The word *geometry* is Greek for *geos* (meaning earth) and* metron* (meaning measure). Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 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 and was called Elements. Euclid's text has been used for over 2000 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.

### Terms in Geometry

### Point

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

### 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, I could name one of the lines above simply by indicating an *e.*

### More 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 labeled Ray, A is the endpoint and this ray means that all points starting from A are included in the ray.

### Terms in Geometry - 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 Image 1 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.

In Image 2, this angle would be called angle 3. *OR*, you can also name the vertex by using a letter. For instance, angle 3 could also be named angle B if you choose to change the number to a letter.

In Image 3, this angle would be named angle ABC or angle CBA or angle B.**Note:** 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, a 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.

### Types of Angles - Acute

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 that 90° and can look something like the angles between the grey rays in the image above.

### Types of Angles - Right Angle

A right angle measures exactly 90° and will look something like the angle in the image. A right angle equals 1/4 of a circle.

### Types of Angles - Obtuse Angle

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

### Types of Angles - Reflex

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

### Types of Angles - Complementary Angles

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

In the image shown angles ABD and DBC are complementary.

### Types of Angles - Supplementary Angles

Two angles adding up to 180° 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 is by subtracting angle ABD from 180 degrees.

### Basic and Important Postulates in Geometry

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, 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. Here are a few of the basics and postulates that are intended for entry-level Geometry. (Note:* there are many more postulates that are stated here, these postulates are intended for beginner geometry)*

### Basic and Important Postulates in Geometry - Unique Segment

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

### Basic and Important Postulates in Geometry - Circle Measurement

There are 360° around a circle.

### Basic and Important Postulates in Geometry - Line Intersection

Two lines can intersect at ONLY one point. S is the only intersection of AB and CD in the figure shown.

### Basic and Important Postulates in Geometry - Midpoint

A line segment has ONLY one midpoint. M is the only midpoint of AB in the figure shown.

### Basic and Important Postulates in Geometry - 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.

### Basic and Important Postulates in Geometry - Conservation of Shape

Any geometric shape can be moved without changing its shape.

### Basic and Important Postulates in Geometry - 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 further in distance between A and B.

2. If two points lie in a plane, the line containing the points lie in 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)

### Measuring Angles - Basic Sections

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

Think of a whole pie as 360°, if you eat a quarter (1/4) of it the measure would be 90°. If you ate 1/2 of the pie? Well, as stated above, 180° is half, or you can add 90° and 90° -- the two pieces you ate.

### Measuring Angles - The Protractor

If you cut the whole pie into 8 equal pieces. What angle would one piece of the pie make? To answer this question, you can divide 360° by 8 *(the total by the number of pieces).* This will tell you that each piece of the pie has a measure of 45°.

Usually, when measuring an angle, you will use a protractor, each unit of measure on a protractor is a degree °.**Note**: The size of the angle **is not** dependent on the lengths of the sides of the angle.

In the above example, the protractor is used to show you that the measure of angle ABC is 66°

### Measuring Angles - Estimation

Try a few best guesses, the angles shown are approximately 10°, 50 °, 150°,

**Answers**:

1. = approximately 150°

2. = approximately 50°

3 = approximately 10°

### More about Angles - Congruency

Congruent angles are angles that have the same number of degrees. For instance, 2 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 by as noted in the image above. Segment AB is congruent to segment OP.

### More about Angles - 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.

### More about Angles - 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*to each obtuse angle.

### More about Angles - Important Theorem #1

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

### More about Angles - Important Theorem #2

The measure of the exterior angle will always equal the sum of the measure of the 2 *remote* interior angles. NOTE: the remote angles in the figure below 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 angle B and angle C then you automatically know what angle RAB is.

### More about Angles - Important Theorem #3

If a transversal intersects two lines such that corresponding angles are congruent, then the lines are parallel. AND, 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.