Fundamental Geometry Concepts

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.

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

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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."

Two lines will intersect at a point

Image: Kevin Olson · CC BY-NC-SA

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

Image: Kevin Olson · CC BY-NC-SA

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

Image: Kevin Olson · CC BY-NC-SA

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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.

Image: Kevin Olson · CC BY-NC-SA

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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.

Image: Kevin Olson · CC BY-NC-SA

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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 $\angle 3$.

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

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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.

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Solutions 1.4

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1. 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: $LN=40$.

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