Pre-AP GeometryUnit 1Points, Lines, Planes, and Angles

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Introduction to Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space.

Geometry is one of the oldest sciences.

Introduction to GeometryGeometry is a type of math used to measure things that are impossible to measure with devices.

For example, no one has been able take a tape measure around the earth, yet we are pretty confident that the circumference of the planet at the equator is 24,901.473 miles.

The first known case of calculating the distance around the earth was done by Eratosthenes around 240 BC.

Introduction to GeometryThe study of Geometry includes proofs.

Proofs are not unique to Geometry. They could have been done in Algebra or delayed until Calculus.

The reason that high school geometry almost always spends a lot of time with proofs is that the first great Geometry textbook, "The Elements," was written by Euclid, exclusively with proofs.

Introduction to Geometry

This course, for the most part, is based on Euclidean geometry.

"Euclidean" (or "elementary") refers to the book "The Elements" written over 2,000 years ago by a man named Euclid.

Introduction to Geometry

Euclid started with some basic concepts.

He built upon those concepts to create more and more concepts.

His structure and method influence the way that geometry is taught today.

Introduction to Geometry

This course will include more than just facts about geometric objects; the ability to "prove" that a particular answer is correct using logic and reason is the most important part of this course.

Basic Figures

Point ▫A point is a geometric element that has

position but no dimensions and is used to define an exact location in space.

▫A point has no volume, area, or length, making it a zero dimensional object.

▫A point is defined by its coordinates.

Basic Figures

Line▫A line can be described as an ideal zero-

width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points.

▫In Euclidean geometry, exactly one line can be found that passes through any two points.

Basic Figures

Equidistant - equally distant from any two or more points.

Basic Figures

10m

X

Equal distances

Suppose that you and a friend are partners in a game in which you must locate various clues to win. You are told to pick up your next clue at a point that:1.Is as far from the fountain as from the oak tree and2. Is 10 m from the telephone pole.

You locate X, which satisfies both requirements, but grumble because there simply isn’t any clue to be found at X.

Is there another

location that satisfies both requirements?

Discussion in GSP

Review Quiz

1. Which language does the word “geometry” come from and what does it mean?

2. What is the circumference of the Earth at the equator? (exact value is preferred, but the nearest 1000 is acceptable)

3. Who wrote the book “The Elements”?4. Which geometric element has dimensions

of zero?5. How many points does it take to define a

line?

Written Exercises 1.1, p. 3: # 1 - 5

Points, Lines and PlanesLesson 1.2Pre-AP Geometry

Points, Lines, and Planes

The three most basic figures in geometry are points, lines, and planes.

This lesson illustrates how these three basic figures relate to one another.

Objectives

1. Use the undefined terms point, line, and plane.

2. Draw representations of points, lines, and planes.

3. Use the terms collinear, coplanar, and intersection.

VocabularyPoint

A geometric element that has position but no dimension. A point is defined by its coordinates. Symbol: · A

LineA line can be described as an ideal zero-width, infinitely long, perfectly straight curve containing an infinite number of points.

Symbol:

(The term curve in mathematics includes "straight curves")

AB

Vocabulary

PlaneA plane is a two-dimensional surface that is perfectly flat, is infinitely vast, and infinitesimally thin.

Undefined term A term, such as point, line, plane, and space, that is accepted without definition.

Vocabulary

SpaceThe set of all points. The unlimited area which extends in all directions and within which all things exist.

IntersectTo meet or cross at a point.

VocabularyCollinear

Lying on the same straight line.Collinear points lie along a straight line.Any two points are always collinear.

CoplanarLying in the same plane.A set of points in space is coplanar if the points all lie in the same plane.

Note: Collinear points are automatically coplanar, but coplanar points are not necessarily collinear.

Points

Things that we can use to represent a point:

a marble

Lines

Things that we can use to represent a line:

a taut piece of string

Plane

Things that we can use to represent a plane:

a sheet of papera poster board

Intersecting Planes

Things that we can use to represent intersecting planes:

the sides of a cardboard box

Practice Quiz - True or False

1. ends at P.2. Point S is on an infinite number of lines.3. A plane has no thickness.4. Collinear points are coplanar.5. Planes have edges.6. Two planes intersect in a line segment.7. Two intersecting lines meet in exactly

one point.8. Points have no size.

PF

Problem Set 1.2Written Exercises

p.7: # 2 – 26 even, 27 - 36

Lesson 1.3: Segments, Rays, and Distance

Pre-AP Geometry

Points, Lines, and Planes• Line Segment

▫ Two points (called the endpoints) and all the points between them that are collinear with those two points

Named line segment AB, AB, or BA

line AB segment AB 

A B A B

Length of a segment

•Length of BC is stated as BC. It is the distance between points B and C.

•On a number line, length of a segment is found by subtracting the coordinates of the endpoints.

•On a coordinate plane, length of a segment is found using the distance formula D =

2 22 1 2 1( ) ( )x x y y

Examples

•Find the length between 5 and -3 on the number line

Find the distance of segment AB if A(-3, 5) and B(2, -7)

Postulates• Postulate: statement that is accepted without

proof• Segment Addition Postulate

▫If B is between A and C, then AB + BC = AC• Ruler Postulate

1.The points on a line can be paired with the real numbers in such a way that any two points can have the coordinates 0 and 1

2.Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

Examples

•EG = 7x + 3 EF = 3x + 8 FG = 2x + 1

1.Find x:2.Find EG:3.Find EF:4.Find FG:

EF G

Segment Length terms• Congruent- two objects that have the same size and shape. We use

the symbol to show that two objects are congruent.≅• Congruent segments- two segments with equal lengths.

Example: DE FG≅Midpoint of a segment: a point that divides a segment into two

congruent segments.Midpoint formula: M = ( )

Segment bisector: A line, segment, ray, or plane which intersects a segment at its midpoint.

2 1 2 1,2 2

x x y y

Points, Lines, and Planes• Ray

▫ Part of a line that starts at a point and extends infinitely in one direction.

• Initial Point▫ Starting point for a ray.

• Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D▫ “It begins at C and goes through D and on forever”

Points, Lines, and Planes

•Opposite Rays▫If C is between A and B, then CA and CB

are opposite rays.▫Together they make a line.

A BC

Lesson 1.3 homework

•P. 15 # 2-40 evens

Lesson 1.4: Angles

Parts of an angle

•Sides of an angle are made up of rays•The rays meet at a point called the vertex

vertexsides

Naming an angle• An angle can be named by the vertex, by the 3 points

on the angle: the side, the vertex and the other side, or a number inside the angle.

The angle can be named ∠GHI, ∠IHG, ∠H, or ∠1

G

I

H

1

Classifying angles

•Acute angle: Angle measuring greater than 0° and less than 90°.

•Obtuse angle: Angle measuring greater than 90° and less than 180°

•Right angle: An angle measuring exactly 90°

•Straight angle: An angle measuring exactly 180°

Angle Postulates• Protractor Postulate:On AB in a given plane, choose any point O between A

and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with real numbers from 0 to 180 in a way such that:a. OA is paired with 0, and OB with 180b. If OP is paired with x, and OQ with y, the m∠POQ = │x - y │

• Angle addition postulate:-If B lies on the interior of ∠AOC, then m ∠AOB +

m∠BOC = m∠AOC-If ∠AOC is a straight angle, then m∠AOB+m ∠BOC =

180.

Angle Vocabulary

•Congruent Angles▫Two angles with equal measures

•Adjacent angles▫Angles which share a vertex and a common

side, but no common interior points•Angle bisector

▫A ray which divides an angle into two congruent, adjacent angles

Congruence symbols and drawing conclusions•Do not assume anything in geometry. Just

because two segments look equal does not mean that they are.

Postulates and Theorems Relating Points, Lines, and Planes

Lesson 1.5Pre-AP Geometry

Postulates

A point is defined by its location.

A line contains at least two points.

A plane contains at least three points not all in one line.

Space contains at least four points not all in one plane.

PostulatesThrough any two points there is exactly one line.

Through any three points there is at least one plane and through any three non-collinear points there is exactly one plane.

If two points are in a plane, then the line that contains the point is in that plane.

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

TheoremIf two lines intersect, then they intersect in exactly one point.

TheoremThrough a line and a point not in the line there is exactly one plane.

TheoremIf two lines intersect, then exactly one plane contains the lines.

Review Quiz – True or False

1. A given triangle can lie in more than one plane.

2. Any two points are collinear.

3. Two planes can intersect in only one point.

4. Two lines can intersect in two points.

Review Quiz – True or False

1. A given triangle can lie in more than one plane. False. Through a line and a point not in the line there is exactly one plane.

2. Any two points are collinear. True. 3. Two planes can intersect in only one point.

False. If two planes intersect, then they intersection is a line.

4. Two lines can intersect in two points. False. If two lines intersect, then they intersect in exactly one point.

Problem Set 1.5Written Exercises

p.25: # 1 –20