# Fundamentals of Geometry

This article covers all the basics of geometry, including points, lines, segments, rays, planes, and angles. You'll also learn a bit about postulates, theorems, and proofs. The primary focus is on lines and related geometric figures.

**Geometry** is the mathematical study of the properties and relations of points, lines, angles, surfaces, and solids. Essentially, it's the study of shapes and their parts.

## 1.1: Points, Lines, and Planes

Start by familiarizing yourself with the concepts below.

### Points

A **point** is an exact location on a plane. They are usually named with a single letter, such as point A, B, C, and so on.

Remember that points are places, not things. They are represented by a dot, but take up no space themselves.

### Plane

A **plane** is a flat two-dimensional surface that extends infinitely in every direction.

Planes may be named by giving them a capital letter, or by listing any three points on the plane in any order.

The plane in the example could be called Plane M, Plane ABC, Place CBA, Plane BCA, or any other combination of the three point names.

Points that are on the same plane are called **coplanar points**.

### Lines

A **line** is a straight path between two points, which extends infinitely in both directions. It has no width and therefore takes up no actual space. It's flat.

Lines can be named by drawing a horizontal arrow above two points in the line. For example, a line with points C and D would be written like \(\overleftrightarrow{CD}\) or \(\overleftrightarrow{DC}\)

They may also be identified using a single lower case italic letter, often written in cursive. So the line in this example could be called line \(\overleftrightarrow{CD}\), \(\overleftrightarrow{DC}\), or line *l*.

*l*

### Coordinate Plane

A **coordinate plane**, also known as a Cartesian plane, after a French mathematician, is a two-dimensional plane arranged in a grid-like structure. It consists of a vertical line called the y-axis and a horizontal line called the x-axis.

Points on a coordinate plane are assigned **coordinates**, which is a pair of two values given as an **ordered pair** (x,y), where x is the distance left or right of the y-axis and y is the distance above or below the x-axis.

A coordinate plane is arranged into four different **quadrants,** which are the different sections of the plane divided by the axes.

Coordinate planes must have some sort of **scale**, which is how many units one space on the grid represents. The scale in the example image is 1, since each block in the grid represents one unit. The scale is written along the axes of the plane.

In the example image, point A is at (4,2) because it's located four spaces to the right (positive x), and two spaces up (positive y). Point B is at (-4,3), because it's 4 spaces to the left of the y axis (negative x), and 3 spaces up (positive y).

### Collinear Points

**Collinear points** are points that are on the same line. In this example, points A, B, and C are collinear. Points C, D, and E are also collinear. Points A, B, and D are NOT collinear, nor are points D, E, and B.

### Line Segments

A **line segment** is a part of a line with two endpoints.

Line segments can be represented by drawing a horizontal line above two points in the line. For example, a line segment with endpoints C and D would be written like \(\overline{CD}\) or \(\overline{DC}\)

The length between two points in a line segment is simply the two points listed next to each other. So the length from C to D would be written: CD.

### Congruent Line Segments

Line segments are **congruent** if they have the same length.

For example, a one foot line segment would be congruent with another one foot long line segment. In the example shown, \(\overline{AB} \cong \overline{CD}\). The equals sign with the little squiggle over it (tilde) means "is congruent to."

The little line going through the middle of each line segment are called **tick marks**. When tick marks match, it means the lines are congruent. Since line segments \(\overline{EF}\) and \(\overline{GH}\) both have *two* tick marks, we know they're congruent, even though I never specified an exact length.

Statements like \(\overline{AB} \cong \overline{CD}\) or \(\overline{EF} \cong \overline{GH}\), are known as **congruency statements.** We may need to write these in proofs, which we'll get into later on.

### Rays

A **ray** is a part of a line that starts with one endpoint and extends infinitely in one direction.

Rays can be represented by drawing a horizontal arrow above the points in one direction. For example, \(\overrightarrow{AB}\)

It is best to start with the endpoint and extend the arrow right towards the other point. While \(\overleftarrow{BA}\) is technically correct, \(\overrightarrow{AB}\) is just a little easier to read since it goes left to right. So stick with notating it this way rather than right to left. In this example, \(\overleftarrow{AB}\) and \(\overrightarrow{BA}\) would be incorrect, as the arrow would go the wrong direction from A to B.

**Opposite rays** are two rays that start from the same point and go off in opposite directions. If point C lies between points A and B on \(\overleftrightarrow{AB}\), then \(\overrightarrow{CA}\) and \(\overrightarrow{CB}\) are opposite rays.

### Parallel Lines

**Parallel lines** are lines that are always the same distance apart and never meet. They have the same slope.

They are shown by drawing two vertical lines between each line name, for example, \(l \parallel m\) means "*l* is parallel to *m*."

In this example, it could also be written \(\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}\)

### Perpendicular Lines

**Perpendicular lines** are lines that intersect at four right angles (ninety-degree angles).

They are shown by drawing a perpendicular symbol between each line name, for example, \(l \bot m\) means "*l* is perpendicular to *m*."

#### Knowledge Check 1.1

Use the following diagram to answer the questions:

- What are
**all**the possible names for the line containing the points A, B, and C (in both directions)? There are 6. - Is the geometric figure containing the points Q and R a Ray, a Line, or a Line Segment?
- True or false: Lines
*l*and*m*are perpendicular. - Identify the two opposite rays starting at point B in \(\overleftrightarrow{AC}\)

1. Possible names for this line include: \(\overleftrightarrow{AB}\) \(\overleftrightarrow{BA}\) \(\overleftrightarrow{AC}\) \(\overleftrightarrow{CA}\) \(\overleftrightarrow{BC}\) or \(\overleftrightarrow{CB}\)

2. \(\overline{QR}\) is a line segment.

3. This is false. Lines *l* and *m* are not perpendicular, because they do not intersect at a right angle.

4. The two opposite rays coming off point B are \(\overrightarrow{BA}\) and \(\overleftrightarrow{BC}\)

## 1.2: Intersections, Midpoints, and Bisectors

### Intersections

An **intersection** occurs when two geometric figures pass through each other. When this happens, the two figures are said to have intersected.

Two lines will intersect at a point

Two planes will intersect along a line. In this example, line l intersects the two planes.

A plane and a line will intersect at a single point. In this example, line l intersects plane M at point X.

### Midpoints

The halfway point in a line segment is called its **midpoint.** It divides the line segment into two equal length (congruent) segments. Here are some examples of midpoints.

In the example above, B is a midpoint of \(\overline{AC}\), so \(\overline{AB}\cong\overline{BC}\). E is a midpoint of \(\overline{DF}\), so \(\overline{DE}\cong\overline{EF}\). H is a midpoint of \(\overline{GI}\), so \(\overline{GH}\cong\overline{HI}\).

### Bisectors

A **segment bisector** is something that passes through a segment at the midpoint and "bisects" it. A segment bisector could be a line, ray, plane, or another segment.

## 1.3: Verticies and Angles

### Vertex

A **vertex** is a point where two or more lines, line segments, or rays intersect (cross or connect with) each other. Since a vertex is a point, it follows the same naming guidelines as points. This would be called vertex A.

### Angle

An **angle** is formed by two rays with the same endpoint (the vertex).

They can be named by drawing the angle symbol \(\angle \) followed by the vertex point, by naming all 3 points involved in the angle, with the vertex in the center, or by using a numerical name assigned to a particular angle.

So the angle in this example could be called \(\angle A\), \(\angle BAC\), \(\angle CAB\), or \(\angle 1\).

You may also see some people give angles letter names instead of numerical names. To prevent confusion with lines, we prefer to use numbers.

To visually show an angle is present, you can draw a curve between the two sides of the angle, near the vertex.

### Multiple Angles

If multiple angles share the same vertex, you may not name the angle using only the vertex, as there's no way of knowing which angle you're referring to.

The angle in this example cannot be called \(\angle A\) because we don't know which angle it's referring to. If we wanted to identify the angle in the top portion, it must be called \(\angle BAC\), \(\angle CAB\), or \(\angle 3\).

## 1.4: Postulates and Theorems

A **proof** in math is an argument used to show that a mathematical statement is true using postulates, theorems, and definitions. It "proves" that something is true.

A **postulate**, also known as an axiom, is a statement that is known to be true, without evidence. We don't have to prove it.

A **theorem** is a statement that has been proven to be true using other theorems, definitions, or postulates. If you're working on a proof, you don't need to prove these statements individually, as they've already been proven or accepted.

### Segment Addition Postulate

A simple example of a postulate is the segment addition postulate, which states that "If point B is on the line segment \(\overline{AC}\), then AB + BC = AC.

Remember, \(\overline{AC}\) is the name of the line segment itself, and \(AC\) is the length of the line segment.

This one (hopefully) makes sense, even if it seems obvious. If you perfectly cut a piece of rope at any point, the two pieces will have the same combined length as the original rope.

### The Ruler Postulate

The **Ruler Postulate** says that every point on a line can be paired with a real number. The real number paired with a point is the coordinate of the point. Just like points on a ruler.

The **distance** between two points is the absolute value of the difference between them.

Look at the example ruler, measured in centimeters. There are two points, O and P. O is at 2cm, and P is at 6.5cm. The difference between 6.5cm and 2cm is 4.5cm, so the length of \(\overline{OP}\) is 4.5cm.

### Knowledge Check 1.4

1. Assume the example segment \(\overline{AC}\) is not to scale. Given that \(AB=10\) and \(BC=15\), what is \(AC\)?

2. You are given a line segment \(\overline{LN}\) with point M somewhere on the line between L and N. Given that \(LM=15, LM=3x, MN=5x\), what is x and what is the length of total segment \(\overline{LN}\)?

You can always draw it out, sometimes visualizing it may help.

View Answers1. The answer is: \(AC=25\) because \(AB+BC=10+15=25\)

2. *x* must be equal to 5 because \(LM=15=3x\), so \(x=15/3=5\). We know \(MN=5x\), so \(MN\) must equal \(5\times5\) which is 25. With the segment addition postulate, we know that \(LM+MN=LN\), so \(LN=15+25=40\). The total length of the segment is: \(LN=40\).

## 1.5: Polygons

A **polygon** is a closed two-dimensional figure on a plane with at least three straight sides and no curved sides. Essentially, it's any two dimensional shape without curves. So a triangle is a polygon, but a circle is not.

Shapes with equal length sides and interior angles are known as **regular polygons**. If one of the side lengths or interior angles does not match the others, it is an **irregular polygon**.

An **interior angle** is an angle formed on the inside of a polygon where two sides meet.

### Polygons By Number of Sides

### Triangle

A **triangle** is a polygon with three sides.

The three angle measurements angles of a triangle add up to 180°

### Quadrilateral

A **Quadrilateral** is a polygon with four sides.

The four interior angle measurements of a quadrilateral add up to 360°

### Pentagon

A **Pentagon** is a polygon with five sides.

The five interior angle measurements of a pentagon add up to 540°

### Hexagon

A **Hexagon** is a polygon with six sides.

The five interior angle measurements of a hexagon add up to 720°

### Heptagon

A **Heptagon** is a polygon with seven sides.

The seven interior angle measurements of a heptagon add up to 900°

### Octagon

A **Octagon** is a polygon with eight sides.

The eight interior angle measurements of a octagon add up to 1080°

### Nonagon

A **Nonagon** is a polygon with nine sides.

The nine interior angle measurements of a nonagon add up to 1260°

### Decagon

A **Decagon** is a polygon with ten sides.

The ten interior angle measurements of a decagon add up to 1440°

### Hendecagon

A **Hendecagon** is a polygon with eleven sides.

The eleven interior angle measurements of a hendecagon add up to 1620°

### Dodecagon

A **Dodecagon** is a polygon with twelve sides.

The twelve interior angle measurements of a hendecagon add up to 1860°

### n-gon

An **n-gon** is a polygon with any number of sides, given n is the number of sides.

For example, a hendecagon could be called an 11-gon. A 20 sided polygon would be called a 20-gon.

The formula for total interior angle measurement is \((n-2)\times180^{\circ}\)

Notice how with each additional side, the regular polygons seem to look more like circles. In the 5000-gon, there's so many congruent sides, it looks like a circle on the screen. If we zoomed in *really* far and counted, there should be 5000 straight sides.

## 1.6: Congruence vs Equality

In this chapter, we have mentioned that congruent line segments have the same length. **Congruence** is the quality of two or more items having corresponding or agreeing properties. Congruent polygons are polygons with the same shape and size. **Equality** is the quality of two or more items being *exactly* the same. Two line segments may be congruent, but this does not mean they are equal. The measure of two congruent line segments is equal, they're the same number on the number line, but the line segments themselves are not equal.

Shapes with the same sizes and angles are also considered congruent. This does not mean two identical triangles are equal - they are still two different triangles.

Consider the following example with two triangles. They are identical in size, side lengths, and angle measurements, but are at two different locations on the plane. These two triangles are congruent, but they are not equal.

In this example, \(\triangle ABC \cong \triangle DEF\), but \(\triangle ABC \) is not equal to \(\triangle DEF\). \(\overline{AB}\cong\overline{DE}\) and \(AB=DE\), but \(\overline{AB}\) is not \(= \overline{DE}\)

## Review Worksheet

Here is a downloadable, printable PDF worksheet with problems relating to this chapter.

Download Worksheet Solutions