Question Video: Identifying a Line Segment Mathematics • First Year of Preparatory School

Video Transcript

Which of the following is represented by the line segment between 𝐴 and 𝐵?

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 𝐴 and 𝐵 will have these two points as endpoints. We can see that only the figure in option (A) has 𝐴 and 𝐵 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 𝐵 through 𝐴 and that option (E) is the ray from 𝐴 through 𝐵.

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 𝐴 and 𝐵 or the line between 𝐵 and 𝐴.

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 𝐷𝐸 onto the side 𝐷𝐹 than it takes to rotate 𝐴𝐵 onto 𝐴𝐶. 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, let’s consider the following square 𝐴𝐵𝐶𝐷. 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 𝑚 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 𝐴 through 𝐵 and the ray from 𝐴 through 𝐶, 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 𝐴.

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 𝐴 shown has a measure of 360 degrees. The superscript circle represents the units of degrees.

It is also worth noting that the ray from 𝐴 through 𝐵 is coincident with the ray from 𝐴 through 𝐶. 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 𝐶 through 𝐷 onto the ray from 𝐶 through 𝐵. 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 𝐴𝐵𝐶 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 𝑚 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.