Topics
8. Lines and Angles
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United States of America
Grades 9-12

8.01 Introduction to geometric notation

Lesson
Practice

Introduction

Geometry uses standard vocabulary and symbols to communicate facts and relationships about geometric figures. We will recall geometric terminology and use geometric notation to identify geometric figures.

Introduction to geometric notation

Geometry relies on three undefined terms which form the foundation for all other geometric terms are built. Although no formal definition exists, these undefined terms can still be described.

Point

A precise location, usually labeled with a dot, that has no size or dimension. A point is usually denoted by a capital letter- in this case, A.

A dot labeled A.
Line

A straight path that extends infinitely in both directions, usually represented with two arrows on each end. A line is denoted as a pair of points with the line symbol over them- in this case, \overleftrightarrow{AB}.

A straight line drawn through two points labeled A and B.
Plane

A flat surface with no width that extends infinitely in all directions.

A parallelogram with an arrow pointing away from each side to show that it extends infinitely.
Distance along a line

When a ruler is placed along the line and values lined up with two points on the line, the distance between the points is the absolute value of the difference of the values of the points

A ruler labeled with millimeters. Whole millimeters are labeled with numbers, half millimeters are shown with long ticks, fourth millimeters are shown with short ticks. A line containing points A and B is above the ruler. A lines up with the half millimeter mark and B lines up with 2 millimeter mark.
Distance along a circular arc

A fraction of the circumference of a circle that lies between two points on the circle

Circle C with points A and B on the circle. The fraction of the circumference of circle C that lies between A and B is highlighted.

These undefined terms can now be used to define other geometric figures.

Line segment

The set of all points on a line bounded by two other points called endpoints. A line segment is denoted by its endpoints with the line segment symbol over them- in this case, \overline{AB}.

A straight figure that connects two points, and does not continue in either direction.
Ray

The set of all points on a line bounded by an endpoint and extending to infinity in one direction. A ray is denoted by its endpoint and another point on the line, with the ray symbol pointing from the endpoint- in this case, \overrightarrow{AB}.

A straight figure that starts with a point A then extends forever in one direction going through point B.
Angle

Angles are formed wherever two lines, segments or rays intersect. Angles are measured in degrees. The point of intersection is called the vertex of the angle.

Two rays that share an endpoint.
Circle

A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center).

A circle with a point drawn directly in the center.

The intersection of two geometric figures is the set of all points they share in common. Points are considered collinear if they lie on the same line and geometric figures are considered coplanar if they lie on the same plane. We can then build the following definitions:

Collinear

Points that lie on the same line.

Left diagram titled Collinear Points shows a line with 3 points plotted on the line. Right diagram titled Non Collinear Points shows 4 points arranged in a diamond shape.
Coplanar

Figures that lie in the same plane.

Left diagram titled Coplanar points shows a plane with 3 points plotted on the same plane. Right diagram titled Non Coplanar Points shows 3 points plotted on the same plane and one point not on the plane.
Parallel planes

Planes that do not intersect.

A pair of non intersecting planes.

Examples

Example 1

Use the diagram to identify the following geometric figures.

A plane P containing point E, and line l. Points A, B, and C lie on line l. A point M lies outside the plane. A line connects B and M.
a

A plane

Worked Solution
Create a strategy

Planes are named with three noncollinear, coplanar points. Identify three points on the plane in the diagram that do not form a line.

Apply the idea

A, B, and E is one possible solution.

Reflect and check

There are four points labeled on the plane: A, B, C, and E. Any combination except ABC would work to name the plane.

b

A line segment, ray, and line that contain point A and point C

Worked Solution
Create a strategy

The correct notation for a line includes two arrows, for a ray includes a right arrow, and for a segment includes a line with no arrows.

Apply the idea

Line: \overleftrightarrow{AC}

Ray: \overrightarrow{AC}

Segment: \overline{AC}

Reflect and check

We know from the postulates that through any two points there exists exactly one line. Since segments and rays are both parts of a line the two points that define the line can also define a segment and rays in either direction.

Example 2

Use the diagram to identify geometric figures.

Three intersecting lines A J, E F, and G M. A J intersects G M at point L. E F intersects G M at point I. A J intersects E F at point J. A fourth line C D is drawn. C D intersects G M at point H. C D intersects A B at point K.
a

Three collinear points

Worked Solution
Create a strategy

Collinear points are those that lie on the same line. Since there are multiple lines with three points labeled, there are multiple correct answers.

Apply the idea

E, J, and I is one possible answer.

b

The intersection of \overleftrightarrow{LH} and \overleftrightarrow{CD}

Worked Solution
Create a strategy

The line intersection postulate states that "if two distinct lines intersect, then they intersect in exactly one point", so we know the answer will be a single point. Which point do the two lines have in common?

Apply the idea

H

c

All angles with I as its vertex.

Worked Solution
Create a strategy

There are a total of four angles that contain I as their vertex.

Apply the idea

\angle EIM

\angle EIH

\angle HIJ

\angle JIM

Example 3

Jitka is planting flowers in the garden bed 3 feet equidistant from a birdbath. Describe the figure Jitka is creating in her garden.

Worked Solution
Create a strategy

Use the shape of a birdbath to determine the arrangement of the planted flowers.

Apply the idea

Jitka is creating a circle by planting flowers 3 feet from a birdbath in her garden.

Idea summary

We can define all figures based on the undefined terms point, line, plane, distance along a line, and distance along a circular arc and use these ideas throughout geometry.

Outcomes

G.CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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