### Video Transcript

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.