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Lesson Explainer: Geometric Concepts Mathematics

In this explainer, we will learn how to identify and define line segments, straight lines, rays, and types of angles.

In geometry, there are many different types of objects, and we often represent these using points. For example, we can represent the side of a shape using the vertices of the shape, also called the endpoints.

In the above shape, we can describe one of the sides with the vertices, for example the side from 𝐴 to 𝐡.

However, this is not the only object that we can describe with two points. We can also represent the straight line that passes through 𝐴 and 𝐡 using these two points.

This means we need notation to differentiate between these two objects. We use the notation 𝐴𝐡 to represent the side of a shape. It is the portion of the straight line between 𝐴 and 𝐡. We call this the line segment between 𝐴 and 𝐡, and 𝐴 and 𝐡 are called its endpoints. We use the notation ⃖⃗𝐴𝐡 to represent the straight line passing through 𝐴 and 𝐡. The double-sided arrow tells us that the line continues indefinitely in both directions.

In fact, there is a third geometric object that can be described by two points. We could also have a straight line with a single endpoint; this is called a ray. Consider the following diagram of a ray from 𝐴 to 𝐡.

We see that the line continues indefinitely in one direction, so we represent this with the notation 𝐴𝐡. The arrow tells us the direction of the ray, and the first point is the starting point of the ray. It is worth noting that the order of the points matters in the notation of a ray, but it does not matter in the notations of lines or line segments. We see that 𝐴𝐡 and 𝐡𝐴 are the same and ⃖⃗𝐴𝐡 and ⃖⃗𝐡𝐴 are the same; however, we can sketch 𝐡𝐴 to see that it is distinct from 𝐴𝐡.

We can formalize these definitions as follows.

Definition: Straight Line, Line Segment, Rays, and Notations

𝐴𝐡 is the line segment with endpoints 𝐴 and 𝐡. It is a set containing all of the points that lie on this finite portion of a straight line. Since line segments are finite, we can measure their lengths.

⃖⃗𝐴𝐡 is the straight line passing through the distinct points 𝐴 and 𝐡. It is a set containing all of the points that lie on this straight line that extends infinitely in both directions. Since straight lines extend indefinitely in both directions, we cannot measure their length.

𝐴𝐡 is the ray passing through the distinct points 𝐴 and 𝐡, which ends at 𝐴. It is a set containing all of the points that lie on this ray that extends infinitely in the direction from 𝐴 to 𝐡. Since rays extend indefinitely in one direction, we cannot measure their length.

Since each of these objects are sets, we can use set notation to describe points that lie on lines, line segments, and rays. For example, consider the following diagram.

We can see that point 𝐢 lies on the ray from 𝐴 to 𝐡. We can write this as πΆβˆˆοƒ«π΄π΅. We can also note that π΄π΅βŠ‚οƒ«π΄π΅βŠ‚βƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΄π΅ since every point on the line segment is also on both the ray and line, and every point on the ray also lies on the line.

Let’s see some examples of representing and identifying the different forms of lines.

Example 1: Representing a Line given in a Figure Mathematically

How can the line shown in the figure be represented mathematically?

Answer

We first note that the line shown in the diagram has arrows on both sides. This tells us that the line extends indefinitely in both directions. We can also see that 𝐴 and 𝐡 are distinct points on this straight line.

We then recall that ⃖⃗𝐴𝐡 is the straight line passing through the distinct points 𝐴 and 𝐡 that extends indefinitely in both directions. Hence, the diagram shows the straight line ⃖⃗𝐴𝐡.

In our next example, we will match a diagram with a geometrical object’s notation.

Example 2: Identifying the Correct Diagram from the Object’s Notation

Which of the following is represented by 𝐴𝐡?

Answer

We begin by recalling that 𝐴𝐡 is the line segment with endpoints 𝐴 and 𝐡. It is a set containing all of the points that lie on this finite portion of a straight line. This means the diagram representing 𝐴𝐡 will be a portion of a straight line with endpoints 𝐴 and 𝐡. This is called a line segment.

We see that options A and E contain an arrow on one side of the line. This means that the line extends infinitely in this direction. So, they are examples of rays.

We see that options B and C contain arrows both sides of the line; this means that the line extends infinitely in both directions. Thus, they are examples of straight lines.

Finally, we see that option D is a finite line with endpoints 𝐴 and 𝐡. Hence, option D is represented by 𝐴𝐡.

In our next example, we will determine the correct notation for a geometrical object given graphically.

Example 3: Representing a Line given in a Figure Mathematically

How can the ray shown in the figure be represented mathematically?

Answer

We start by noting that the line shown in the diagram has an arrow on one side. This tells us that the line extends infinitely in the direction. We then recall that lines that extend infinitely in a single direction are called rays

In this diagram, we see that the starting point of the ray is the point 𝐴 and we are given a second point 𝐡 on the ray. We can say that the ray extends infinitely in the direction from 𝐴 to 𝐡 and has the starting point 𝐴. We write this as 𝐴𝐡.

We can combine line segments together in many ways, for example to make a shape called a polygon.

This is the triangle 𝐴𝐡𝐢. We can see that this triangle is different to the following triangle since they have different side lengths.

Another way of noting they are different is to consider how far apart the sides are at each vertex; these are called angles.

More formally, we call the rotation it takes to rotate one side onto another side the measure of the angle at this vertex. In the triangles above, we can represent this rotation at vertex 𝐡 with the following arrows.

We can see that the rotation required to rotate 𝐴𝐡 onto 𝐡𝐢 is smaller in the first triangle than in the second triangle. So, we can say that the measure of the angle at 𝐡 is smaller in the first triangle than the second.

There are a few important distinctions to make.

First, we differentiate between the angle and the measure of an angle in the same way we differentiate between a side and a side length. For example, consider the following square 𝐴𝐡𝐢𝐷.

We can recall that all of the sides of a square have the same length; however, we cannot say that 𝐴𝐡 and 𝐢𝐷 are the same since they are sets containing different points. In the same way, we can note that rotating one side onto another in a square is 14 of a full rotation. So, all of the angles have the same measure. However, we do not say that the angles themselves are equal; only their measures are equal.

Second, there are actually two angles between any two line segments sharing an endpoint. For example, consider the angles in the first triangle at vertex 𝐡.

Since we can rotate the sides either clockwise or counterclockwise, we actually have two angles at each vertex. If we do not specify, then we are talking about the angle with the smaller measure.

Third, both the triangle 𝐴𝐡𝐢 and the angles of the triangle are defined by three points, so we require additional notation to distinguish between these objects. We use the notation △𝐴𝐡𝐢 for the triangle with vertices at 𝐴,𝐡, and 𝐢, and the notations ∠𝐴𝐡𝐢 or 𝐴𝐡𝐢 for the angle between 𝐴𝐡 and 𝐡𝐢.

We can define angles and their notations formally as follows.

Definition: Angles

A union of two rays 𝐡𝐴 and οƒͺ𝐡𝐢 sharing the starting point 𝐡 and a rotation that takes the ray 𝐡𝐴 onto the ray οƒͺ𝐡𝐢 are called an angle at 𝐡. This can be written ∠𝐴𝐡𝐢 or 𝐴𝐡𝐢. The common point between the two rays is called the vertex of the angle, and the two rays are called the sides of the angle. We can also say that the angle at 𝐡 is the included angle between 𝐴𝐡 and 𝐡𝐢.

Since there are two rotations that take 𝐡𝐴 onto οƒͺ𝐡𝐢, the larger rotation is called the reflex angle, and, if it is not specified, then we mean the angle with the smaller rotation.

We can similarly define the measure of an angle, but, to do this, we first need to decide on units for the rotation and a baseline value for these units. The units of the measure of an angle are called degrees, written as π‘₯∘, and we describe the measure of a full rotation as 360∘.

This is actually the larger of the two angles at 𝐡, so we say that the reflex angle has measure 360∘. Since 𝐡𝐴 and οƒͺ𝐡𝐢 are coincident, no rotation is needed to rotate one onto the other, so we can also say that the measure of this angle is 0∘, written as π‘šβˆ π΄π΅πΆ=0∘. This is called a zero angle.

We can split these rotations into equal angles. For example, we can think of half of a full rotation where we note that 12Γ—360=180∘∘.

A rotation of half a turn clockwise or counterclockwise will take 𝐡𝐴 onto οƒͺ𝐡𝐢, so we can say that π‘šβˆ π΄π΅πΆ=180∘. This is called a straight angle.

Similarly, we can think about quarter turns. In this case, we see that 14Γ—360=90∘∘, so the measure of a quarter turn will be 90∘. We call this a right angle, and, since it is the angle in a square, we often use a square turn in diagrams to represent angles of this measure as shown.

We call any angle with a measure

  • smaller than 90∘ an acute angle,
  • between 90∘ and 180∘ an obtuse angle,
  • above 180∘ a reflex angle.

We can summarize this information as follows.

Definition: Measure of an Angle

The measure of an angle is the size of the rotation to take one edge of the angle onto the other. It is measured in degrees, and has the notation π‘šβˆ π΄π΅πΆ.

We can describe angles by their measures as follows:

  • A full turn has a measure of 360∘.
  • A half turn is called a straight angle, and it has a measure of 180∘.
  • A quarter turn is called a right angle, and it has a measure of 90∘.
  • An angle with a measure smaller than 90∘ is called an acute angle.
  • An angle with a measure between 90∘ and 180∘ is called an obtuse angle.
  • An angle with a measure greater than 180∘ is called a reflex angle.
  • An angle with a measure of 0∘ is called a zero angle.

Let’s now see an example of determining the correct definition of an angle.

Example 4: Defining an Angle

Which of the following is the definition of an angle?

  1. It is the rotation of a point from one position to another around a line segment.
  2. It is the rotation of a ray from one position to another around the starting point.
  3. It is the measure of the distance between two points.
  4. It is a quantity that measures how many points two lines intersect at.
  5. It is the rotation of a line segment from one position to another around a ray.

Answer

We recall that an angle is defined to be the rotation of one ray (or line segment) onto another about a point called its vertex. This is option B.

We can note that, in general, we can only rotate around a point, so the option involving rotating around a ray does not make sense. Option C would be the length of a line segment and not an angle.

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

Example 5: Determining Which Angle is Acute

Which of the following is an acute angle?

Answer

We start by recalling that an acute angle is an angle with a measure less than 90∘. We can see that only option A has a measure less than 90∘ since it has a measure of 25∘.

We could stop here; however, it is worth noting that we can answer this question without knowing the exact measures of the angles. We recall that an angle with a measure of 90∘ is a quarter turn or a right angle.

For an angle to be acute, it must be a smaller angle than this, we can see that only option A has a smaller rotation than this, so it is the only acute angle.

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

Example 6: Finding the Measure of a Reflex Angle from a Diagram

Find the measure of reflex angle ∠𝐴𝐡𝐢.

Answer

To answer this question, we first recall that a full turn is defined to have a measure of 360∘. We can add a full turn onto the diagram as shown.

We see that the full turn is the sum of the reflex angle and the smaller given angle. Since the full turn has a measure of 360∘ and the acute angle at the vertex has a measure of 25∘, if we call the measure of the reflex angle π‘₯, we have 360=π‘₯+25.∘∘

We can subtract 25∘ from both sides of the equation to get π‘₯=360βˆ’25=335.∘∘∘

In the previous example, we showed a useful propertyβ€”the sum of both angles at the same vertex is equal to 360∘.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • 𝐴𝐡 is the line segment with endpoints 𝐴 and 𝐡. It is a set containing all of the points that lie on this finite straight line.
  • ⃖⃗𝐴𝐡 is the straight line passing through the distinct points 𝐴 and 𝐡. It is a set containing all of the points that lie on this straight line that extends infinitely in both directions.
  • 𝐴𝐡 is the straight line (called a ray) passing through the distinct points 𝐴 and 𝐡 that starts at 𝐴. It is a set containing all of the points that lie on this ray that extends infinitely in the direction from 𝐴 to 𝐡.
  • The union of two rays 𝐡𝐴 and οƒͺ𝐡𝐢 that share a starting point and the rotation that takes the ray 𝐡𝐴 onto the ray οƒͺ𝐡𝐢 is called the angle at 𝐡. It can be written ∠𝐴𝐡𝐢 or 𝐴𝐡𝐢. The common starting point between the two rays is called the vertex of the angle, and the two rays are called the sides of the angle. We can also say that the angle at 𝐡 is the included angle between 𝐴𝐡 and 𝐡𝐢. The larger angle is called the reflex angle.
  • The measure of an angle is the size of the rotation to take one edge of the angle onto the other; it is measured in degrees and has the notation π‘šβˆ π΄π΅πΆ.
  • A half turn is called a straight angle, and it has a measure of 180∘.
  • A quarter turn is called a right angle, and it has a measure of 90∘.
  • An angle with a measure smaller than 90∘ is called an acute angle.
  • An angle with a measure between 90∘ and 180∘ is called an obtuse angle.
  • An angle with a measure greater than 180∘ is called a reflex angle.
  • The sum of both angles at the same vertex is equal to 360∘.

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