MTH 232

Section 9.1Figures in the Plane

Overview• In this section we consider the most basic shapes

of geometry:1. Points2. Lines3. Segments4. Rays5. 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 angle180 straight90 right

Between 0 and 90 acuteBetween 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 non-overlapping 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.