int math 2 section 5-1 1011
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Elements of GeometryTRANSCRIPT
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Chapter 5Logic and Geometry
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Monday, February 6, 2012
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Getting to know Geometry
Section 5-1Elements of Geometry
Monday, February 6, 2012
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Essential Questions
What are the fundamental geometric concepts?
What are the basic geometric postulates and how do you use them?
Where you’ll see this:
Construction, art, photography, navigation
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Vocabulary1. Geometry:
2. Point:
3. Line:
4. Plane:
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point:
3. Line:
4. Plane:
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point: A location in space that has no dimensions
3. Line:
4. Plane:
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point: A location in space that has no dimensions
3. Line: A set of points that goes on forever in both directions; has one dimension (length)
4. Plane:
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point: A location in space that has no dimensions
3. Line: A set of points that goes on forever in both directions; has one dimension (length)
*A line segment is a finite portion of a line
4. Plane:
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point: A location in space that has no dimensions
3. Line: A set of points that goes on forever in both directions; has one dimension (length)
*A line segment is a finite portion of a line
4. Plane: A flat surface that goes on forever in all directions; has two dimensions (length and width)
5. Space:
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Vocabulary1. Geometry: The study of points (and collections of
points) in space
2. Point: A location in space that has no dimensions
3. Line: A set of points that goes on forever in both directions; has one dimension (length)
*A line segment is a finite portion of a line
4. Plane: A flat surface that goes on forever in all directions; has two dimensions (length and width)
5. Space: The set of all points anywhere and everywhere
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More Vocabulary6. Collinear Points:
7. Noncollinear Points:
8. Coplanar Points:
9. Noncoplanar Points:
10. Intersection:
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points:
8. Coplanar Points:
9. Noncoplanar Points:
10. Intersection:
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points: Points not on the same line
8. Coplanar Points:
9. Noncoplanar Points:
10. Intersection:
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points: Points not on the same line
8. Coplanar Points: Points that are all on the same plane
9. Noncoplanar Points:
10. Intersection:
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points: Points not on the same line
8. Coplanar Points: Points that are all on the same plane
9. Noncoplanar Points: Points not on the same plane
10. Intersection:
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points: Points not on the same line
8. Coplanar Points: Points that are all on the same plane
9. Noncoplanar Points: Points not on the same plane
10. Intersection: The point or points where two things meet (figures, lines, etc.)
11. Congruent Line Segments:
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More Vocabulary6. Collinear Points: Points that lay on the same line
7. Noncollinear Points: Points not on the same line
8. Coplanar Points: Points that are all on the same plane
9. Noncoplanar Points: Points not on the same plane
10. Intersection: The point or points where two things meet (figures, lines, etc.)
11. Congruent Line Segments: Two or more line segments that have the same length
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Even More Vocab12. Midpoint of a Segment:
13. Bisector of a Segment:
14. Postulate:
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Even More Vocab12. Midpoint of a Segment: The point that divides the
line segment into two equal line segments (It’s in the middle!)
13. Bisector of a Segment:
14. Postulate:
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Even More Vocab12. Midpoint of a Segment: The point that divides the
line segment into two equal line segments (It’s in the middle!)
13. Bisector of a Segment: A line, ray, or segment that goes through the midpoint of another segment
14. Postulate:
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Even More Vocab12. Midpoint of a Segment: The point that divides the
line segment into two equal line segments (It’s in the middle!)
13. Bisector of a Segment: A line, ray, or segment that goes through the midpoint of another segment
14. Postulate: A rule in geometry that is assumed to be true
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Example 1In the figure, AB and CD bisect each other at point
E. Name congruent segments. D
EAB
C
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Example 1In the figure, AB and CD bisect each other at point
E. Name congruent segments. D
EAB
C
AE ≅ EB
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Example 1In the figure, AB and CD bisect each other at point
E. Name congruent segments. D
EAB
C
AE ≅ EB CE ≅ ED
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Point, Line, and Plane Postulates
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Point, Line, and Plane Postulates
Postulate 1: There is exactly one line that goes through any two points
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Point, Line, and Plane Postulates
Postulate 1: There is exactly one line that goes through any two points
Postulate 2: There is exactly one plane that goes through any three noncollinear points
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Point, Line, and Plane Postulates
Postulate 1: There is exactly one line that goes through any two points
Postulate 2: There is exactly one plane that goes through any three noncollinear points
Postulate 3: The line connecting any two points on a plane also lies in that same plane
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Point, Line, and Plane Postulates
Postulate 1: There is exactly one line that goes through any two points
Postulate 2: There is exactly one plane that goes through any three noncollinear points
Postulate 3: The line connecting any two points on a plane also lies in that same plane
Postulate 4: The intersection of any two planes is a line
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Example 2Refer to the diagram to name the types of points
indicated.
S
R
A
D
G
F
E
H
B
C
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Example 2
S
R
A
D
G
F
E
H
B
C
a. Three or more collinear points
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Example 2
S
R
A
D
G
F
E
H
B
C
a. Three or more collinear points
A, B, C, D
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Example 2
S
R
A
D
G
F
E
H
B
C
b. All points that are coplanar with point F
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Example 2
S
R
A
D
G
F
E
H
B
C
b. All points that are coplanar with point F
A, B, C, D, E
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Example 2
S
R
A
D
G
F
E
H
B
C
c. All points that are coplanar with E and noncollinear with D
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Example 2
S
R
A
D
G
F
E
H
B
C
c. All points that are coplanar with E and noncollinear with D
F
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Example 3
S
R
A
D
G
F
E
H
B
C
a. How would you classify the intersection of the two planes?
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Example 3
S
R
A
D
G
F
E
H
B
C
a. How would you classify the intersection of the two planes?
The intersection is AD
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Example 3
S
R
A
D
G
F
E
H
B
C
a. How would you classify the intersection of the two planes?
The intersection is AD (Postulate 4)
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Example 3
S
R
A
D
G
F
E
H
B
C
b. If B and E were to be connected by a line, which plane would they be in?
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Example 3
S
R
A
D
G
F
E
H
B
C
b. If B and E were to be connected by a line, which plane would they be in?
The would be in plane S
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Example 3
S
R
A
D
G
F
E
H
B
C
b. If B and E were to be connected by a line, which plane would they be in?
The would be in plane S (Postulate 3)
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Example 3
S
R
A
D
G
F
E
H
B
C
c. Are F and G able to be connected by a line?
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Example 3
S
R
A
D
G
F
E
H
B
C
c. Are F and G able to be connected by a line?
Yes
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Example 3
S
R
A
D
G
F
E
H
B
C
c. Are F and G able to be connected by a line?
Yes (Postulate 1)
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Problem Set
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Problem Set
p. 194 #1-29
“Nothing in life is to be feared. It is only to be understood.” - Marie Curie
Monday, February 6, 2012