### Video Transcript

Geometric Concepts

In this video, we will learn how to
identify and define various geometric objects, including line segments, rays,
straight lines, angles, and types of angles.

In geometry, there are many
different objects that we want to classify and analyze. For instance, consider the
following triangle. We usually describe a triangle by
using its vertices. In this case, we call it the
triangle π΄π΅πΆ since we can transverse the triangle in this order. If we wanted to describe a single
side of this triangle, we might follow a similar line of reasoning to describe it by
its endpoints.

However, there are actually many
objects that we can describe using two points. For example, consider the given
straight line passing through π· and πΈ. This straight line continues
indefinitely in both directions, which we can represent using a double-sided
arrow. We note that the finite line
between the two points would have a length, whereas this line does not. This means that we need notation to
differentiate between these two possibilities.

We use a double-sided arrow above
the vertices to represent a line that extends indefinitely in both directions. We can think of this as the arrow
pointing in the directions that the line extends. We use a bar over the vertices to
show that the line does not extend indefinitely in either direction. This is called a line segment.

There is one more object that we
can describe using two points. This time, letβs consider a line
starting at a point π» that passes through a point πΌ and continues indefinitely in
this direction. This is called a ray. And we represent these using an
arrow above the vertices. It is worth noting that the order
we write the vertices matter when we are talking about rays since the ray will only
extend in one direction. The first vertex is the initial
point of the ray, and the second is any other point on the ray. The same is not true for the other
two objects; we can list the vertices in any order since it will still describe the
same object.

We can define each of these objects
more formally as follows. First, we define the line segment
between π΄ and π΅ to be the set of all points on the finite portion of the straight
line between these two points. This definition allows us to
consider a few properties of line segments. We can note that we have defined
this to be a set of points. So we can say that any point on the
line segment is an element of this set.

Another property we can note is
that line segments are a finite portion of a line. This means that we can consider
their lengths. We write the length of a line
segment between two points π΄ and π΅ as just π΄π΅.

Next, we use a double-sided arrow
above the vertices to represent the straight line between two points. This is the set of all points that
lie on this line. And it extends indefinitely in both
directions. We cannot represent the length of
straight lines since they extend indefinitely.

Finally, we use an arrow to
represent rays. The ray from π΄ to π΅ is the set of
all points that lie on the ray starting at π΄ that extends indefinitely in the
direction through π΅.

The last property we can note about
these objects is that since they are sets, we consider if there is any overlap in
these sets. In general, we can see that the
line segment from π΄ to π΅ is a subset of the ray from π΄ through π΅. And this in turn is a subset of the
line between π΄ and π΅. Or is it option (E) the line segment π΄π΅?

Letβs now move onto an example
where we need to represent a given geometric object using this notation.

How can the line shown in the
figure be represented mathematically? Option (A) π΄π΅. Option (B) the ray from π΄
through π΅. Option (C) the ray from π΅
through π΄. Option (D) the line through π΄
and π΅. Or is it option (E) the line
segment π΄π΅?

In this question, we are given
a line and two points on this line. And we are asked to represent
the figure mathematically.

To do this, we first note that
there are arrows on both sides of the line. So this line extends
indefinitely in both directions. We can also note that we have
two distinct points π΄ and π΅ that lie on the line. We can then recall that we can
represent that a line extends indefinitely in both directions by using a
double-sided arrow above two distinct points which lie on the line. This gives us two different
equivalent ways of representing this line. We can see that this only
matches option (D).

We can also check what is
represented by the four other options. First, we recall that π΄π΅
without any further notation means the length of the line segment between these
two points. Second, we recall that a
one-sided arrow represents a ray, that is, a straight line that extends
indefinitely in a single direction. The first point is the initial
point of the ray. We know that the given figure
is not a ray, since it extends indefinitely in both directions.

Finally, we recall that a line
above the points represents a line segment, that is, the finite line segment
between the two points. This is not the same as the
given line, since the figure extends indefinitely in both directions. Hence, the answer is option
(D).

In our next example, we will
determine the correct figure of a geometric object from its mathematical
notation.

Which of the following is
represented by the line segment between π΄ and π΅?

To answer this question, we can
begin by recalling that a horizontal bar above two distinct points represents a
line segment, that is, a finite portion of a straight line between the two
points. This means that the line
segment between π΄ and π΅ will have these two points as endpoints. We can see that only the figure
in option (A) has π΄ and π΅ as the endpoints. The other options all extend
indefinitely in one direction or both directions.

We could stop here. However, we can also represent
the other options. First, we recall that we can
represent a ray by using an arrow above the points, that is, a line starting at
one point and extending indefinitely in the direction of another point. We see that option (B) is the
ray from π΅ through π΄ and that option (E) is the ray from π΄ through π΅.

We can also recall that we can
represent the straight line between two points using a double-sided arrow above
the points. Since this line extends
indefinitely in both directions, the order of the points does not matter. So we can represent both of
these lines as either the line between π΄ and π΅ or the line between π΅ and
π΄.

We are now ready to start
considering polygons. These are shapes that are made
of line segments, called sides, and points, called vertices. If we have two triangles as
shown, then we can see that these two triangles are not the exact same shape in
a few different ways. One way is to compare the side
lengths of the two triangles. However, we can also compare
how far apart the sides of each triangle are at each vertex. This is known as an angle.

More formally, the measure of
the angle at a vertex is the rotation required to rotate one side of the angle
onto the other side of the angle. We see that it takes more
rotation to rotate the side π·πΈ onto the side π·πΉ than it takes to rotate π΄π΅
onto π΄πΆ. So we can say that this angle
has a larger measure.

There are a lot of important
points to note about these definitions. First, we differentiate between
the angle at a vertex and the measure of the angle in the same way we
differentiate between sides and lengths.

To see this, letβs consider the
following square π΄π΅πΆπ·. In a square, all of the sides
have the same length. This means that we can equate
their lengths. However, we cannot say that the
sides themselves are equal, since these are sets with different elements. In the same way, we know that
it will take one-quarter of a full turn to rotate one side of the square onto an
adjacent side. So we know that all of the
angles have the same measure. However, we cannot equate the
angles themselves. We can represent this using
angle notation. The π represents that we are
talking about the measure. And the symbol at the start of
the angle shows that we are talking about an angle. The middle point is the vertex
of the angle.

Another thing worth noting is
that there are always two angles between two lines that share a vertex, since we
can rotate either side onto the other and in either direction. If we do not specify which
angle we are talking about, then we are talking about the angle with the smaller
measure.

We can now use these ideas to
formally define angles. An angle is the union of two
rays that share an initial point, say, the ray from π΄ through π΅ and the ray
from π΄ through πΆ, and the rotation needed to take one ray onto the other
ray. We call the shared initial
point of the rays the vertex of the angle and the two rays that form the angle
the sides of the angle. It is worth noting that we
often use line segments or lines instead of rays. And the result is the same.

We can represent this angle
using either the angle symbol or a hat over the vertex of the angle as
shown. Since there are two angles
between any two rays, we refer to the larger rotation as the reflex angle at
π΄.

We can formally define angle
measure in a similar way. However, we first need to
decide on units for the measure of an angle. We can do this by defining one
full rotation to have a value of 360. And we use the units of degrees
to represent angle measure. So the reflex angle at π΄ shown
has a measure of 360 degrees. The superscript circle
represents the units of degrees.

It is also worth noting that
the ray from π΄ through π΅ is coincident with the ray from π΄ through πΆ. So no rotation is needed to
rotate the rays onto each other. So the other angle at the
vertex has a measure of zero degrees. We call this a zero angle.

If we were to split a full turn
in half, then its measure will also half. For instance, half a turn is
needed to rotate the ray from πΆ through π· onto the ray from πΆ through π΅. So we can say that this angle
has measure 180 degrees. This is called a straight
angle, since it occurs with points on a straight line. We can extend this further by
halving the rotation required once again.

This time, we have a quarter of
a full rotation. So the angle measure will be
one-quarter of 360 degrees, which is 90 degrees. We have seen angles of this
measure when considering the interior angles in a square. These are called right
angles. Since these are the angles in a
square, we often represent right angles using rectangular sides rather than a
curve as shown.

We can compare the measure of
any angle to these angles to categorize them. We say that any angle with
measure smaller than 90 degrees is an acute angle. Any angle with measure between
90 degrees and 180 degrees is an obtuse angle. And any angle with measure
greater than 180 degrees is a reflex angle. We can define these more
formally as follows. We define a measure of an angle
π΄π΅πΆ to be the size of the rotation required to align the sides of the
angle. The center of the rotation is
the vertex of the angle. And a full rotation is said to
have a measure of 360 degrees.

We use an π before the angle
to show that we are talking about its measure. We call an angle with measure
360 degrees a full turn, an angle with measure 180 degrees a straight angle, and
an angle of measure 90 degrees a right angle. We also call an angle with
measure zero degrees a zero angle. Finally, we call an angle whose
measure is between zero and 90 degrees an acute angle, an angle whose measure is
between 90 and 180 degrees an obtuse angle, and an angle whose measure is
greater than 180 degrees a reflex angle.

Letβs now see an example of using
these ideas to correctly identify the definition of an angle.

Which of the following is the
definition of an angle? Option (A) it is the rotation
of a point from one position to another around a line segment. Option (B) it is the rotation
of a ray from one position to another around the starting point. Option (C) it is a measure of
the distance between two points. Option (D) it is a quantity
that measures how many points two lines intersect at. Or is it option (E) it is the
rotation of a line segment from one position to another around a ray?

In this question, we are asked
to identify which of five given options is the correct definition of an
angle. We can do this by recalling
that an angle is the rotation needed to rotate one ray onto another ray that
share the same initial point. We call the shared initial
point the vertex of the angle. In particular, we call the
rotation needed to rotate one ray onto the other the angle at the vertex. We can see that this matches
option (B).

We can also note that we only
define rotations in the plane around a point, so options (A) and (E) cannot be
correct. Similarly, option (C) is the
length of a line segment from its endpoints.

In our next example, we will
determine which of five given options is an acute angle.

Which of the following is an
acute angle?

In this question, we need to
identify which of the five given angles is an acute angle. We can do this by recalling
that we say that an angle π΄π΅πΆ is acute if its measure is between zero and 90
degrees. We can see that of the five
given options, only option (D) has a measure between these values. Thus, it is the only acute
angle. We can also classify the other
angles using their measures. First, we recall that angles
with measure between 90 and 180 degrees are obtuse angles. So option (B) is an obtuse
angle, since its measure is 120 degrees.

Next, we recall that an angle
of measure greater than 180 degrees is a reflex angle. So option (A) is a reflex
angle. We can also recall that an
angle of measure 90 degrees is referred to as a right angle and that an angle of
measure 180 degrees is called a straight angle.

It is also worth noting that we
can answer this question without knowing the exact measures by noting that acute
angles must have smaller measure than a right angle. So we can compare the size of
the rotations to a quarter turn to see that only option (D) is an acute
angle.

In our final example, we will
determine the measure of a reflex angle by using the measure of the smaller angle at
the same vertex.

Find the measure of the reflex
angle π΄π΅πΆ.

In this question, we are asked
to find the measure of the reflex angle π΄π΅πΆ. We can first recall that a
reflex angle is one with measure greater than 180 degrees. So in this case, it is the red
angle at vertex π΅. We can also recall that the
measure of an angle is a measure of the amount of rotation required to rotate
one ray onto the other using the vertex as the center of the rotation and that a
full rotation has a measure of 360 degrees. This means if we were to rotate
the ray from π΄ through π΅ a full turn as shown, then the measure of this angle
is 360 degrees.

We can see that rotating a full
turn is equivalent to rotating through the angle of measure 25 degrees and then
through the red reflex angle. If we call the measure of this
unknown reflex angle π₯, then we can say that 25 degrees plus π₯ must be equal
to 360 degrees. We can then subtract 25 degrees
from both sides of the equation and evaluate to obtain that π₯ is 335
degrees.

We can use the same reasoning
to prove a useful general result. All of the angles around a
point will be equivalent to a full turn. So the sum of the measures of
all of the angles around a point must be 360 degrees.

Letβs now go over the key points we
found in this video.

First, we saw that there are many
ways of representing different geometric objects in the plane. For instance, we can use a
horizontal bar over two points to represent the line segment between two points. It is a finite line with the two
points as its endpoints and it is a set containing all of the points on this line
segment.

Second, we saw that we can use a
double-sided arrow to represent the straight line between two points. This line extends indefinitely in
both directions. And it is a set containing all
points that lie on this line.

Third, we can use an arrow above
two points to represent a ray, that is, a line starting at the first point that goes
through the second point and extends indefinitely in this one direction. It is the set containing all points
on this ray.

Next, we defined a rotation needed
to rotate one ray onto another ray that share an initial point as an angle. We called the initial point that
they share the vertex of the angle. And the rays are often called the
sides of the angle. We also saw that we called the size
of the rotation needed to rotate one ray onto the other ray in an angle its measure,
where if a full turn is used in the rotation, then we say that the angle has a
measure of 360 degrees. If half a turn is used in the
rotation, then we say its measure is 180 degrees. This is called a straight
angle. An angle that requires a quarter
turn in the rotation will have a measure of 90 degrees and is called a right
angle. Finally, if there is no rotation in
the angle, then its measure is zero degrees. And we call this a zero angle.

We also saw that we classify angles
based on the size of their measures. If an angle has measure between
zero and 90 degrees, we say it is an acute angle. If an angle has a measure between
90 and 180 degrees, then we say it is an obtuse angle. And if the measure of the angle is
greater than 180 degrees, then we call it a reflex angle.