Chapter 1 Essentials of Geometry

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Chapter 1Essentials of Geometry

• Section 1.1
• Identify Points, Lines, and Planes

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In 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.

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• 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!
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• 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.
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EXAMPLE 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.

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for Example 1
GUIDED PRACTICE
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Ray 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.
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• 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.

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EXAMPLE 2
Name segments, rays, and opposite rays
b. Name all rays with endpoint J .
Which of these rays
are opposite rays?
SOLUTION
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for Example 2
GUIDED PRACTICE
ANSWER
Yes points J and G lie on the same side of H, so
they are the same ray
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Intersection

• 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.
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EXAMPLE 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
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EXAMPLE 4
Sketch intersections of planes
Sketch two planes that intersect in a line.
SOLUTION
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for Examples 3 and 4
GUIDED PRACTICE

• Sketch two different lines that intersect a plane
  at the same point.

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for 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.
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Homework

• p. 5-7
• 3-22all, 44
• Heading Skip a space
• First and Last Name between each
• Date problem!
• Period
• Assignment