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MTH 232
Section 9.1
Figures in the Plane
Overview
• In this section we consider the most basic shapes
of geometry:
1. Points
2. Lines
3. Segments
4. Rays
5. Angles
• We also introduce a large number of notations
and terms essential for the communication of
geometric concepts and relationships.
Points, Lines, and Planes
• A point is a location in space. Points are represented by
dots and labeled with uppercase letters.
• A line is made up of points (a minimum of two different
points is required). Lines extend infinitely in two
directions. They can be drawn with a ruler or
straightedge. Lines can be named either by lowercase
letters or by identifying two points that belong to that
line.
• Three or more points that lie on (belong to) the same
line are said to be collinear. Three noncollinear points
determine a plane, which is a set of points that idealize
a flat surface.
More Definitions
• If two lines have a point in common, that
common point is said to be a point of intersection
for the two lines.
• Lines that do not have a point of intersection, or
are the same line, are called parallel.
• If there is a point B on each of lines i, j, and k,
then the three lines are said to be concurrent.
• A transversal to lines r and s is a line t that
intersects both r and s but not at the same point.
Still More Definitions
• A line segment consists of two endpoints and all
the points between them.
• The length of a line segment is the distance
between the endpoints.
• Two line segments are congruent if they have the
same length.
• The midpoint of a line segment is the point on
the line segment that is the same distance from
one endpoint as it is from the other endpoint.
Rays and Angles
• A ray is a subset of a line that contains an
endpoint and all the points that lie of one side
or the other of that endpoint.
• The union of two rays with a common
endpoint is called an angle (the common
endpoint is called a vertex). The two rays are
the sides of the angle.
• Angles in the plane partition (divide) the plane
into two regions: the interior and the exterior.
More About Angles
• Angles are classified by their measure (the
number of degrees required to rotate one side of
the angles onto the other side).
Number of degrees
Type of angle
180
straight
90
right
Between 0 and 90
acute
Between 90 and 180
obtuse
• Two angels are congruent if they have the same
measure.
Still More About Angles
• Two angles are complimentary if their
measures add to equal 90 degrees.
• Two angles are supplementary if their
measures add to equal 180 degrees.
• Adjacent angles have a common side and nonoverlapping interiors.
• Intersecting lines form vertical angles. Vertical
angles are congruent.
Parallel Lines and Transverals
• If two parallel lines are cut by a transversal,
several types of angle pairs are formed:
1. Corresponding (congruent)
2. Alternate Interior (congruent)
3. Alternate Exterior (congruent)
4. Same Side Interior (supplementary)
5. Same Side Exterior (supplementary)
Finally…
• The sum of the measures of the angles in a
triangle is 180 degrees.