1.1:identify points, lines, & planes 1.2:use segments & congruence objectives: 1.to learn...

1.1:1.1: Identify Points, Lines, & PlanesIdentify Points, Lines, & Planes1.2:1.2: Use Segments & CongruenceUse Segments & Congruence

Objectives:

1. To learn the terminology and notation of the basic building blocks of geometry

2. To use the Ruler and Segment Addition Postulates

3. To construct congruent segments with compass and straightedge

VocabularyVocabulary

Point Line Segment

Line Ray

Plane

In your notes, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

Undefined Terms?Undefined Terms?

What the Ancient Greeks said:“A point is that which has

no part. A line is breadthless length.”

In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined undefined termsterms, however, and cannot be made any simpler, so we just describe them.

Undefined Terms?Undefined Terms?

In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined undefined termsterms, however, and cannot be made any simpler, so we just describe them.

What the Ancient Chinese said:“The line is divided into

parts, and that part which has no remaining part is a point.”

PointsPoints

• Basic unit of geometry

• No size, only location

• ZERO dimensions

• Represented by a dot and named by a CAPITAL letter

P

Mathematical model of a point

PointsPoints

P

Mathematical model of a point

A star is a physical model of a point

LinesLines• Straight arrangement of

points• No width, only length• Extends forever in 2

directions• ONE dimension• Named by two points on

the line: line AB or BA or• or

• It can be named with a

lower-case script letter: l

A

B

Mathematical model of a line

AB BA

l

LinesLines

A

B


Spaghetti is a physical model of a line

LinesLines

• How many lines can you draw through any two points?

A

B


ONE

Collinear PointsCollinear Points

• Collinear pointsCollinear points are points that --?--.– Points A, B, and C are collinear

A

B

C

Lie on a line

What do you supposed NON-collinear means?

PlanesPlanes

• Flat surface that extends forever

• Length and width but no height (2-D)

• Represented by a 4-sided figure and named by a capital script letter or 3 (non-collinear) letters on the same plane.

Mathematical model of a plane

PlanesPlanes


Flattened dough is a physical model of a plane

PlanesPlanes

• How many points does it take to define a plane?

• (tell where the plane is in space?)


THREE

Coplanar PointsCoplanar Points

• Coplanar pointsCoplanar points are points that --?--.– Points A, B, and C are coplanar

Lie on the same plane

What do you suppose NON-coplanar means?

Hierarchy of Building BlocksHierarchy of Building Blocks

0-D

1-D

2-D

3-D

SpaceSpace is the set of all points

A Romance of Many A Romance of Many DimensionsDimensionsAre there more than three spatial dimensions?

?

Point Segment Square Cube

0-D 1-D (Length) 2-D (Area) 3-D (Volume)

1 point 2 points 4 points 8 points

0 “sides” 2 “sides” 4 “sides” 6 “sides”


Point Segment Square Cube Hypercube

0-D 1-D (Length) 2-D (Area) 3-D (Volume)





0-D 1-D (Length) 2-D (Area) 3-D (Volume) 4-D (Hypervolume)






1 point 2 points 4 points 8 points 16 points





1 point 2 points 4 points 8 points 16 points

0 “sides” 2 “sides” 4 “sides” 6 “sides” 8 “sides”

WATCH the following

Computer animation of hypercube

Explanation of 4th dimension

https://www.youtube.com/watch?v=iXYXuHVTS_k

https://www.youtube.com/watch?v=9DoSqeJNG74

Example 1Example 1

1. Give two other names for and plane R.

2. Name three points that are collinear.

3. Name four points that are coplanar.

PQ

Line SegmentLine Segment

• A line segmentline segment consists of two endpoints and all the collinear points between them.– Line segment AB or

A

B

AB

Endpoints

Congruent SegmentsCongruent Segments

• Congruent segmentsCongruent segments are line segments that have the same length.

Symbol for congruent

Copying a SegmentCopying a Segment

We’re going to try making two congruent segments using only a compass and a straightedge. Here, we’re not using a ruler to measure the length of the segment!

1. Draw segment AB.


2. Draw a line with point A’ on one end.


3. Put point of compass on A and the pencil on B. Make a small arc.


4. Put point of compass on A’ and use the compass setting from Step 3 to make an arc that intersects the line. This is B’.



Click on the image to watch a video of the construction.

http://www.mathopenref.com/constcopysegment.html

RayRay

• A rayray consists of an endpoint and all of the collinear points to one side of that endpoint.– Ray AB or AB

A laser is a physical model of a ray

Example 2Example 2

Ray BA and ray BC are considered opposite rays. Use the picture to explain why.

At what time would the hands of a clock form opposite rays?

A

B

C

3:15, 6:00, 1:35,…

Example 3Example 3

1. Give another name for .

2. Name all rays with endpoint J. Which of these rays are opposite rays?

GH

J

E

FH

G

HG

JGJHJFJE

IntersectionIntersection

• Two or more geometric figures intersectintersect if they have one or more points in common. The intersectionintersection of the figures is the set of points the figures have in common.

The intersection of two lines is a point.

The intersection of two planes is a

line.

Example 4Example 4

What is the length of segment AB?

AB

Example 4Example 4

You basically used the Ruler Postulate to find the length of the segment, where A corresponds to 0 and B corresponds to 6.5. So AB = |6.5 – 0| = 6.5 cm

AB

Example 5Example 5

Now what is the length of ?

AB

AB

8.5 – 2 = 6.5

Ruler PostulateRuler Postulate

The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is its coordinatecoordinate.

The distancedistance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

Example 6Example 6

When asked to measure the segment below, Kenny gave the answer 2.7 inches. Explain what is wrong with Kenny’s measurement. Inches are not divided into tenths

Give Them an Inch…Give Them an Inch…

A Standard English ruler has 12 inches. Each inch is divided into parts. • Cut an inch in half, and you’ve got 1/2 an inch.

• Cut that in half, and you’ve got 1/4 an inch. • Cut that in half, and you’ve got 1/8 inch. • Cut that in half, and you’ve got 1/16 inch.

Click the ruler and practice measuring both inches and centimeters

http://www.rulergame.net/

Example 7Example 7

Let’s say you found the length of a segment to be 6’ 7” using your dad’s tape measure. Convert this measurement to the nearest tenth of a centimeter (1” ≈ 2.54 cm).

6 7/12 = 6.58 1 = 2.546.58 XCross-multiplyX= 16.71

Example 8Example 8

Use the diagram to find GH.

15

Example 8Example 8

Use the diagram to find GH.

Could you as easily find GH if G was not collinear with F and H? Why or why not?

No, the parts wouldn’t equal the whole. FG + GH = FH

Segment Addition PostulateSegment Addition Postulate

• If B is betweenbetween A and C, then AB + BC = AC.• If AB + BC = AC, then B is betweenbetween A and C.

BETWEEN BETWEEN indicates collinear points

Example 9Example 9Point A is between S and M. Find x if SA = 2x – 5, AM = 7x + 3, and SM = 25.

2x – 5 + 7x+3 = 259x = 27X=3

A MS

25

2x – 5 7x + 3

Example 10Example 10

Point E is between J and R. Find JE given that JE = x2, ER = 2x, and JR = 8.

WORK IT OUT WITH A PICTURE!!

X = 2, why does it not equal -4?

Example 11: SATExample 11: SAT

Points E, F, and G all lie on line m, with E to the left of F. EF = 10, FG = 8, and EG > FG. What is EG?

Work it out with a picture.

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1.1:identify points, lines, & planes 1.2:use segments & congruence objectives: 1.to learn...

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