introduction to geometry – points, lines, and planes points - have no height or width -...
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Introduction to Geometry – Points, Lines, and Planes
Points - have no height or width- represented using a dot and labeled using a capital letter
A
B
C
D
Introduction to Geometry – Points, Lines, and Planes
Points - have no height or width- represented using a dot and labeled using a capital letter
A
B
C
D
Line - has no thickness. Its only dimension is length.
- has no endpoints. It extends forever in each direction. ( arrows )
- it is ALWAYS straight
- labeled with a lower case script letter ( line a )
- if there are points on the line, you can
also name it using the points. ( line AB )
- the part of the line between A and B
is called a line segment
- line segments have definite endpoints
a
A
B
AB
A
B
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear
AB
C
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear
AB
C
D
E
F
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear - points A and F are non-collinear
AB
C
D
E
F
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear - points A and F are non-collinear - points B, E, and F are non-collinear
AB
C
D
E
F
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear - points A and F are non-collinear - points B, E, and F are non-collinear
AB
C
D
E
F
Ray – starts at a fixed point and continues in one direction forever
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear - points A and F are non-collinear - points B, E, and F are non-collinear
AB
C
D
E
F
Ray - starts at a fixed point and continues in one direction forever
- named from their starting point and one other point on the ray
( ray AC or )
A
C
AC
Introduction to Geometry – Points, Lines, and Planes
Collinear - points that are on the SAME line - a line is really a collection of points - points A, B, and C are collinear - points D, E, and F are also collinear - points A and F are non-collinear - points B, E, and F are non-collinear
AB
C
D
E
F
Ray - starts at a fixed point and continues in one direction forever
- named from their starting point and one other point on the ray
( ray AC or )
- two rays connected at the same starting point create an angle
A
C
AC
Introduction to Geometry – Points, Lines, and Planes
Geometric plane – a flat surface that has height and width. No thickness.
Introduction to Geometry – Points, Lines, and Planes
Geometric plane - a flat surface that has height and width. No thickness.
- it is two – dimensional and extends in all directions into infinity
- named with capital script letters ( plane K )
K
Introduction to Geometry – Points, Lines, and Planes
Geometric plane - a flat surface that has height and width. No thickness.
- it is two – dimensional and extends in all directions into infinity
- named with capital script letters ( plane K )
- also can be named using three points NOT on the same line
( plane DEF )
DE
F
Introduction to Geometry – Points, Lines, and Planes
Geometric plane - a flat surface that has height and width. No thickness.
- it is two – dimensional and extends in all directions into infinity
- named with capital script letters ( plane K )
- also can be named using three points NOT on the same line
( plane DEF )
DE F
K
C
B
A
H
I
Points lie on planes and are in planes.
Introduction to Geometry – Points, Lines, and Planes
Geometric plane - a flat surface that has height and width. No thickness.
- it is two – dimensional and extends in all directions into infinity
- named with capital script letters ( plane K )
- also can be named using three points NOT on the same line
( plane DEF )
DE F
K
C
B
A
H
I
Points lie on planes and are in planes.
Lines contain points and are in planes
Introduction to Geometry – Points, Lines, and Planes
Geometric plane - a flat surface that has height and width. No thickness.
- it is two – dimensional and extends in all directions into infinity
- named with capital script letters ( plane K )
- also can be named using three points NOT on the same line
( plane DEF )
DE F
K
C
B
A
H
I
Points lie on planes and are in planes.
Lines contain points and are in planes
Planes contain points and lines
Introduction to Geometry – Points, Lines, and Planes
Planes exist in three – dimensional space and contain their own points and lines.
In geometry, you can have multiple planes. ( Planes A and B )
AC
B
ED
F
A
B
Introduction to Geometry – Points, Lines, and Planes
Planes exist in three – dimensional space and contain their own points and lines.
In geometry, you can have multiple planes. ( Planes A and B )
Line AB is in plane A but not in plane B.
AC
B
ED
F
A
B
Introduction to Geometry – Points, Lines, and Planes
Planes exist in three – dimensional space and contain their own points and lines.
In geometry, you can have multiple planes. ( Planes A and B )
Line AB is in plane A but not in plane B.
Point F is in plane B but not in plane A.
AC
B
ED
F
A
B
Introduction to Geometry – Points, Lines, and Planes
Planes exist in three – dimensional space and contain their own points and lines.
In geometry, you can have multiple planes. ( Planes A and B )
Line AB is in plane A but not in plane B.
Point F is in plane B but not in plane A.
Points or lines that lie in the same plane are called coplanar
AC
B
ED
F
A
B
Introduction to Geometry – Points, Lines, and Planes
Here is an example of points and lines that DO NOT lie in plane A.
A CB
E
DF
A
Introduction to Geometry – Points, Lines, and Planes
Here is an example of points and lines that DO NOT lie in plane A.
Points F, D and E are not in plane A.
AB
E
DF
A
C
Introduction to Geometry – Points, Lines, and Planes
Here is an example of points and lines that DO NOT lie in plane A.
Points F, D and E are not in plane A.
Line DE is not in plane A.
AB
E
DF
A
C
Introduction to Geometry – Points, Lines, and Planes
Here is an example of points and lines that DO NOT lie in plane A.
Points F, D and E are not in plane A.
Line DE is not in plane A.
Lines intersect planes they are not contained in at exactly one point ( point C )
AB
E
DF
A
C
Introduction to Geometry – Points, Lines, and Planes
Line Postulate – for any two points, there is exactly one line containing them.
A
B
- a straight line is the shortest distance between two points.
Introduction to Geometry – Points, Lines, and Planes
Line Postulate – for any two points, there is exactly one line containing them.
A
B
- a straight line is the shortest distance between two points.
Theorem – if two lines intersect, they intersect at exactly one point.
E
Introduction to Geometry – Points, Lines, and Planes
Term Dimensions Must Contain
Line One 2 points
Plane Two 3 or more non collinear points
Space Three 4 or more noncoplanar points
Introduction To Geometry – Intersecting Lines and Planes
In geometry you have :
- lines intersecting lines
Introduction To Geometry – Intersecting Lines and Planes
In geometry you have :
- lines intersecting lines
- planes intersecting planes
- lines intersecting planes
Introduction To Geometry – Intersecting Lines and Planes
Postulate – if two planes intersect, their intersection
forms a line.
Postulate – a line not contained in a plane will intersect that plane at exactly one point
C
Introduction To Geometry – Intersecting Lines and Planes
A line in a plane divides the plane into two half – planes.
half – plane
half – plane
a
Introduction To Geometry – Intersecting Lines and Planes
A line in a plane divides the plane into two half – planes.
half – plane
half – plane
aPoints A and B lie in the same half – plane.
AB