Advanced Geometry

Objectives After studying this chapter, you will be able to: 6.1 Relating Lines to Planes Understand basic concepts relating to planes Understand basic concepts relating to planes Identify four methods for determining a plane Identify four methods for determining a plane Apply two postulates concerning lines and planes 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 Recognize when a line is perpendicular to a plane Apply the basic theorem concerning perpendicularity of a line and 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 Recognize lines parallel to planes, parallel planes, and skew lines Use properties relating parallel lines and planes Use properties relating parallel lines and planes

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)

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

In Chapter 3, you learned that two points determine a line... A B AB,or line “m” m

In 6.1, you will learn the Four Ways to Determine a Plane (you must memorize and know these!!!) 1.Postulate: Three non-collinear points determine a plane. 2. Thm 45: A line and a point not on the line determine a plane. 3.Thm 46: Two intersecting lines determine a plane. 4.Thm 47: Two parallel lines determine a plane. A C m A B n m n m Did you notice how all three Theorems stem from the postulate: “Three Noncollinear points determine PLANE?”

Postulate: If a line intersects a plane not containing it, then the intersection is exactly one point. Postulate: If two planes intersect, their intersection is exactly one line. m P S m nB A one POINT a line intersects a plane one LINE two planes intersect

6.1 Relating Lines to Planes n m S P R V A B W m ∩ n = ___?___ AB

6.1 Relating Lines to Planes n m S P R V A B W A, B & V determine plane ___?___ m

6.1 Relating Lines to Planes n m S P R V A B W Name the foot of RS in m: ___?___ P

6.1 Relating Lines to Planes n m S P R V A B W AB & RS determine plane: ___?___ n

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

6.1 Relating Lines to Planes n m S P R V A B W Does W lie in plane n ? NO!

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

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

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

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

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. Two types of Perpendicularity: Definition: TWO LINES are perpendicular if they intersect at right angles. m E A C D B Given: AB  m Then: AB  BC, and AB  BD, and AB  BE Points E, C, and D determine plane m. intersect EVERY ONE of the lines FOOT foot

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. m A B C F Given: BF and CF lie in plane m AF  FB AF  FC Prove: AF  m Hint: See Theorem 48!

Sample Problem #1: m S If ∡STR is a right angle, can we conclude that ST  m ? ST  TR Rt ∡  segs ST  m ? Can’t be done with only one  line on m! See Theorem 48 again! T R ∡ S TR is Rt ∡ Given n 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!

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

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

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

Theorem 49: If a plane intersects two PARALLEL PLANES, the lines of intersection are parallel.

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

Parallelism of Lines and Planes – 1.If two planes are perpendicular to the same line, they are parallel to each other.

Parallelism of Lines and Planes – 2.If a line is perpendicular to one of two parallel planes, it is perpendicular to the other plane as well. n || m m n

Parallelism of Lines and Planes – 3.If two planes are parallel to the same plane, they are parallel to each other. n p m m || n p || n m || p

Parallelism of Lines and Planes – 4.If two lines are perpendicular to the same plane, they are parallel to each other. m

Parallelism of Lines and Planes – 5.If a plane is perpendicular to one of two parallel lines, the plane is perpendicular to the other line as well. m

6.3 Basic Facts about Parallel Planes m n A B CD Given: m || n AB lies in plane m CD lies in plane n AC || BD PROVE: AD bisects BC.

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

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) (1, 2, 6, 8 – 10)