# Unit 1 Points, Lines, Planes, and Angles

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Unit 1 Points, Lines, Planes, and Angles
Pre-AP Geometry Unit 1 Points, Lines, Planes, and Angles

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Introduction to 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 Geometry
Geometry 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, miles. The first known case of calculating the distance around the earth was done by Eratosthenes around 240 BC.

Introduction to Geometry
The 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.

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: Is as far from the fountain as from the oak tree and 2. 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? Basic Figures 10m X Equal distances Discussion in GSP

Review Quiz Which language does the word “geometry” come from and what does it mean? What is the circumference of the Earth at the equator? (exact value is preferred, but the nearest 1000 is acceptable) Who wrote the book “The Elements”? Which geometric element has dimensions of zero? How many points does it take to define a line?

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

Points, Lines and Planes
Lesson 1.2 Pre-AP Geometry 19

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. 20

Objectives Use the undefined terms point, line, and plane.
Draw representations of points, lines, and planes. Use the terms collinear, coplanar, and intersection. 21

Vocabulary Point A geometric element that has position but no dimension. A point is defined by its coordinates. Symbol: · A Line A 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") 22

Vocabulary Plane A 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. 23

Vocabulary Space The set of all points. The unlimited area which extends in all directions and within which all things exist. Intersect To meet or cross at a point. 24

Vocabulary Collinear Lying on the same straight line.
Collinear points lie along a straight line. Any two points are always collinear. Coplanar Lying 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. 25

Points Things that we can use to represent a point: a marble 26

Lines Things that we can use to represent a line: a taut piece of string 27

Plane Things that we can use to represent a plane: a sheet of paper a poster board 28

Intersecting Planes Things that we can use to represent intersecting planes: the sides of a cardboard box 29

Practice Quiz - True or False
ends at P. Point S is on an infinite number of lines. A plane has no thickness. Collinear points are coplanar. Planes have edges. Two planes intersect in a line segment. Two intersecting lines meet in exactly one point. Points have no size. 30

Problem Set 1.2 Written Exercises p.7: # 2 – 26 even, 27 - 36
31

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 Always name a segment using the endpoints. 33

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 =

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 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 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 Find x: Find EG: Find EF:
Find FG:

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.

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. Use yarn and have students go to front and show this. A B C 40

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 vertex sides

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 1 H

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 180 b. 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 Adjacent angles Angle bisector
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 1.5 Pre-AP Geometry 49

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. 50

Postulates Through 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. 51

Theorem If two lines intersect, then they intersect in exactly one point. 52

Theorem Through a line and a point not in the line there is exactly one plane. 53

Theorem If two lines intersect, then exactly one plane contains the lines. 54

Review Quiz – True or False
A given triangle can lie in more than one plane. Any two points are collinear. Two planes can intersect in only one point. Two lines can intersect in two points. 55

Review Quiz – True or False
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. Any two points are collinear. True. Two planes can intersect in only one point. False. If two planes intersect, then they intersection is a line. Two lines can intersect in two points. False. If two lines intersect, then they intersect in exactly one point. 56

Problem Set 1.5 Written Exercises p.25: # 1 –20
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