Geometry: Lesson 2.1 â Points, Lines, and Planes
Geometry: Lesson 2.1 – Points, Lines, and Planes Geometry Oklahoma Academic Standards: N/A
Lesson Objectives:
1. To define and name basic structures of Geometry 2. To understand how basic structures interact with each other.
Introduction: Now that we have an understand of how Geometry (and all types of
Mathematics), we can begin to look at the basic fundamental structures of Geometry itself. We can then start to build definitions with these structures. This will lead to postulates and theorems to be used to create and solve problems that occur in and outside of real-life situations.
Let’s start with the most basic structure–the point. Vocabulary: Undefined Term – Point A point has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Since a point has no definition, the best we can do is just look at points and describe their characteristics.
Point Characteristic 1: A point has no ______________ or ________________. Point Characteristic 2: A point describes a _____________________ in space.
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Depending on what type of scale you are looking at will depend what the point will look like.
For example, a planet in the solar system could be described as a point in space. Also, an a cell in your body could also be described as a point in space. Both of these things have vastly different sizes and structure, but are also points. To make things simple, let’s draw points like: It would probably be a good idea to give a point a name so that we are not confused by which point in space we are talking about.
Let’s label all of our points with a _____________________________________________. Examples of points: Another basic structure of Geometry is formed by connecting two points in space together. This structure is called the line.
Vocabulary: Undefined Term – Line A line has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Like a point, lines have no definition. However, we do know characteristics about it to give us an idea what it is.
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Line Characteristic 1: A line will extend forever in only two different directions.
Line Characteristic 2: A line contains an __________________ amount of points on it. Using the above information, we can construct (draw) a line like so:
(To make our lives easier we will only reference lines as being straight lines only. Other math subjects will introduce curved lines.)
Like points, we will need to name lines to properly communicate which line we are referencing. Since we know a line is made up of at least 2 points, it would make since to name a line using those 2 points on the line. Also, since we’re making reference to a line specifically, we should probably label those 2 points with an indication of a line.
Long story short: The above line would be labeled as __________________________.
Sometimes lines will have a ____________________________________ off to the side of it. You could simply name a line with that as well.
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Example 1: Answer the following questions given the diagram.
a.) Name line m in another way.
_____________________________________ b.) Name a point on line SU.
_____________________________________ c.) Name line RU in another way.
_____________________________________ As you can see from the above diagram, not all points are on the same line. If a point is on the same line as another point, we say the points are collinear.
Vocabulary: Definition – Collinear A collinear point is a point on the same line as another point.
For example, Points A and B are collinear; but, point C is not collinear with both Points A and B.
Example 2: Draw a diagram that meets the following criteria (Answers may vary.) Criterion 1: There must be at least 3 lines labeled k, l, and m.
Criterion 2: There must be three points labeled R, X, and W on line m. Criterion 3: The lines k and l will share a point labeled V. Criterion 4: The points V, X, and Z are collinear.
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Assignment 2.1a You have the choice of the following:
1. Go on a scavenger hunt for 4 different “spaces”. Draw the “spaces” on a piece of notebook
paper. Label the points and the lines in a manner like we have in today’s lesson. For example,
one of my “spaces” would be a piano with the keys as points (labeled as C-D-E, etc.) and the chords on the piano (like the C-chord: C-E-G) as a line.
2. Complete the Worksheet: “Geometry: Handout 2.1a” and turn it in.
Now that we’ve identified points and lines, we can extend both of them to our third basic structure of Geometry: the plane.
Vocabulary: Undefined Term – Plane A plane has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Again like the point and line, a plane has no definition. However, the characteristics of a plane are known.
Plane Characteristic 1: A plane has no ___________________. Therefore, they are flat surfaces. Plane Characteristic 2: A plane must be made up of 3 ________________ points. Plane Characteristic 3: A plane extends in ____________ direction forever.
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Plane Characteristic 4: A plane contains an ____________________ amount of ______________ and _______________________.
Using the above characteristics, a plane would have to be drawn like so:
Like a point and a line, we will need to name a plane in order to communicate which plane we are referring to. Since a plane must have at least 3 points, it only makes sense to name the plane with any of the three points contained in the plane.
For example, the plane we drew above would be labeled as ______________________. Like a line, planes sometime are labeled in a different way. Occasionally, a drawing of a plane
will contain an ________________________________________________________ in a corner of the diagram. This would indicate the name of the plane.
For example, we could also label the above plane as _________________________. Sometime lines and points will be inside a plane together. When this happens, we will refer to them as being coplanar.
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Vocabulary: Definition – Coplanar A coplanar point or line is a point or line on the same plane as another.
For example, Line BC and Point D are coplanar; but, Line AE is not. Example 3: Use the following diagram to answer the questions. a.) Name all points coplanar with point O
______________________________________ b.) Name all points coplanar with point A
______________________________________ Example 4: Use the following diagram to answer the questions.
a.) Name a point on the back plane.
______________________________________ b.) Name the plane on the bottom.
______________________________________
c.) Name the intersection of the planes FEH and GFD. ______________________________________
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Now that we have an understanding of how the basic structures interact with each other, we can make some postulates (facts without proof).
Postulate 1: Given any two points, there will be exactly one ___________________. Example:
Postulate 2: Given any three non-collinear points, there will be exactly one ______________. Example:
Postulate 3: Two lines will intersect in exactly one _____________________. Example:
Postulate 4: Two planes will intersect in exactly one _______________________. Example:
Assignment 2.1b You have the choice of the following:
1. Complete with a partner the Worksheet: “Geometry: Handout 2.1b” and turn it in. 2. Complete on your own the Worksheet: “Geometry: Handout 2.1b” and turn it in.
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Lesson Objectives:
1. To define and name basic structures of Geometry 2. To understand how basic structures interact with each other.
Introduction: Now that we have an understand of how Geometry (and all types of
Mathematics), we can begin to look at the basic fundamental structures of Geometry itself. We can then start to build definitions with these structures. This will lead to postulates and theorems to be used to create and solve problems that occur in and outside of real-life situations.
Let’s start with the most basic structure–the point. Vocabulary: Undefined Term – Point A point has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Since a point has no definition, the best we can do is just look at points and describe their characteristics.
Point Characteristic 1: A point has no ______________ or ________________. Point Characteristic 2: A point describes a _____________________ in space.
Geometry: Lesson 2.1
1
Depending on what type of scale you are looking at will depend what the point will look like.
For example, a planet in the solar system could be described as a point in space. Also, an a cell in your body could also be described as a point in space. Both of these things have vastly different sizes and structure, but are also points. To make things simple, let’s draw points like: It would probably be a good idea to give a point a name so that we are not confused by which point in space we are talking about.
Let’s label all of our points with a _____________________________________________. Examples of points: Another basic structure of Geometry is formed by connecting two points in space together. This structure is called the line.
Vocabulary: Undefined Term – Line A line has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Like a point, lines have no definition. However, we do know characteristics about it to give us an idea what it is.
Geometry: Lesson 2.1
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Line Characteristic 1: A line will extend forever in only two different directions.
Line Characteristic 2: A line contains an __________________ amount of points on it. Using the above information, we can construct (draw) a line like so:
(To make our lives easier we will only reference lines as being straight lines only. Other math subjects will introduce curved lines.)
Like points, we will need to name lines to properly communicate which line we are referencing. Since we know a line is made up of at least 2 points, it would make since to name a line using those 2 points on the line. Also, since we’re making reference to a line specifically, we should probably label those 2 points with an indication of a line.
Long story short: The above line would be labeled as __________________________.
Sometimes lines will have a ____________________________________ off to the side of it. You could simply name a line with that as well.
Geometry: Lesson 2.1
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Example 1: Answer the following questions given the diagram.
a.) Name line m in another way.
_____________________________________ b.) Name a point on line SU.
_____________________________________ c.) Name line RU in another way.
_____________________________________ As you can see from the above diagram, not all points are on the same line. If a point is on the same line as another point, we say the points are collinear.
Vocabulary: Definition – Collinear A collinear point is a point on the same line as another point.
For example, Points A and B are collinear; but, point C is not collinear with both Points A and B.
Example 2: Draw a diagram that meets the following criteria (Answers may vary.) Criterion 1: There must be at least 3 lines labeled k, l, and m.
Criterion 2: There must be three points labeled R, X, and W on line m. Criterion 3: The lines k and l will share a point labeled V. Criterion 4: The points V, X, and Z are collinear.
Geometry: Lesson 2.1
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Assignment 2.1a You have the choice of the following:
1. Go on a scavenger hunt for 4 different “spaces”. Draw the “spaces” on a piece of notebook
paper. Label the points and the lines in a manner like we have in today’s lesson. For example,
one of my “spaces” would be a piano with the keys as points (labeled as C-D-E, etc.) and the chords on the piano (like the C-chord: C-E-G) as a line.
2. Complete the Worksheet: “Geometry: Handout 2.1a” and turn it in.
Now that we’ve identified points and lines, we can extend both of them to our third basic structure of Geometry: the plane.
Vocabulary: Undefined Term – Plane A plane has no definition. However, we do know characteristics about it like how to name it and how it interacts with other structures.
Again like the point and line, a plane has no definition. However, the characteristics of a plane are known.
Plane Characteristic 1: A plane has no ___________________. Therefore, they are flat surfaces. Plane Characteristic 2: A plane must be made up of 3 ________________ points. Plane Characteristic 3: A plane extends in ____________ direction forever.
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Plane Characteristic 4: A plane contains an ____________________ amount of ______________ and _______________________.
Using the above characteristics, a plane would have to be drawn like so:
Like a point and a line, we will need to name a plane in order to communicate which plane we are referring to. Since a plane must have at least 3 points, it only makes sense to name the plane with any of the three points contained in the plane.
For example, the plane we drew above would be labeled as ______________________. Like a line, planes sometime are labeled in a different way. Occasionally, a drawing of a plane
will contain an ________________________________________________________ in a corner of the diagram. This would indicate the name of the plane.
For example, we could also label the above plane as _________________________. Sometime lines and points will be inside a plane together. When this happens, we will refer to them as being coplanar.
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Vocabulary: Definition – Coplanar A coplanar point or line is a point or line on the same plane as another.
For example, Line BC and Point D are coplanar; but, Line AE is not. Example 3: Use the following diagram to answer the questions. a.) Name all points coplanar with point O
______________________________________ b.) Name all points coplanar with point A
______________________________________ Example 4: Use the following diagram to answer the questions.
a.) Name a point on the back plane.
______________________________________ b.) Name the plane on the bottom.
______________________________________
c.) Name the intersection of the planes FEH and GFD. ______________________________________
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Now that we have an understanding of how the basic structures interact with each other, we can make some postulates (facts without proof).
Postulate 1: Given any two points, there will be exactly one ___________________. Example:
Postulate 2: Given any three non-collinear points, there will be exactly one ______________. Example:
Postulate 3: Two lines will intersect in exactly one _____________________. Example:
Postulate 4: Two planes will intersect in exactly one _______________________. Example:
Assignment 2.1b You have the choice of the following:
1. Complete with a partner the Worksheet: “Geometry: Handout 2.1b” and turn it in. 2. Complete on your own the Worksheet: “Geometry: Handout 2.1b” and turn it in.
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