Title: Chapter 1 Essentials of Geometry
1Chapter 1Essentials of Geometry
- Section 1.1
- Identify Points, Lines, and Planes
2In geometry, the words point, line, and plane are
undefined terms. This means there is no formal
meaning, but there is agreement about what they
mean.
- Point a point represents a certain location in
space. It has no dimension and is represented by
a dot. - Line a line extends in opposite directions
without end. It has one dimension and is
represented by a line with two arrowheads. - Plane a plane is a flat surface that extends in
all directions without end. It has two
dimensions and is represented by a shape that
looks like a parallelogram.
3- Through any two points, there is exactly one
line. - You can use any two points on a line to name it.
m
A B
- Through any three points not on the same line,
there is exactly one plane. You can use three
points that are not all on the same line to name
a plane.
Plane F or plane ABC
NOTE If the three points are all on the same
line, an INFINITE number of planes will pass
through the line!
4- Collinear points are points that lie on the same
line.
- Noncollinear points are points that do not lie on
the same line.
Points A, B, C, and D are noncollinear.
- Coplanar points are points that lie in the same
plane.
- Noncoplanar points are points that do not lie on
the same plane.
Points A, B, C, and D are noncoplanar.
5EXAMPLE 1
Name points, lines, and planes
SOLUTION
- Points S, P, and T lie on the same line, so they
are collinear. Points S, P, T,and V lie
in the same plane, so they are coplanar.
6 for Example 1
GUIDED PRACTICE
7Ray part of a line consisting of one endpoint
and all points of the line on one side of the
endpoint, extending forever. NOTE AB and BA are
different rays because the first letter names the
endpoint.
8- Opposite Rays are two collinear rays with the
same endpoint.
- Two opposite rays always form a line.
- Segments and rays are collinear if they lie on
the same line.
- Lines, segments, and rays are coplanar if they
lie in the same plane.
9EXAMPLE 2
Name segments, rays, and opposite rays
b. Name all rays with endpoint J .
Which of these rays
are opposite rays?
SOLUTION
10for Example 2
GUIDED PRACTICE
ANSWER
Yes points J and G lie on the same side of H, so
they are the same ray
11Intersection
- 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 figures have in common.
Lines m and n intersect at point A
The intersection of two different lines is a
point.
Planes Q and R intersect at line m
The intersection of two different planes is a
line.
12EXAMPLE 3
Sketch intersections of lines and planes
a. Sketch a plane and a line that is in the
plane.
b. Sketch a plane and a line that does not
intersect the plane.
c. Sketch a plane and a line that intersects
the plane at a point.
SOLUTION
13EXAMPLE 4
Sketch intersections of planes
Sketch two planes that intersect in a line.
SOLUTION
14for Examples 3 and 4
GUIDED PRACTICE
- Sketch two different lines that intersect a plane
at the same point.
15for Examples 3 and 4
GUIDED PRACTICE
6. Name the intersection of plane A and plane
B.
7. Name the intersection of line k and plane
A.
16Homework
- p. 5-7
- 3-22all, 44
- Heading Skip a space
- First and Last Name between each
- Date problem!
- Period
- Assignment