determine plane n. S

23. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P B Does W lie in plane n ? S NO!

24. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P B VW Line AB and line ___?___ determine plane m. S

25. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P B W or P A, B, V and ___?___ are coplanar points. S

26. Ch 6 Sample Problems 6.1 Relating Lines to Planes n R A m W V P B R or S A, B, V and ___?___ are NONcoplanar points. S

27. Ch 6 Sample Problems See Sample Problem #2 on page 272 for a Proof! 6.1 Relating Lines to Planes If R & S lie in plane n, what can be said about RS ? n R A m W V P RS lies in plane n! B S

28. Ch 6 Sample Problems 6.2 Perpendicularity of a Line and a Plane A Given: B, C, D, and E lie in plane n. AB  n BE  bisector of CD B C n E PROVE: Δ ADC is isosceles. D

29. Ch 6 Sample Problems 6.2 Perpendicularity of a Line and a Plane A Given: B, C, D, and E lie in plane n. AB  n BE  bisector of CD B C n E PROVE: Δ ADC is isosceles. D

30. A Ch 6 Sample Problems 6.2 Perpendicularity of a Line and a Plane B n Statements Reasons C Given 1. AB  n E D If a line is  to a plane, it is  to every line in the plane that passes through its foot 2. AB  BD 3. AB  BC Same as #2  Segs form Rt ∡s 4. ∡ABC is Rt ∡ Same as #4 5. ∡ABD is Rt ∡ All Rt ∡s are  6. ∡ABC  ∡ABD Angle Given 7. BE  bisector of CD If a point is on  bisector, it is =dist from the segment’s endpoints 8. BC  BD Side Reflexive 9. AB  AB Side SAS (8, 6, 9) 10. ΔABC ΔABD CPCTC 11. AD  AC 12. Δ ADC is isosceles If a Δ has at least two  sides, then it is isosceles

31. Ch 6 Sample Problems 6.3 Basic Facts about Parallel Planes B A Given: m || n AB lies in plane m CD lies in plane n AC || BD m n PROVE: AD bisects BC. C D

32. Ch 6 Sample Problems B A m 6.3 Basic Facts about Parallel Planes Statements Reasons 1. m || n given n 2. AB lies in m given given 3. CD lies in n C D given 4. AC || BD 5. AC and BD det plane ACDB Two || lines determine a plane If a plane intersects two || planes, the lines of intersection are || 6. AB || CD 7. ACDB is a parallelogram If both pairs of opp sides of a a quad are ||, it is a parallelogram 8. AD bisects BC. In a parallelogram the diagonals bisect each other

33. Ch 6 Assignments 6.1 Pp 273 – 274 (2, 5, 7, 8, 15); 6.2 Pp 278 – 279 (1 – 8, 10); 6.3 Pp Pp 284 – 285 (1, 4, 6); Ch 6 Review Pp 288 – 289 (1, 2, 6, 8 – 10)