math 7 geometry 01 undefined terms rev 2
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The session shall begin shortly…
A Mathematics 7 Lecture
The Basics of Geometry
QUESTION!
What words come to mind when you hear
the word
GEOMETRY?
The Basics of Geometry for High School
Geometry
OH MY!
Geometry Geometry comes from the Greek words geo (earth) and metron (measure), so geometry literally means “earth measure”Geometry is the branch of mathematics that defines and relates the basic properties and measurements of shapes
Geometry The ancient Egyptians, Chinese,
Babylonians, Romans, and Greeks used geometry for surveying,
navigation, astronomy, and other practical occupations. The Greeks
sought to systematize the geometric facts they knew by establishing
logical reasons for them and relationships among them.
The systematizing of geometric facts and
principles culminated in the geometry text
Elements, written in approximately 325 B.C.
by Euclid. This most remarkable text has been
in use for over 2000 years.
This recognizes Euclid as the Father of Geometry.
Euclid
Father of Geometry
Geometry
Geometry in Art and Nature
Geometry
• What are the undefined terms in geometry?
• What concepts present the foundations of geometry?
• Can you sketch the intersection of lines and planes?
QUESTIONS!
The Basics of Geometry for High School
THE UNDEFINED TERMS IN
GEOMETRY
The Basics of Geometry for High School
The terms points, lines, and planes are the foundations of geometry, but point, line, and plane are all what we call undefined terms.
How can that be?
Any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular!
Undefined Terms?
The Basics of Geometry for High School
The Three Undefined Terms
The Basics of Geometry for High School
POINT
LINE
PLANE
• Has no dimension• Indicates location• Usually represented by a small dot
A
A point is represented with a capital letter.
The above is called point A.
The Basics of Geometry for High School
Point
• Extends in one dimension.• Represented with a straight line with two
arrowheads to indicate that the line extends without end in two directions.
Line
The Basics of Geometry for High School
Naming a Line
l
A
B
Line
The Basics of Geometry for High School
1. Using a small letter (e.g. l or m)
2. Using two points on the line
The given line can be named as
AB (line AB) l (line l)
Note that the letters can be interchanged, so we haveBA9::::::::::::: :
• Extends in two dimensions.• Represented by a slanted 4 sided
figure, but you must imagine it extends without end, even though the representation has edges.
Plane
The Basics of Geometry for High School
A plane can be named using a capital letter or three points on the plane
A
B C
M
This is plane M or plane ABC
Plane
The Basics of Geometry for High School
Undefined Terms: A SummaryUndefined
TermRepresen-
tation Naming Other Information
Point Dot
•• P
Read: point P
Has no length, width or thickness; occupies no space
Line Straight mark with two arrow heads
Two capital letters with a double arrowhead above them or a lower case letter
Has infinite length but no width and thickness; is straight
Plane Slanted four-sided figure
Single capital letter
Read: Plane M
Has infinite length and width but no thickness; is a flat surface
M
The Basics of Geometry for High School
Determine the undefined term suggested by each of the following.1. the tip of a pencil2. the top of the teacher’s table3. telephone wires4. the wall of your classroom5. the surface of the page of your book
Check your understanding
point
plane
line
plane
plane
Determine the undefined term suggested by each of the following.6. the ruler’s edge7. the tip of a needle8. a corner of a piece of paper 9. the floor of your bedroom10. the string on a guitar
Check your understanding
line
point
point
plane
line
ADDITIONAL CONCEPTS RELATED TO THE
UNDEFINED TERMS
The Basics of Geometry for High School
• Collinear points are points that lie on the same line.
l
CA
B Points A, B and C are collinear.
The Basics of Geometry for High School
Collinear and Coplanar Points
• Coplanar points are points that lie on the same plane.
A
BC Points A, B and C are
coplanar.
Examples
COLLINEAR POINTS
- B, A, L, K- S, M, K- Q, M, ? - T, ? , D
S
M
Q
B
D H
T
KLA
Collinear and Coplanar Points
The Basics of Geometry for High School
D or B
A or S
NONCOLLINEAR POINTS- B, A, M, K- D, M, K- Q, M, ? - T, ? , D
S
M
Q
B
D H
T
KLA
Examples
Collinear and Coplanar Points
The Basics of Geometry for High School
Any point except B and D
Any point except A and S
Examples
COPLANAR POINTS
- G, E, O- F, L, I- O, E, ? - A, ? , C
Collinear and Coplanar Points
The Basics of Geometry for High School
G or B
F, L, or I
Examples NONCOPLANAR POINTS
- F, A, G- G, O, A, L- O, E, ? - A, ? , C
Collinear and Coplanar Points
The Basics of Geometry for High School
Any point except G or B
Any point except F, L, or I
1. 2 points that are collinear.
2. 3 points that are coplanar.
3. 2 points that are noncollinear.
MGD F
B
Check your understandingGiven the figure shown, name the following:
4. 3 points that are noncoplanar.
A B D
What happens if you cut the
line shown below at
points B and C? What
figures will you form?
C
The Basics of Geometry for High School
Subsets of a Line
QUESTION!
A B
DCA B
C DB C
RAY RAYSEGMENT
The Basics of Geometry for High School
Subsets of a Line
B
A LINE SEGMENT (or segment) is a set of points consisting of two points on a line, and all the points on the line between the two points
C
LINE SEGMENT
The Basics of Geometry for High School
Subsets of a Line
The two fixed points at the end of a line segment are called ENDPOINTS.
A line segment is named by its endpoints. The points can be
interchanged in the nameB C
This is line segment BC or CB
In symbols:
BC or CB
The Basics of Geometry for High School
Subsets of a Line
A RAY is the part of the line which consists of a fixed point called the ENDPOINT and the set of all points on one side of the endpoint.
The endpoint is the starting point and the arrow determine the direction of the ray.
A B
RAY
The Basics of Geometry for High School
Subsets of a Line
A
ray BA but not ray AB
DB C
In naming a ray, the endpoint always comes first, followed by the second point.
ray CD but not ray DC
BA::::::::::::::
CD::::::::::::::
The Basics of Geometry for High School
Subsets of a Line
Not all symbols are created equal!
Symbol alert!
BAAB is the same as
AB BAis the same as
BUT…
A B
A B
The Basics of Geometry for High School
Subsets of a Line
The ray is different!
ABis NOT the same asBAA B AB
A B BA
Initial point 1st
Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray.
Symbol alert!
The Basics of Geometry for High School
Subsets of a Line
then and are opposite rays.
If C is between A and B,
Opposite Rays
CA
A BC
CB
C is the common initial point for the rays.
The Basics of Geometry for High School
Subsets of a Line
• Two or more geometric figures intersect if they have one or more points in common.
• The intersection of the figures is the set of points the figure has in common
Think!!
How do 2 lines intersect?
How do 2 planes intersect?
What about a line and a plane?
The Basics of Geometry for High School
Intersection of Lines and Planes
To think about the questions on the last slide let’s look at the following…
B A
ETwo lines intersect at a
point, like here at point
A.F
Line BF is the intersection of the planes G and H.
GH
Point E is the
intersection of plane H and line
ECD
C
The Basics of Geometry for High School
Intersection of Lines and Planes
If two lines intersect, then they intersect in exactly one (one and only one) point.
The point exists (there is at least one point) and is unique (no more than one point exists).
The Basics of Geometry for High School
Intersection of Lines and Planes
A
If two planes intersect, then their intersection is a unique line.
The Basics of Geometry for High School
Intersection of Lines and Planes
A plane and a line not on the plane intersect at a unique point.
The Basics of Geometry for High School
Intersection of Lines and Planes
E
Summing it up!The Basics of Geometry for High School
The Basics of Geometry 2: Introduction to
Postulates and Theorems
What’s Next?
The Basics of Geometry for High School
Thank you!
Thank you!
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