Advanced Geometry

ObjectivesObjectivesAfter studying this chapter, you will be able After studying this chapter, you will be able to:to:6.1 Relating Lines to Planes6.1 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 Plane6.2 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 Apply the basic theorem concerning perpendicularity of a line and plane a line and plane

6.3 Basic Facts about Parallel Planes6.3 Basic Facts about Parallel Planes•Recognize lines parallel to planes, parallel planes, and Recognize lines parallel to planes, parallel planes, and skew lines skew lines •Use properties relating parallel lines and planes Use properties relating parallel lines and planes

New New VocabularyVocabulary6.1 Foot 6.1 Foot [of a line] [of a line] - - The point of intersection of a line The point of intersection of a line and a plane. (p 270)and a plane. (p 270)

6.3 Skew 6.3 Skew [lines] [lines] – – Two lines that are not coplanar. Two lines that are not coplanar. (p 283)(p 283)

Important Related Vocabulary from Important Related Vocabulary from previous chapters:previous chapters:CollinearCollinear – – lying on the lying on the same LINE same LINE (1.3, P 18)(1.3, P 18)PlanePlane – – SURFACESURFACE such that if any two points on the such that if any two points on the surface are connected bysurface are connected by a line, a line, all points of the lineall points of the line are also on the are also on the surface (4.5, P 192)surface (4.5, P 192)CoplanarCoplanar – – Lying on the Lying on the same PLANE same PLANE (4.5, P 192)(4.5, P 192)NoncoplanarNoncoplanar - - points, lines, segments (etc.) that points, lines, segments (etc.) that DO NOTDO NOT lie on the lie on the same PLANE same PLANE (4.5, P 192)(4.5, P 192)IntersectIntersect - - to to OVERLAPOVERLAP a figure or figures geometrically a figure or figures geometrically so as to have so as to have a pointa point or or set of pointsset of points in in COMMONCOMMON (1.1, Pp 5 - 6) (1.1, Pp 5 - 6)Parallel Parallel [lines] - [lines] - COPLANARCOPLANAR lines that lines that DO NOTDO NOT intersect (4.5, intersect (4.5, P 195)P 195)

In Chapter 3, you learned that two points determine a line . . .

AA

BBAB, AB,

or or

line “m”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.

AA

CC

mm

AA

BB

nn

mm

nn

mm

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.

mmPP

SS

mm

nnBB

AA

one POINT

a line intersects a plane

one LINE

two planes intersect

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWWm ∩ n = ___?___ AB

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWWA, B & V determine

plane ___?___m

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWWName the foot of RS in m: ___?___ P

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWWAB & RS determine

plane: ___?___nn

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

AB & point ___?___determine plane n.R or R or SS

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

Does W lie in plane n ? NO!NO!

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

Line AB and line Line AB and line ___?______?___determine plane determine plane mm..VWVW

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

A, B, V and A, B, V and ___?______?___are coplanar points.are coplanar points.W or PW or P

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

A, B, V and A, B, V and ___?______?___are NONcoplanar are NONcoplanar points.points.R or SR or S

6.1 Relating Lines to Planes nn

mm

SSPPRR

VVAA

BBWW

If R & S lie in plane n, If R & S lie in plane n, what can be said what can be said about RS ?about RS ?RS lies in RS lies in plane n!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.

mm

EE

AA

CC

DDBB

Given: AB m

Then: AB BC, and AB BD,

andAB BE

Points E, C, and D determine plane m.

intersect

EVERY ONE of the linesFOO

T

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.

mm

AA

BBCC

FF

Given: BF and CF lie in plane mAF

FBAF FCProve: AF m

Hint: See Theorem 48!

Sample Problem #1:

mmSS

If ∡STR is a right angle, can we conclude that ST m ?ST TR Rt ∡ segsST m ? Can’t be done with only one line on m!See Theorem 48 again!

TT

RR

∡ STR is Rt ∡ Givennn

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

nnDD

BBEE CC

AAGiven:Given:B, C, D, and E lie in B, C, D, and E lie in plane n.plane n.AB AB n nBE BE bisector of CD bisector of CD PROVE:Δ ADC is isosceles.

6.2 Perpendicularity of a Line and a Plane

nDD

BBEE CC

AAGiven:Given:B, C, D, and E lie in B, C, D, and E lie in plane n.plane n.AB AB n nBE BE bisector of CD bisector of CD PROVE:Δ ADC is isosceles.

Reasons6.2 Perpendicularity of a Line and a Plane nDD EE CC

AA

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 ∡ABD8. BC BD 9. AB AB 10. ΔABC ΔABD 11. AD AC

GivenIf a line is to a plane, it is to every line in the plane that passes through its foot

BBSame as #2 Segs form Rt ∡s Same as #4All Rt ∡s are GivenIf a point is on bisector, it is =dist from the segment’s endpoints ReflexiveSAS (8, 6, 9)CPCTCIf 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 intersectDefinition: Two planes are PARALLEL if they do not intersectDefinition: 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 || mmm

nn

Parallelism of Lines and Planes –

3.If two planes are parallel to the same plane, they are parallel to each other.

nn

pp

mm m || np || nm || p

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

mm

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.

mm

6.3 Basic Facts about Parallel Planesmm

nn

AA BB

CC DD

Given:Given:m || nm || nAB lies in plane mAB lies in plane mCD lies in plane nCD lies in plane nAC || BD AC || BD PROVE: AD bisects PROVE: AD bisects BC.BC.

6.3 Basic Facts about Parallel Planes mm

nn

AA BB

CC DD

1. m || n1. m || n

8. AD bisects 8. AD bisects BC.BC.

2. AB lies in 2. AB lies in mm3. CD lies in 3. CD lies in nn4. AC || BD 4. AC || BD 5. AC and BD det plane 5. AC and BD det plane ACDBACDB6. AB || 6. AB || CD CD 7. ACDB is a 7. ACDB is a parallelogramparallelogram

givengivengivengivengivengivengivengivenIf a plane intersects two || planes, the If a plane intersects two || planes, the lines of intersection are ||lines of intersection are ||Two || lines determine a planeTwo || lines determine a plane

If both pairs of opp sides of a a quad are ||, it If both pairs of opp sides of a a quad are ||, it is a parallelogramis a parallelogramIn a parallelogram the diagonals bisect In a parallelogram the diagonals bisect each othereach other

StatementsStatements ReasonsReasons

6.1 Pp 273 – 274 (2, 5, 7, 8, 6.1 Pp 273 – 274 (2, 5, 7, 8, 15);15);6.2 Pp 278 – 279 (1 – 8, 10); 6.2 Pp 278 – 279 (1 – 8, 10); 6.3 Pp Pp 284 – 285 (1, 4, 6); 6.3 Pp Pp 284 – 285 (1, 4, 6);

Ch 6 Review Pp 288 – 289 Ch 6 Review Pp 288 – 289 (1, 2, 6, 8 – (1, 2, 6, 8 – 10)10)