1. Module 1 Lesson 1 The Foundation of Geometry
Welcome to Geometry! In this lesson, we will discuss the basic terms that are the foundation of all of geometry. Geometry is useful in many areas of life, especially in art and design. The Spanish artist Joan Miro used many geometric shapes in his paintings, as in the one shown here.
Module 1 Lesson 1 The Foundation of Geometry

2. Euclid was a Greek mathematician. He was born about 300 B.C.
This term, the Geometry you will be studying is called Euclidean Geometry.
Euclid was a Greek mathematician.
He was born about 300 B.C.
He is considered the “father of Geometry”.
There are several different types of Geometry, but we will be studying Euclidean geometry. It is named after the Greek mathematician Euclid, who was born around 300 B.C. He is considered the father of Geometry since he compiled all of the knowledge about Geometry and put it into one book.
Image source:

3. 3 undefined terms of Geometry
Point – just a location
Line – made up of an infinite number of points but has no thickness or width
Note: Although there are an infinite number of points on a line, you only need two(2) points to be able to draw (determine) the line
Plane – a flat surface made up of an infinite number of points, has no depth and extends infinitely in all directions
Note: Although there are an infinite number of points on a plane, you only need three(3) points to be able to draw (determine) the plane.
There are 3 undefined terms that are the basis for Geometry. Although they technically do not have definitions, we do use words to describe them so that we all agree on their meaning. The first term is a point , which represents a location (as on a map).
The second term is a line. A line is made by connecting two points, but it extends infinitely so it really has an infinite number of points. It has no thickness or width. If you had one point drawn, you would not know how to draw the line through that point (the line could go in many different directions and still go through one point). However, if you have two points, that is all you need to determine how the line is to be drawn.
The third undefined term is plane. A plane is a flat surface that extends infinitely in all directions. Even though it is made up of an infinite number of points, you only need 3 points to determine exactly how the plane is drawn.

4. How to draw and name objects
Point
Draw with a dot and name with a capital letter
Line
Draw with a line and an arrow at each end
Named as line BC or line m
Plane
Draw as a slanted 4-sided figure
Name as plane WXY
or as plane R P m B C
To draw a point, just use a dot. Name it with a capital letter like point P is shown here.
To draw a line, connect two points and put an arrow on each end. There are two different ways to name a line. You can name it with a small cursive letter, like m is shown here. So you would call this “line m”. Notice that m is not a point (since it is not a capital letter), just the name of the line written to the side. Another way to name a line is to use two points on the line. So the line here could be called “line BC” or “line CB”. The order in which you put the letters does not matter.
A plane is drawn as a slanted 4-sided figure. There are also two different ways to name a plane. You can name it with a capital cursive letter, like R shown here. Remember R is not a point because it is a script letter. It is just the name of the plane. So you would call this “Plane R “. Another way to name it is to use three (3) points that are on the plane. You could call this “Plane WXY” or Plane “XYW”, etc. The order of the letters does not matter. W X Y R

5. Note about naming planes
For example, this box shows several planes.
The top of the box is one plane.
There are four points shown
on the top plane: A, D, B, and C.
You could name it plane
ADC or ABC or BCD, etc.
However, many people use the
four corner points to name it so another acceptable name would be plane ABCD (or plane CBDA, etc.) The order of the points does not matter.
You will see a box like this in a lot of problems because it shows several planes at once. The top of the box is a plane, the front is a plane, etc. You can use 3 points to name each plane, so for the top, the name could be ADC or ABC or BCD, etc. You can use any 3 points on the top and the order does not matter. However, since there are 4 corner points, many people name the plane with all 4 points and that is ok, too. So other good names would be plane ABCD or CBDA, etc.

6. These terms can model real-life objects.
plane
There are points, lines, and planes all around you. For example, look at this picture of part of a room. The bottom corner where the two baseboards touch is modeled with a point. One of the baseboards is modeled by a line. One of the walls is modeled by a plane.
line
point

7. A couple of definitions:
Collinear: on the same line Noncollinear: not on the same line Ex. Points A and B are collinear but points A, B and C are noncollinear. Tricky question: Are points B and C collinear?? A B C
The word collinear means that points are on the same line. Noncollinear is the same word but has “non” in front of it so it means that the points are not on the same line. In this figure, points A and B are on the same line so they are collinear. Look at points A, B, and C. If you connected them, they would make a line with a bend in it and that is not really a line. A line must be straight with no turns. So the points A, B, and C are noncollinear.
Here is a tricky question: What about B and C? They are not on the line shown here but could you draw a straight line and have B and C on it. Yes, you could. So, B and C are collinear. Remember that collinear means that they are on the same straight line even if the line is not in the picture.

8. Coplanar – on the same plane
Points A, D, and B are coplanar (all are on the “top” plane). Points A, D, E and F are noncoplanar. Another tricky question: Are points B, C and H coplanar?
The word coplanar means that points or lines are on the same plane. For example, points A, D, and B are all coplanar since they are on the “top” plane of this box. Points A, D, E, and F are noncoplanar. A, D, and E are on the “left side” of the box, but F is on the “right side”.
Here is a very tricky question so pay close attention: What about points B, C, and H? At first, it looks like the answer would be no. B and C are on the “right side” and H is on the “left side”. BUT, could you draw a new plane (rectangle) that connected B, C, and H? Yes, you could. You could draw a slanted rectangle that connected B and C and then H and E. So, the points B, C, and H are coplanar.

9. Key points to remember Any two (2) points will always be collinear.
Any three (3) points will always be coplanar.
Just remember these two facts:
Any two points will always be collinear. If I draw two dots anywhere, you could connect them to make a line (even if the line is not shown in the diagram).
Likewise, any three points will always be coplanar.

10. Lines and Planes can intersect (share common points)
Lines and planes can intersect, which just means that they touch at common points. For example, line AE is on the front left of the box and line DA is on the top left. They will touch (intersect) at the corner at point A.
Line HG is on the bottom back of the box and line GF is on the bottom right side. Where do they touch? At the back corner at point G.
Finally, what about line EF and line DC? EF is at the front bottom of the box. Line DC is on the top back of the box. Will they ever touch? No, so they do not intersect.
Where do AE and DA intersect? Where do HG and GF intersect? Where do EF and DC intersect?

11. How do planes intersect?
There are 6 planes shown here on the box (top side, bottom side, front, back, right side, left side).
Where do planes ADC and plane DHE intersect? Plane ADC is the top of the box. Plane DHE is the left side of the box. So the top and the left side will intersect at the top left edge, which is DA. But remember, that they planes really keep going so the answer is line DA (or line AD).
Where do planes AEB and HGE intersect? Plane AEB is the front side of the box. Plane HGE is the bottom of the box. The front and the bottom will touch at the bottom front edge so that is line EF.
Finally, where do planes BFG and ADH intersect? Plane BFG is the right side of the box. Plane ADH is the left side of the box. So they do not intersect.
Where do planes ADC and plane DHE intersect? Where do planes AEB and HGE intersect? Where do planes BFG and ADH intersect?

12. Two lines intersect in a point or not at all.
KEY FACT
Two lines intersect in a point or not at all.
Two planes intersect in a line or not at all.
Just remember, two lines either intersect in a point or they do not intersect at all.
Two planes intersect in a line or they do not intersect at all. Planes never intersect in line segments because the planes are really infinite.

13. Two important formulas to remember
Distance Formula
Midpoint Formula
When you are working with points, you often want to know the distance between them and where the midpoint is. These two formulas will help you do that. In the upcoming video, you will see examples of these formulas worked out to help you learn how to use them.