Lesson Video: Geometric Concepts Mathematics

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 𝐴𝐵𝐶 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 𝐷 and 𝐸. 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, let’s consider a line starting at a point 𝐻 that passes through a point 𝐼 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 𝐴 and 𝐵 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 𝐴 and 𝐵 as just 𝐴𝐵.

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 𝐴 to 𝐵 is the set of all points that lie on the ray starting at 𝐴 that extends indefinitely in the direction through 𝐵.

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 𝐴 to 𝐵 is a subset of the ray from 𝐴 through 𝐵. And this in turn is a subset of the line between 𝐴 and 𝐵. Or is it option (E) the line segment 𝐴𝐵?

Let’s 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) 𝐴𝐵. Option (B) the ray from 𝐴 through 𝐵. Option (C) the ray from 𝐵 through 𝐴. Option (D) the line through 𝐴 and 𝐵. Or is it option (E) the line segment 𝐴𝐵?

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 𝐴 and 𝐵 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 𝐴𝐵 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 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.

Let’s 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 𝐴𝐵𝐶 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 𝐴𝐵𝐶.

In this question, we are asked to find the measure of the reflex angle 𝐴𝐵𝐶. 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 𝐵. 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 𝐴 through 𝐵 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 𝑥, then we can say that 25 degrees plus 𝑥 must be equal to 360 degrees. We can then subtract 25 degrees from both sides of the equation and evaluate to obtain that 𝑥 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.

Let’s 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.