What if you were given a picture of a figure or object, like a map with cities and roads marked on it? How could you explain that picture geometrically? After completing this Concept, you'll be able to describe such a map using geometric terms.
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CK-12 Basic Geometric Definitions - Guidance
James Sousa: Definitions of and Postulates Involving Points, Lines, and Planes
Guidance
A point is an exact location in space. It describes a location , but has no size. Examples are shown below:
Label It | Say It |
---|---|
point |
A line is infinitely many points that extend forever in both directions. Lines have direction and location and are always straight .
Label It | Say It |
---|---|
line | line |
line |
A plane is infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper that goes on forever.
Label It | Say It |
---|---|
Plane | Plane |
Plane | Plane |
We can use point , line , and plane to define new terms.
Space is the set of all points extending in three dimensions. Think back to the plane. It extended in two dimensions, what we think of as up/down and left/right. If we add a third dimension, one that is perpendicular to the other two, we arrive at three-dimensional space.
Points that lie on the same line are collinear . , and are collinear because they are all on line . If a point were located above or below line , it would be non-collinear .
Points and/or lines within the same plane are coplanar . Lines and and points , and are coplanar in Plane . Line and point are non-coplanar with Plane .
An endpoint is a point at the end of a line segment. A line segment is a portion of a line with two endpoints. Or, it is a finite part of a line that stops at both ends. Line segments are labeled by their endpoints. Order does not matter.
Label It | Say It |
---|---|
Segment | |
Segment |
A ray is a part of a line with one endpoint that extends forever in the direction opposite that endpoint. A ray is labeled by its endpoint and one other point on the ray. For rays, order matters. When labeling, put the endpoint under the side WITHOUT the arrow.
Label It | Say It |
---|---|
Ray | |
Ray |
An intersection is a point or set of points where lines, planes, segments, or rays cross.
Postulates
A postulate is a basic rule of geometry. Postulates are assumed to be true (rather than proven), much like definitions. The following is a list of some basic postulates.
Postulate #1: Given any two distinct points, there is exactly one (straight) line containing those two points.
Postulate #2: Given any three non-collinear points, there is exactly one plane containing those three points.
Postulate #3: If a line and a plane share two points, then the entire line lies within the plane.
Postulate #4: If two distinct lines intersect, the intersection will be one point.
Lines and intersect at point .
Postulate #5: If two distinct planes intersect, the intersection will be a line.
When making geometric drawings, be sure to be clear and label all points and lines.
Example A
What best describes San Diego, California on a globe?
A. point
B. line
C. plane
Answer: A city is usually labeled with a dot, or point, on a globe.
Example B
Use the picture below to answer these questions.
a) List another way to label Plane .
b) List another way to label line .
c) Are and collinear?
d) Are and coplanar?
Answer:
a) Plane . Any combination of three coplanar points that are not collinear would be correct.
b) . Any combination of two of the letters , or would also work.
c) Yes
d) Yes
Example C
What best describes a straight road connecting two cities?
A. ray
B. line
C. segment
D. plane
Answer: The straight road connects two cities, which are like endpoints. The best term is segment, or .
CK-12 Basic Geometric Definitions E
Guided Practice
1. What best describes the surface of a movie screen?
A. point
B. line
C. plane
2. Answer the following questions about the picture.
a) Is line coplanar with Plane , Plane , both, or neither?
b) Are and collinear?
c) What point belongs to neither Plane nor Plane ?
d) List three points in Plane .
3. Draw and label a figure matching the following description: Line and ray intersect at point . Then, redraw so that the figure looks different but is still true to the description.
4. Describe the picture below using the geometric terms you have learned.
Answers:
1. The surface of a movie screen is most like a plane.
2. a) Neither
b) Yes
c)
d) Any combination of , and would work.
3. Neither the position of or on the line, nor the direction that points matter.
For the second part:
4. and are coplanar in Plane , while and intersect at point .
Practice
For questions 1-5, draw and label a figure to fit the descriptions.
- intersecting and Plane containing but not .
- Three collinear points , and and is also collinear with points and .
- , and such that and are coplanar, but is not.
- Two intersecting planes, and , with where is in plane and is in plane .
- Four non-collinear points, , and , with line segments connecting all points to each other.
- Name this line in five ways.
- Name the geometric figure in three different ways.
- Name the geometric figure below in two different ways.
- What is the best possible geometric model for a soccer field? Explain your answer.
- List two examples of where you see rays in real life.
- What type of geometric object is the intersection of a line and a plane? Draw your answer.
- What is the difference between a postulate and a theorem?
For 13-16, use geometric notation to explain each picture in as much detail as possible.
For 17-25, determine if the following statements are true or false.
- Any two points are collinear.
- Any three points determine a plane.
- A line is to two rays with a common endpoint.
- A line segment is infinitely many points between two endpoints.
- A point takes up space.
- A line is one-dimensional.
- Any four points are coplanar.
- could be read “ray ” or “ray .”
- could be read “line ” or “line .”