The word *geometry* is Greek for *geos* (meaning Earth) and* metron* (meaning measure). Geometry was extremely important to ancient societies, and it was used for surveying, astronomy, navigation, and building. Geometry as we know it is actually Euclidean geometry, which was written well over 2,000 years ago in ancient Greece by Euclid, Pythagoras, Thales, Plato, and Aristotle — just to mention a few. The most fascinating and accurate geometry text was written by Euclid, called "Elements." Euclid's text has been used for over 2,000 years.

Geometry is the study of angles and triangles, perimeter, area, and volume. It differs from algebra in that one develops a logical structure where mathematical relationships are proved and applied. Start by learning the basic terms associated with geometry.

## Geometry Terms

#### Point

Points show position. A point is shown by one capital letter. In this example, A, B, and C are all points. Notice that points are on the line.

#### Naming a Line

A line is infinite and straight. If you look at the picture above, AB is a line, AC is also a line and BC is a line. A line is identified when you name two points on the line and draw a line over the letters. A line is a set of continuous points that extend indefinitely in either of its direction. Lines are also named with lowercase letters or a single lower case letter. For instance, one of the lines above could be named simply by indicating an *e.*

## Important Geometry Definitions

#### Line Segment

A line segment is a straight line segment which is part of the straight line between two points. To identify a line segment, one can write AB. The points on each side of the line segment are referred to as the endpoints.

#### Ray

A ray is the part of the line which consists of the given point and the set of all points on one side of the endpoint.

In the image, A is the endpoint and this ray means that all points starting from A are included in the ray.

## Angles

An angle can be defined as two rays or two line segments having a common endpoint. The endpoint becomes known as the vertex. An angle occurs when two rays meet or unite at the same endpoint.

The angles pictured in the image can be identified as angle ABC or angle CBA. You can also write this angle as angle B which names the vertex. (common endpoint of the two rays.)

The vertex (in this case B) is always written as the middle letter. It matters not where you place the letter or number of your vertex. It is acceptable to place it on the inside or the outside of your angle.

When you are referring to your textbook and completing homework, make sure you are consistent. If the angles you refer to in your homework use numbers, use numbers in your answers. Whichever naming convention your text uses is the one you should use.

#### Plane

A plane is often represented by a blackboard, bulletin board, the side of a box, or the top of a table. These plane surfaces are used to connect any two or more points on a straight line. A plane is a flat surface.

You are now ready to move to types of angles.

## Acute Angles

An angle is defined as where two rays or two line segments join at a common endpoint called the vertex. See part 1 for additional information.

#### Acute Angle

An acute angle measures less than 90 degrees and can look something like the angles between the gray rays in the image.

## Right Angles

A right angle measures exactly 90 degrees and will look something like the angle in the image. A right angle equals one-fourth of a circle.

## Obtuse Angles

An obtuse angle measures more than 90 degrees, but less than 180 degrees, and will look something like the example in the image.

## Reflex Angles

A reflex angle is more than 180 degrees, but less than 360 degrees, and will look something like the image above.

## Complementary Angles

Two angles adding up to 90 degrees are called complementary angles.

In the image shown, angles ABD and DBC are complementary.

## Supplementary Angles

Two angles adding up to 180 degrees are called supplementary angles.

In the image, angle ABD + angle DBC are supplementary.

If you know the angle of angle ABD, you can easily determine what the angle DBC measures by subtracting angle ABD from 180 degrees.

## Basic and Important Postulates

Euclid of Alexandria wrote 13 books called "The Elements" around 300 BC. These books laid the foundation of geometry. Some of the postulates below were actually posed by Euclid in his 13 books. They were assumed as axioms but without proof. Euclid's postulates have been slightly corrected over a period of time. Some are listed here and continue to be part of Euclidean geometry. Know this stuff. Learn it, memorize it, and keep this page as a handy reference if you expect to understand geometry.

There are some basic facts, information, and postulates that are very important to know in geometry. Not everything is proved in geometry, thus we use some *postulates,* which are basic assumptions or unproved general statements that we accept. Following are a few of the basics and postulates that are intended for entry-level geometry. There are many more postulates than those that are stated here. The following postulates are intended for beginner geometry.

## Unique Segments

You can only draw one line between two points. You will not be able to draw a second line through points A and B.

## Line Intersection

Two lines can intersect at only one point. In the figure shown, *S* is the only intersection of AB and CD.

## Midpoint

A line segment has only one midpoint. In the figure shown, *M* is the only midpoint of AB.

## Bisector

An angle can only have one bisector. A bisector is a ray that's in the interior of an angle and forms two equal angles with the sides of that angle. Ray AD is the bisector of angle A.

## Conservation of Shape

The conservation of shape postulate applies to any geometric shape that can be moved without changing its shape.

## Important Ideas

1. A line segment will always be the shortest distance between two points on a plane. The curved line and the broken line segments are a farther distance between A and B.

2. If two points are on a plane, the line containing the points is on the plane.

3. When two planes intersect, their intersection is a line.

4. All lines and planes are sets of points.

5. Every line has a coordinate system (the Ruler Postulate).

## Basic Sections

The size of an angle will depend on the opening between the two sides of the angle and is measured in units that are referred to as *degrees,* which are indicated by the ° symbol. To remember approximate sizes of angles, remember that a circle once around measures 360 degrees. To remember approximations of angles, it will be helpful to remember the above image.

Think of a whole pie as 360 degrees. If you eat a quarter (one-fourth) of the pie, the measure would be 90 degrees. What if you ate one-half of the pie? As stated above, 180 degrees is half, or you can add 90 degrees and 90 degrees — the two pieces you ate.

## The Protractor

If you cut the whole pie into eight equal pieces, what angle would one piece of the pie make? To answer this question, divide 360 degrees by eight (the total divided by the number of pieces)*.* This will tell you that each piece of the pie has a measure of 45 degrees.

Usually, when measuring an angle, you will use a protractor. Each unit of measure on a protractor is a degree.

The size of the angle is not dependent upon the lengths of the sides of the angle.

## Measuring Angles

The angles shown are approximately 10 degrees, 50 degrees, and 150 degrees.

#### Answers

1 = approximately 150 degrees

2 = approximately 50 degrees

3 = approximately 10 degrees

## Congruence

Congruent angles are angles that have the same number of degrees. For instance, two line segments are congruent if they are the same in length. If two angles have the same measure, they, too, are considered congruent. Symbolically, this can be shown as noted in the image above. Segment AB is congruent to segment OP.

## Bisectors

Bisectors refer to the line, ray, or line segment that passes through the midpoint. The bisector divides a segment into two congruent segments, as demonstrated above.

A ray that is in the interior of an angle and divides the original angle into two congruent angles is the bisector of that angle.

## Transversal

A transversal is a line that crosses two parallel lines. In the figure above, A and B are parallel lines. Note the following when a transversal cuts two parallel lines:

- The four acute angles will be equal.
- The four obtuse angles will also be equal.
- Each acute angle is supplementary

## Important Theorem #1

The sum of the measures of triangles always equals 180 degrees. You can prove this by using your protractor to measure the three angles, then total the three angles. See triangle shown to see that 90 degrees + 45 degrees + 45 degrees = 180 degrees.

## Important Theorem #2

The measure of the exterior angle will always equal the sum of the measure of the two remote interior angles. The remote angles in the figure are angle B and angle C. Therefore, the measure of angle RAB will be equal to the sum of angle B and angle C. If you know the measures of angle B and angle C, then you automatically know what angle RAB is.

## Important Theorem #3

If a transversal intersects two lines such that corresponding angles are congruent, then the lines are parallel. Also, if two lines are intersected by a transversal such that interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Edited by Anne Marie Helmenstine, Ph.D.