Get documents about "1.2 Points_ Lines and Planes
Document Sample
1.1 Points, Lines and
Planes
Honors Modern Geometry
Mr. Sgroi
Using Undefined terms and
definition
A definition uses known words to describe a
new word. In geometry, some words
such as point, line and plane are
undefined terms or not formally defined.
Using Undefined terms and
definition
• A point has no
dimension. It is A
usually represented
Point A
by a small dot.
Using Undefined terms and
definition
• A line extends in one
dimension. It is l
usually represented
by a straight line with A
two arrowheads to
indicate that the line
extends without end
B
in two directions. In Line l or AB
this book, lines are
always straight lines.
Using Undefined terms and
definition
A plane extends in two
dimensions. It is usually
represented by a shape
that looks like a tabletop A
or wall. You must
imagine that the plane
C
extends without end even B
though the drawing of a
plane appears to have
edges.
Plane M or plane ABC
A few basic concepts . . .
Must be commonly understood without
being defined. One such concept is the
idea that a point lies on a line or a plane.
Collinear points are points that lie on the
same line.
Coplanar points are points that lie on the
same plane.
Ex. 1: Naming Collinear and
Coplanar Points
a. Name three points
H
that are collinear
G
Solution: D E
D, E and F lie on the
same line, so they
are collinear.
Ex. 1: Naming Collinear and
Coplanar Points
b. Name four points that
are coplanar. H
G
Solution:
D E
D, E, F, and G lie on the
same plane, so they are
coplanar. Also D, E, F,
and H are coplanar;
although, the plane
containing them is not
drawn.
Ex. 1: Naming Collinear and
Coplanar Points
c. Name three points
that are not H
collinear. G
E
Solution: D
There are many correct
answers. For
instance, points H,
E, and G do not lie
on the same line.
More . . .
Another undefined
concept in geometry is l
the idea that a point on a
line is between two other
points on the line. You
can use this idea to
define other important
terms in geometry.
Consider the line AB
Line l or AB
(symbolized by AB).
More . . . A B
The line segment or Segment AB
segment AB l
(symbolized by AB) B
consists of the
endpoints A and B,
and all points on AB
that are between A A
and B. Line l or AB
More . . . A B
The ray AB Ray AB
(symbolized by AB) l
consists of the initial B
point A and all points
on AB that lie on the
same side of A as
point B. A
Line l or AB
More . . . A B
Note that AB is the Ray BA
same as BA and AB is l
the same as BA. B
However, AB and BA
are not the same.
They have different
initial points and A
extend in different Line l or AB
directions.
More . . .
If C is between A and B,
then CA and CB are
l
opposite rays.
Like points, segments and B
rays are collinear if they C
lie on the same line. So,
any two opposite rays are
collinear. Segments, rays A
and lines are coplanar if Line l or AB
they lie on the same
plane.
Ex. 2: Drawing lines, segments
and rays
Draw three noncollinear points J, K, and L.
Then draw JK, KL and LJ.
Draw J, K and L
K
Then draw JK
J
L
Ex. 2: Drawing lines, segments
and rays
Draw three noncollinear points J, K, and L.
Then draw JK, KL and LJ.
K
Draw KL
J
L
Ex. 2: Drawing lines, segments
and rays
Draw three noncollinear points J, K, and L.
Then draw JK, KL and LJ.
K
Draw LJ
J
L
Ex. 3: Drawing Opposite Rays
Draw two lines. Label
points on the lines and
name two pairs of M
opposite rays. Q
Solution: Points M, N, and X
X are collinear and X is
between M and N. So XM
and XN are opposite rays. P N
Ex. 3: Drawing Opposite Rays
Draw two lines. Label
points on the lines and
name two pairs of M
opposite rays. Q
Solution: Points P, Q, and X X
are collinear and X is
between P and Q. So XP
and XQ are opposite rays. P N
Goal 2: Sketching Intersections of
Lines and Planes
Two or more geometric intersect if they
have one or more points in common. The
intersection of the figures is the set of
points the figures have in common.
Activity: p. 12 – Modeling intersections.
– Use two index cards. Label them as shown
and cut slots along each card.
Complete the exercise and place the completed
questions in your lab section labeling this Lab 1.2.
Ex. 4: Sketching intersections
Sketch the figure
described.
a. A line that intersects a
plane in one point
Draw a plane and a
line.
Emphasize the point
where they meet.
Dashes indicate where
the line is hidden by
the plane
Ex. 4: Sketching intersections
Sketch the figure
described.
b. Two planes that
intersect in a line
Draw two planes.
Emphasize the line
where they meet.
Dashes indicate where
one plane is hidden by
the other plane.
Union
An angle is the union ( ) of two
noncollinear rays with a common
endpoint.
In geometric notation:
AB AC CAB
B Union means
A combining two
sets into a
larger set
C
Intersection
The intersection symbol ( ) means the
objects that figures have in common.
For example: ABC BC BC
Note: Segment BC is
A common with itself
and the triangle.
B C
Now for some practice…
Assignment . .
Pages 7-8 # 1-13 all
