introduction to geometry level 1
geometry is extremely useful for engineers
architects, painters, carpenters
teachers, electricians, machinist
home builders and many more. Geometric principles are important in the
construction of buildings
and roads, the design and use of machinery
the creation of three-dimensional visuals and efects and to better
understand
and unlock the secrets of the universe itself like any language and mathematics
course
the important ideas in geometry must be developed gradually
with understanding practice and effort
alright let's get started with the basics the simplest figure
studied in geometry is called a point although a point doesn't have any size
it is often represented by a dot which does have some size
but keep in mind that for the most part they do not have a size
we use a dot to visually represent a figure
that lack size. We usually name points by using capital letters
for example the following five points are called
point A, point B, point C, point D,
and point E. All geometric figures consist of points.
another familiar geometric figure is called a line.
Lines are made up of points and are straight. The arrows on the end of the figures
show that the line extend infinitely far in both directions
they extend forever and ever. Although a picture of a line has some thickness
the line itself has no measurable thickness so keep that in mind unless it
is explicitly stated
lines have no thickness. The following figures are example lines
there are two different ways we can name lines the first way is by using a single
lower case letter
such as line M. The alternative way to name lines is by using their points
for example the line in the right contains the points
A and B so we can call the line AB
using symbols we can denote it like this AB
with a line on top or as follows: line BA
both ways are acceptable but for the most part
we denote the letters in alphabetical order so AB will be the typical way to
denote this line
to talk about the next geometric figure we need to remember a concept learned
in algebra one. In algebra you learned that a number line is formed when a numerical value
is assigned to each point on a line
for example the following line represents a number line
and point A is located at negative one, likewise
point B is located at one-and-a-half or writen another way
as one point five. We need to understand number lines
in order to understand line segments or simply
segments like lines segments are made up
of points and are straight. What distinguishes them from lines is the
fact that
line segments have a definite beginning and end
in other words they are measurable they do not extend for ever and ever
in both directions Like lines do. The following are examples of line segments
the line segment on the left can be denoted as:
AB or segment BA
notice that we add a segment on top of the letters in order to distinguish it
from that of a line
which contains arrows in both ends also notice that a segment is named
in terms of its two endpoints because a segment has a definite beginning
and end. The figure in the middle contains two line segments
the vertical up and down segment can be called
RS or segment SR
once again we usually denote them in alphabetical order
the horizontal crosswise or from left to right
can be named as segment ST
or segment TS. In the following number line
the segment whose endpoint is located at coordinate of
negative one and one is denoted as
line segment BD or segment DB
likewise a segment that has a coordinate at negative two
and two is denoted as segment AE
or segment EA. So we learn about lines
which extend for ever and ever in both directions and we learn about segments
which have a definite length and do not go on forever and ever
is it possible to create a figure that contains both of these characteristics?
the answer is yes. These hybrids are called rays
rays like lines and segments
are made up of points and are straight. A ray differs from a line
or segment in that it begins at an endpoint
and then extends infinitely far in one direction only
lines extend in two directions, rays extend in one direction only
so make sure you keep that in mind. The following are examples of rays
now when it comes to denoting them we actually need to be careful
and make sure he start denoting a ray by naming its endpoint first
this way it becomes clear where the ray begins. So the ray in the left
is denoted as
ray AB, notice that we add a ray on top going from left to right
this is the appropriate way to denote rays. It would be completely wrong to denote
the ray as
ray BA, remember we want to start with the endpoint first
the ray in the right can be called ray RS or
ray RT as long as the end point is given first
any other point on the ray can be used to denote the ray
now that we know about points, lines, segments, and rays.
we can now talk about angles. In essence
an angle is made up of two rays with a common end point.
this point is called the vertex of the angle. The rays are called sides of the
angle
there are various ways that angles can be described and denoted
for example if we place the number one inside the angle near the vertex
like this, we would denote this angle as, angle one
the symbol that we used to denote angles is drawn as follows:
its essentially a picture of an angle. If the angle only contains one letter
in the vertex, then you denote it as: angle A.
when you denote the angle using one letter make sure you name the point on
the vertex
an alternative way is needed if the angles is described in terms of three
points
for example, if this angle has a vertex labeled as A
and the other points are B and C respectively, then we need to denote the angle
as follows: angle BAC or
angle CAB. Notice that when we use three letters
the vertex, in this case A must be named in the middle
make sure you always do this. If we take the same angle
and draw another ray from point A to point D
we would end up with three new angles, make sure you can see them
the first angle is formed by points BAD
and is denoted as angle BAD or
angle DAB. The second angle is denoted as
angle DAC or angle CAD
and the third angle is denoted as angle BAC
or Angle CAB
notice that if we try to denote any of the angles by using only one letter
such as angle A, we would be totally confused what angle we are referring to
because all the angles share the same vertex, so in this case
it is really convenient to use three letters to name the angles
alright, the final figure we will talk about is formed by using three segments
and combining them in the following way. The figure that we just created is called
a triangle and we denote it as follows
triangle ABC or triangle ACB
or triangle CBA or triangle CAB
or triangle BAC or triangle BCA.
notice that the letters need to start at one point and move either clockwise
or counterclockwise when denoting them. The convenient way is
once again by using letters that are in alphabetical order
whenever possible. So triangle ABC will be the best way to describe this triangle
notice that a triangle has three segments as it sides
notice that there seems to be three angles but hey contain no rays
nevertheless, this figure contains three angles they are angle A
angle B and angle C. If you recall from your
algebra class we often talk about sets
well it turns out, we can think of a triangle as being a union
of three segments and can be denoted using set notation
as follows: triangle ABC is equal to
segment AB union with segment BC
union with segment AC. Likewise
the intersection of any two segments or sides forms the vertex
of a triangle and can be denoted using set notation as follows:
vertex A is equal to segment BA
intersected with segment AC. Vertex B
is equal to segment AB intersected
with segment BC and vertex C is equal to segment AC
intersected with segment CB. Alright, and there you have it
these are the basic building blocks of geometry and this is only the beginning
to recap, we learned about points which have no size
but we draw a point to visually represent one. We also learned about lines
which contain points and extend infinitely in both directions
this is represented by the arrows that point in both directions
we also learned about segments which have a definite length in other words
they are measurable
as a consequence they have a beginning and an end
which are called the endpoints of the segment. We also learned about rays
which share the characteristics of a line and a segment
it contains an endpoint in one side and contains an arrow
in the other side which extends infinitely in the direction it is facing.
When we have two or more rays that share a common point called a vertex.
we form angles and finally when we connect three line segmens together
forming three distinct vertex we create a triangle.
Okay, make sure you're familiar with all these figures
and the appropriate way to denote and name them in our next video we will
practice denoting
and naming figures with various examples.