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1.1: Angles and Basic Geometry

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Learning Objectives
  • Identify acute, obtuse, right, and straight angles.
  • Compute the complement and supplement of an angle.
  • Determine the number of degrees in a partial rotation.
  • Use vertical, alternate interior, or corresponding angles to solve for an angle.
  • Find the circumference and area of a circle.
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Angles

Before we can jump into Trigonometry, we must define angles; however, before defining angles, we need to define what makes up the components of an angle.

Definition: Ray

ray is a line segment that has one endpoint and extends infinitely in one direction. The point from which the ray originates is called the initial point of the ray. A ray can also be described as a half-line.

Figure 1.1.1: A ray with initial point P
1.1 Figure 1.png

Note that we label the initial point P. The choice of letter is arbitrary, so we could just have easily chosen Q, R, or S. The choice of using capital letters for points on a line or ray, however, is tradition and we stick with that convention in this text. We now move on to defining an angle.

Definition: Angle

When two rays share a common initial point, they form an angle, and the shared initial point is called the angle's vertex.

Figure 1.1.2 shows two examples of what are commonly considered as angles.

Figure 1.1.2: An angle with vertex P (left) and an angle with vertex Q (right)
1.1 Figure 2.png

Figure 1.1.3 also depicts angles - albeit these are, in some sense, extreme cases. In the first case, the two rays are directly opposite each other, forming what is known as a straight angle; in the second, the rays are identical, so the "angle" is indistinguishable from the ray itself.

Figure 1.1.3A straight angle (left) and a "closed" angle (right)
1.1 Figure 3.png
Definition: Straight Angle

A straight angle is defined as an angle that is formed when the two rays forming the angle extend in opposite directions.

Now that we have an idea of what makes an angle, and a "beginner's notation," it's time we define the measure of an angle.

Definition: Angle Measure

The measure of an angle is a number that indicates the amount of rotation that separates the rays of the angle.

There is one immediate problem with this definition, as should be evident from Figure 1.1.4 below.

Figure 1.1.4: The ambiguity of angles
Screen Shot 2022-05-06 at 11.04.41 PM.png

Which amount of rotation are we attempting to quantify?

We have just discovered that this diagram describes at least two angles. Clearly, these two angles have different measures because one appears to represent a larger rotation than the other, so we must label them differently.

In Trigonometry, we often use lowercase Greek letters to represent the measure of the angle. For example, we use the Greek letters α (alpha) and β (beta) as representations of the unknown angle measures in Figure 1.1.5.

Figure 1.1.5
Screen Shot 2022-05-06 at 11.07.51 PM.png

Identifying the lowercase Greek letters will be beneficial in this course. Table 1.1.1 summarizes the (lowercase) Greek alphabet. The letters in bold are the ones most commonly used within Trigonometry.

Table 1.1.1: Lowercase Letters in the Greek Alphabet

 Greek Alphabet α alpha β beta γ gamma δ delta ϵ epsilon ζ zeta η eta θ theta ι iota κ kappa λ lambda μ mu ν nu ξ xi o omicron π pi ρ rho σ sigma τ tau v upsilon ϕ phi χ chi ψ psi ω omega 

Caution

The lowercase Greek letter π is never used as a symbolic representation of an unknown angle. π is reserved for the universal constant π3.141592654 (which is an irrational number).

Another ambiguity that sneaks into angular measure is which direction the angle is "opening." We need to extend our notion of "angle" from merely measuring an extent of rotation to quantities associated with real numbers. To that end, we introduce the concept of an oriented angle.

Definition: Oriented Angle

An oriented angle refers to the measure of rotation between two rays with a common vertex, taking into account the direction of rotation (clockwise or counterclockwise). The angle is swept out starting from an initial side and ending at a terminal side. When the rotation is counterclockwise from the initial side to the terminal side, we say that the angle is positive; when the rotation is clockwise, we say that the angle is negative.

As its name suggests, in an oriented angle, the direction of the rotation is essential.

Figure 1.1.6: A positive angle (left) and a negative angle (right)
1.1 Figure 6.png

Alternative Angle Notations

We have already mentioned that we will frequently use lowercase Greek letters when referencing angles; however, this is not the only convention that exists when referencing an angle. Angles may be labeled with a single letter at the vertex as long as it is clear that there is only one such angle. For example, in Figure 1.1.7 (below), it is clear that angle A is the vertex. There is no ambiguity about this fact. Therefore, an acceptable notation to reference this angle is A (some texts might use the notation mA, meaning "the measure of angle A").

Figure 1.1.7
1.1 Figure 7a.png

However, if there is the possibility for confusion, angles are labeled by specifying 3 points, with the center point being the angle's vertex. Therefore, we could reference the angle in Figure 1.1.7 as BAC. The first letter indicates the initial side of the angle, the middle is the vertex, and the last letter indicates the terminal side. It is crucial that the center letter in this notation is the angle's vertex.

Common Angle Notations

The common notations used for angles are as follows:

  • Lowercase Greek letters (most commonly α, β, and θ)
  • A or mA, where A is the vertex of the rays where the angle is formed
  • BAC, where A is the vertex where the rays emanating from A and going through B and C meet.

Degree Measure

One commonly used system to measure angles is degree measure. Quantities measured in degrees are denoted by the familiar "" symbol.

Definition: Degree Measure

We define degree measure by dividing the full rotation around the circle into 360 segments. The angular measure of one of these segments is defined to be one degree, denoted 1. That is, 1 represents the measure of an angle which constitutes 1360 of a revolution.

Figure 1.1.8 shows the degree measures for a full, half, and quarter revolution. Since we define a full revolution as being 360, half of a revolution measures 12(360)=180, a quarter of a revolution measures 14(360)=90, and so on.

Figure 1.1.8: A full revolution (left), one-half a revolution (middle), and one-quarter of a revolution (right)
Screen Shot 2022-05-06 at 11.17.04 PM.png

The following theorem is a direct result of Figure 8.

Theorem: Degree Measure of a Straight Angle

The degree measure of a straight angle is 180.

In Figure 1.1.8, we have used the small square "" to denote a very special angle - the right angle.

Definition: Right Angle

right angle is defined to be the angle measure of a quarter-circle, which is 90.

The right angle is a very special angle in Trigonometry. As such, we use it as a sort of "measuring stick" against which we judge other angles. One such judgement is whether the measure of an angle is smaller or larger than the measure of a right angle.

Definition: Acute Angle

An angle that measures strictly between 0 and 90 is called an acute angle.

Definition: Obtuse Angle

An angle that measures strictly between 90 and 180 is called an obtuse angle.

It is important to note that, theoretically, we can know the measure of any angle as long as we know the proportion it represents of the entire revolution. For instance, the measure of an angle which represents a rotation of 23 of a revolution would measure 23(360)=240, the measure of an angle which constitutes only 112 of a revolution measures 112(360)=30, and an angle which indicates no rotation at all is measured as 0 (see Figure 1.1.9).

Figure 1.1.9: An obtuse angle (left), an acute angle (middle), and a "closed" angle (right)
Screen Shot 2022-05-06 at 11.22.35 PM.png

We now investigate the relationship between two angles whose measures add to 90 or 180.

Definition: Complementary Angles

Two acute angles are called complementary angles if their measures add to 90.

Definition: Supplementary Angles

Two angles, either a pair of right angles or one acute angle and one obtuse angle, are called supplementary angles if their measures add to 180.

In the diagram below, the angles α and β are supplementary while the pair γ and θ are complementary angles.

Figure 1.1.10: Supplementary angles, α and β (left) and complementary angles, γ and θ (right)
1.1 Figure 10.png
Caution

In this text, we will restrict the definition of complementary and supplementary angles to positive angles only. That is, despite their sum being 90, we will not say that α=20 and β=110 are complementary. Other textbooks allow for this silliness, but in truth there is no need to expand these definitions to include negative angles.

Finally, in practice, the distinction between the angle itself and its measure is blurred so that the sentence "α is an angle measuring 42" is often abbreviated as "α=42."

Example 1.1.1

Consider Figure 1.1.11 below.

Figure 1.1.11
Screen Shot 2022-09-08 at 10.34.53 PM.png
  1. Which angle pairs are supplementary?
  2. Which angle pairs are complementary?
  3. Which angles are obtuse?
  4. Which angles are acute?
Solutions
  1. AOC and BOC are supplementary, AOD and BOD are supplementary, and AOE and BOE are supplementary.
  2. DOE and BOE are complementary.
  3. AOC, AOE, and DOC are each obtuse angles.
  4. BOC, BOE, and DOE are each acute angles.

It could be argued in Example 1.1.1d that COE is acute; however, there is no valid reference for the measure of COB and BOE - only a visual cue. Therefore, we avoid assuming that COE is acute.

Checkpoint 1.1.1
  1. State the complement of 27.
  2. State the supplement of 27.
  3. State the complement of α.
  4. State the supplement of θ.
Answer
  1. 63
  2. 153
  3. 90α
  4. 180θ

The measuring "system" we use to measure angles is formally called the Decimal Degree (DD) system.

Definition: Decimal Degree

An angle written as a decimal is said to belong to the decimal degree (DD) system of measuring angles.

Therefore, angles like 38, 23.1, and 123.456 belong to the decimal degree system. There are other angle measure systems:

  • Degree-Minute-Second (DMS): While we will eventually use this angular measurement system (specifically, in some applications), it is not necessary to introduce it at this time.
  • Radians: We will introduce this "better" angular measurement system later. Without this system, we could not perform Calculus.

Angles in Rotations

Now that we have some basic information about angles and have reintroduced ourselves to a measurement system for angles, it's time to introduce an application of angular measure. Specifically, we will learn to use angles to describe rotation. For example, think of the minute hand on a clock. The minute hand moves through one complete rotation every hour, or 360. In two hours, the minute hand rotates through 720.

Example 1.1.2

Through how many degrees does the minute hand rotate in an hour and a half? In forty minutes?

Solution
Figure 1.1.12
Screen Shot 2022-10-24 at 1.14.56 PM.png

Look at Figure 1.1.12. An hour and a half represents 1.5 complete rotations, or1.5(360)=540.Forty minutes is two-thirds of an hour, so the minute hand rotates through23(360)=240

Checkpoint 1.1.2

The volume control on an amplifier is a dial with ten settings, as shown below. Through how many degrees would you rotate the dial to increase the volume level from 0 to 7?

Figure 1.1.13
Screen Shot 2022-10-24 at 1.17.48 PM.png
Answer

252

Geometry Necessary for Trigonometry

Trigonometry requires a decent grasp of basic Geometry. As such, let's review some pertinent topics.

Definition: Vertical Angles

Non-adjacent angles formed by the intersection of two straight lines are called vertical angles.

For example, 1 and 3 in Figure 1.1.14 are vertical angles, as are the angles labeled 2 and 4.

Figure 1.1.14
vertical-angles-theorem-1621933197.png

Before introducing a theorem relating vertical angles, we need to define what it means for angles to be equal.

Definition: Equality of Angles

Two angles are defined to be equal if their measures are equal.

Having clarified this, we introduce a theorem that is very useful for deriving other concepts from Geometry.

Theorem: Congruency of Vertical Angles

Vertical angles are equal (also known in Geometry as congruent).

Proof

Suppose two lines intersect, as shown in the figure below.

vertical-angles-theorem-1621933197.png

Since 1+2 forms a straight angle,1+2=180.By a similar argument,1+4=180.Therefore,1+2=180=1+4.That is,2=4.By a similar set of arguments, we can show that 1=3.Hence, it must always be the case that vertical angles are congruent.

A line intersecting two parallel lines forms eight angles, as shown in Figure 1.1.15.

Figure 1.1.15: Two parallel lines "cut" by a line
Screen Shot 2022-09-08 at 10.45.55 PM.png

The line cutting through the parallel lines in Figure 1.1.15 gets a special name.

Definition: Transversal

A line intersecting two parallel lines is called a transversal.

Figure 1.1.15 shows that there are four pairs of vertical angles formed by a transversal - 1 and 4, 2 and 3, 5 and 8, and finally 6 and 7; however, there are other relationships that can be seen in Figure 1.1.15.

Definition: Corresponding Angles

Angles in the same position relative to the transversal on each parallel line are called corresponding angles.

For example, 1 and 5 are corresponding angles, as are 4 and 8. This naturally leads to the following theorem.

Theorem: Congruency of Corresponding Angles

Corresponding angles are congruent.

Our final definition concerning the transversals is one of the most popular from Geometry.

Definition: Alternate Interior Angles

Angles that lie inside two parallel lines and on opposite sides of a transversal line that intersects those parallel lines are called alternate interior angles.

In Figure 1.1.15, 3 and 6 are alternate interior angles, as are 4 and 5.

Theorem: Alternate Interior Angles

If a transversal intersects parallel lines, the alternate interior angles are congruent.

Proof

Suppose a transversal cuts two parallel lines as shown below.

Screen Shot 2022-09-08 at 10.45.55 PM.png

We know that the corresponding and vertical angles are congruent. Therefore,1=5and1=4.Hence,5=4.Since our choice of angles was arbitrary, this shows that alternate interior angles are always congruent.

Example 1.1.3

The parallelogram ABCD shown below is formed by the intersection of two sets of parallel lines. Show that the opposite angles of the parallelogram are equal.

Figure 1.1.16
Screen Shot 2022-09-08 at 10.47.36 PM.png
Solution

Angles 1 and 2 are equal because they are alternate interior angles, and angles 2 and 3 are equal because they are corresponding angles. Therefore, angles 1 and 3, the opposite angles of the parallelogram, are equal. A similar argument shows that angles 4, 5, and 6 are equal.

Checkpoint 1.1.3

In the figure from Example 1.1.3, suppose 1=83. Find each of the following angular measures.

  1. 2
  2. 5
  3. 4
  4. 3+6
Answer
  1. 83
  2. 97
  3. 97
  4. 180

Circles

We end this section with one of the defining geometric structures in all of Mathematics - the circle.

Definition: Circle

The shape created from all points in a plane that are equidistant from a given point is called a circle. The given point is called the center of the circle, and the distance from the center to any point along the circle is called the radius of the circle.

Figure 1.1.17: A circle with center O and radius r
1.1 Circle Fixed.png

Figure 1.1.17 shows a circle of radius r centered at a point O. In fact, the word "radius" plays two roles in Geometry. By definition, the radius is the distance between the center of a circle and its edge; however, any line segment joining the center of a circle with any single point on the circle itself is called a radius. Therefore, the word radius simultaneously refers to a distance and a line segment.

Definition: Radius (as a line segment)

A line segment with one endpoint at the center of a circle and the other on the circle is called a radius of the circle.

Just like the radius, the diameter of a circle, denoted as d in Figure 1.1.17, has two interpretations. Our first is in terms of distance.

Definition: Diameter (as a distance)

The diameter of a circle is the length of any line segment going through the center of a circle and whose endpoints are on the circle.

This definition naturally leads to the following theorem.

Theorem: Diameter in Terms of Radius

Given a circle of radius r, the diameter, d, of the circle satisfiesd=2r.

As with the radius of a circle, the second interpretation of the diameter is in terms of a line segment.

Definition: Diameter (as a line segment)

A line segment going through the center of a circle and whose endpoints are on the circle is called a diameter of the circle.

The terminology from Geometry related to circles extends beyond center, radius, and diameter (e.g., arc, chord, sector, and tangent); however, we will introduce these only as needed.

Circumference

Now that we have the basic definition of a circle, let's dive into some measurements.

Definition: Circumference

The circumference is the distance around a circle.

In Figure 1.1.17, the circumference is denoted using the letter C.

Theorem: Circumference

A circle's circumference, C, is proportional to its radius, r. That is,C=kr.The constant of proportionality is 2π. Hence,C=2πr.

A proof of this fundamental theorem in Mathematics is beyond the scope of this course.

In the theorem presented for the circumference, it would be more impactful to state the circumference is proportional to the diameter. That is,C= some number ×d.It turns out that the value of "some number" (which is commonly called the constant of proportionality) is the universal constant π. That is,C=πdπ=Cd.This fascinating fact tells us that no matter the size of the circle, the ratio of its circumference to its diameter is always this magical number that we call π.

Language Note: "Circumference of a Circle" is Redundant

It is common to hear someone say, "the circumference of a circle." However, the word circumference only applies to circles. Therefore, saying "the circumference of a circle" is redundant. For non-circular two-dimensional shapes, we use the word perimeter instead of circumference.

Example 1.1.4

Loi buys a ring light for her YouTube videos. The light is circular and has a radius of 7 inches. What is the circumference of her ring light?

Solution
We are given the fact that the radius is 7 inches. Therefore, r=7 (inches). Using the formula for circumference, we find that Loi's ring light has a circumference ofC=2πr=2π(7 inches)=14π inches44 inches.
Checkpoint 1.1.4

Find the radius of a 100-meter circular race track. Round your answer to one decimal place.

Answer

r15.9 meters

Area of a Circle

The area of a circle is proportional to the square of its radius.

Theorem: Area of a Circle

The area of a circle of radius r is given byA=πr2.

 Again, the proof of this theorem is beyond the scope of this course.

Example 1.1.5

Phuong is making a circular tablecloth for her grandfather's antique oak table. If she wants the tablecloth to have a diameter of at least 7 feet (the bare minimum to cover the table), but no more than 9 feet (allowing a little bit of tablecloth to hang over the edge of the table), between what two values must the area of the tablecloth be? Round your answers to the nearest tenth of a square foot.

Solution
To barely cover the table, Phuong needs a tablecloth with radius 3.5 feet. Therefore, the minimum area of the tablecloth should beAminimum=πr2minimum=π(3.5)238.5 square feet.At most, if she wants a little bit of the tablecloth to hang over the edge, Phuong will want the radius to be 4.5 feet. This leads to an area ofAmaximum=πr2maximum=π(4.5)263.6 square feet.Any amount of circular tablecloth between 38.5 and 63.6 square feet should work nicely for Phuong's project.
Checkpoint 1.1.5

If Piyali has a circular tablecloth with area 35 square feet, what is the diameter of the tablecloth?

Answer

6.7 feet.


This page titled 1.1: Angles and Basic Geometry is shared under a CC BY-SA 12 license and was authored, remixed, and/or curated by Roy Simpson.

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