Math 310Math 310

Section 9.1Section 9.1

Geometry IntroductionGeometry Introduction

Axiomatic SystemAxiomatic System

A logical system which possesses an A logical system which possesses an explicitly stated set of axioms from explicitly stated set of axioms from which theorems can be derived. which theorems can be derived.

(mathworld.wolfram.com)(mathworld.wolfram.com)

www.xkcd.com

Undefined TermsUndefined Terms

PointPoint LineLine PlanePlane

““One must be able to say One must be able to say at all times—instead of at all times—instead of points, straight lines, points, straight lines, and planes—tables, and planes—tables,

chairs, and beer mugs.” chairs, and beer mugs.” - David Hilbert- David Hilbert

Linear “Notions” Linear “Notions”

Collinear pointsCollinear points Is betweenIs between Line segment (segment)Line segment (segment) RayRay

Collinear points (& non Collinear points (& non collinear)collinear)

B C

DE

Is BetweenIs Between

B CE

Line SegmentLine Segment

F

G

RayRay

H

I

Planar “Notions”Planar “Notions”

Coplanar pointsCoplanar points Noncoplanar pointsNoncoplanar points Coplanar linesCoplanar lines Skew linesSkew lines Intersecting linesIntersecting lines Concurrent linesConcurrent lines Parallel linesParallel lines

Coplanar & Noncoplanar Coplanar & Noncoplanar PointsPoints

A CB

D

Coplanar linesCoplanar lines

F

I

H

GED

Skew LinesSkew Lines

F

I

HE

Intersecting LinesIntersecting Lines

K

LJ

Concurrent LinesConcurrent Lines

K

LJ

M

Parallel LinesParallel Lines

N O

QP

Properties of “tables, chairs Properties of “tables, chairs and beer mugs”and beer mugs”

1.1. There is exactly one line that contains There is exactly one line that contains any two distinct pointsany two distinct points

2.2. If two points lie in a plane, then the line If two points lie in a plane, then the line containing the points lies in the plane.containing the points lies in the plane.

3.3. If two distinct planes intersect, then If two distinct planes intersect, then their intersection is a line.their intersection is a line.

4.4. There is exactly one plane that contains There is exactly one plane that contains any three distinct noncollinear points.any three distinct noncollinear points.

Properties (cont)Properties (cont)

5.5. A line and a point not on the line A line and a point not on the line determine a plane.determine a plane.

6.6. Two parallel lines determine a Two parallel lines determine a planeplane

7.7. Two intersecting lines determine a Two intersecting lines determine a plane.plane.

Property 1Property 1

A B

A

B

Property 2Property 2

Property 3Property 3

A

B

Property 4Property 4

Property 5Property 5

Property 6Property 6

Property 7Property 7

Intersecting PlanesIntersecting Planes

ParallelParallel Along a lineAlong a line

Parallel PlanesParallel Planes

Planes Intersecting Planes Intersecting along a linealong a line

Angle, Vertex, SideAngle, Vertex, Side

DefDefWhen two rays share an endpoint, an When two rays share an endpoint, an angleangle is is

formed. The common initial point of the rays is formed. The common initial point of the rays is the the vertexvertex of the angle. Each ray is called a of the angle. Each ray is called a sideside of the angle. of the angle.

Ex.Ex.

B

A

CVertex

Side

Angle: <ABC

Ex.Ex.

F

EG

B

E

FI

H

J<EBF

<I

<GFE

J K

L

M

N

O

Ex.Ex.

Name all six angles.

<MJK

<NKL

<OLJ

<KJL

<LKJ

<JLK

Angle MeasureAngle Measure

To measure an angle we use the unit To measure an angle we use the unit degreedegree. It measures the “opening” of . It measures the “opening” of the angle. The largest angle measure the angle. The largest angle measure is 360° and the smallest is 0°. A is 360° and the smallest is 0°. A complete rotation about a point is 360°. complete rotation about a point is 360°. For more accuracy, angles can be For more accuracy, angles can be further measure in further measure in minutesminutes, and , and secondsseconds. Each degree is divided into . Each degree is divided into 60 minutes, and each minute is divided 60 minutes, and each minute is divided into 60 seconds.into 60 seconds.

Ex.Ex.

Add: 45°23’47” and 62°36’51”Add: 45°23’47” and 62°36’51”

45°23’47” + 62°36’51” = 108°0’38” or 45°23’47” + 62°36’51” = 108°0’38” or 108°38” 108°38”

Add: 145°17’4” and 220°31’32”Add: 145°17’4” and 220°31’32”

145°17’4” + 220°31’32” = 365°48’36” = 145°17’4” + 220°31’32” = 365°48’36” = 5°48’36” 5°48’36”

Ex.Ex.

Solve for x.Solve for x.

B

C

A D

m<ABC = 80°

m<ABD = 30°

m<DBC = (x – 25)°

m<ABC = 82°

m<ABD = (x – 13)°

m<DBC = (x + 7)°

x = 75

x = 44

ProtractorProtractor

A protractor is a standard tool for A protractor is a standard tool for measuring angles. To use, line the measuring angles. To use, line the vertex vertex up with the center of the up with the center of the base of the protractor and line one base of the protractor and line one sideside of the angle up with the 0° of the angle up with the 0° mark. Now measure from 0°, mark. Now measure from 0°, increasing, until you see the other increasing, until you see the other sideside of the angle and read the mark. of the angle and read the mark.

Types of AnglesTypes of Angles

ObtuseObtuse AcuteAcute RightRight StraightStraight

Ex.Ex.

I

H

J

B

E

F

B

A

C

acute

straight

obtuseF

H

G

right

Perpendicular LinesPerpendicular Lines

Def.Def.

Two lines are perpendicular if they Two lines are perpendicular if they intersect and form right angles.intersect and form right angles.

perpendicular

not

perpendicular

Line Perpendicular to a Line Perpendicular to a Plane Plane

A line is perpendicular to a plane if it is A line is perpendicular to a plane if it is perpendicular to every line, contained perpendicular to every line, contained in the plane, passing through the in the plane, passing through the point of intersection.point of intersection.

Ex.Ex.

Ex.Ex.

QuestionsQuestions

Is it possible for a line intersecting a plane Is it possible for a line intersecting a plane to be perpendicular to exactly one line in to be perpendicular to exactly one line in the plane through its intersection with the the plane through its intersection with the plane?plane?

Can a line intersecting a plane be Can a line intersecting a plane be perpendicular to exactly two distinct lines perpendicular to exactly two distinct lines in the plane going through the point of in the plane going through the point of intersection?intersection?

YesYes NoNo

Line Perpendicular to a Line Perpendicular to a PlanePlane

ThrmThrm

A line perpendicular to two distinct A line perpendicular to two distinct lines in the plane through its lines in the plane through its intersection with the plane is intersection with the plane is perpendicular to the plane.perpendicular to the plane.