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1. Foundations of Geometry
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Geometry

1.01 Introduction to geometric notation

Lesson
Practice
Lesson

Concept summary

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

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.
Opposite rays

Rays with the same endpoint that extend in opposite directions; opposite rays form a line.

Two straight figures sharing a common endpoint, and continues forever in opposite directions.

We can also use the undefined terms to construct axioms and postulates which are both statements accepted as fact without proof. The following statements are axioms:

  • Two point postulate: Through any two points there exists exactly one line

  • Three point postulate: Through any three noncollinear points, there is exactly one plane containing them

  • Line intersection postulate: If two distinct lines intersect, then they intersect in exactly one point

  • Plane intersection postulate: If two distinct planes intersect, then they intersect in exactly one line

  • Plane-line postulate: If two points lie in a plane, then the line containing them also lies in the plane

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. From these axioms we can then build the following definitions:

Parallel planes

Planes that do not intersect.

A pair of non intersecting planes.
Parallel lines

Lines in the same plane that do not intersect.

A pair of non intersecting lines.
Skew lines

Lines that are not coplanar and do not intersect.

A pair of non intersecting lines. Lines lie on different planes.

Worked 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

Approach

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

Solution

A, B, and E is one possible solution.

Reflection

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

Approach

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.

Solution

Line: \overleftrightarrow{AC}

Ray: \overrightarrow{AC}

Segment: \overline{AC}

Reflection

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.

c

A pair of skew lines

Approach

Skew lines do not lie on the same plane so they cannot be parallel and they cannot intersect. Start with the line that lies on the plane. Is there a way to connect two points to create a line off the plane that does not intersect this line?

Solution

\overleftrightarrow{EM} and \overleftrightarrow{AB} is one possible solution.

Reflection

Remember, through any two points there exists exactly one line. Lines exist even when they are not labeled.

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

Approach

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

Solution

E, J, and I is one possible answer.

b

Opposite rays

Approach

Opposite rays form a line. To find a pair of opposite rays look for lines with three points labeled, such as we did in part (a). Since there are multiple lines with three points labeled, there are multiple correct answers.

Solution

\overrightarrow{IE} and \overrightarrow{IJ} is one possible answer.

Reflection

When naming rays pay attention to the order of the points. \overrightarrow{IE} is not the same ray as \overrightarrow{EI}.

c

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

Approach

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?

Solution

H

Outcomes

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

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