Angles Lesson

In this lesson, we will learn about angles. Let's begin by thinking about the definition of a line, a line segment, and a ray. Let's suppose we have two distinct points, which are called A and B. These points can be used to determine a line. When we speak of a line, the line continues indefinitely in both directions. In the picture above, we can see the line AB. Notice how the arrows indicate that our line continues in both directions forever. Next, we will discuss a line segment. If we think about the line AB and only take the portion between A and B, including those points, we have the line segment AB. Some books refer to this just as segment AB. Since this is just a piece of a line, the line segment has a finite or countable length. In the picture above, we can see the line segment AB. Notice how there are no arrows, just endpoints that indicate a fixed length. Lastly, let's discuss a ray. If we take our line AB and start at A, meaning A would be our endpoint, and then continue through the point B out indefinitely, we would form the ray AB. In the picture above, we can see the ray AB. Notice how we have one endpoint and one arrow. This tells us the ray extends indefinitely.

What is an Angle in Trigonometry?

In trigonometry, an angle is made up of either two rays with a common endpoint or two line segments with a common endpoint. The two rays or two line segments are known as the sides of the angle. The common endpoint is known as the vertex of the angle. The angle formed above can be named as angle BAC (∠ BAC), angle CAB (∠ CAB), or just angle A (∠ A) using the vertex letter. Associated with every angle is its measure, which is generated by a rotation about the vertex. This measure will be determined by rotating a ray starting at one side of the angle, which is known as the initial side, to the position of the other side, which is known as the terminal side. When we have a counterclockwise rotation, we obtain a positive measure. Additionally, when we have a clockwise rotation, we obtain a negative measure.

Degree Measure

When working with angles, we often measure our angle using the degree. To use degrees, we assign 360 degrees to one complete rotation of a ray. In other words, the terminal side of the angle will correspond to the initial side when it makes one complete rotation. One degree represents the fraction 1/360 of a rotation. This can be written using the degree symbol as: 1°. Similarly, 5 degrees represents 5/360 of a rotation and can be written as 5°. When we talk about the measurement of an angle, we will see the terms acute angle, right angle, obtuse angle, and straight angle.

Angle	Description
Acute	Between 0° and 90°
Right	Exactly 90°
Obtuse	Between 90° and 180°
Straight	Exactly 180°

In our pictures below, we will use the Greek letter θ to name each angle.
Acute Angle:
0° < θ < 90° Right Angle:
θ = 90° Obtuse Angle:
90° < θ < 180° Straight Angle:
θ = 180°

Complementary and Supplementary Angles

When the sum of the measures of two positive angles is 90°, the angles are known as complementary angles. When the sum of the measures of two positive angles is 180°, the angles are known as supplementary angles.
Let's look at an example.
Example #1: Find the measure of each angle. We know the two angles in our example form a right angle, therefore, they are complementary angles and sum to 90°. Let's set up an equation: $$6x + 12x=90$$ $$18x=90$$ $$x=5$$ Now, we can substitute and find the measure of each angle. $$6(5)=30$$ One angle is 30°. $$12(5)=60$$ The other angle is 60°.
Example #2: Find the measure of each angle. We know the two angles in our example form a straight angle, therefore, they are supplementary angles and sum to 180°. Let's set up an equation: $$11x + 7x=180$$ $$18x=180$$ $$x=10$$ Now, we can substitute and find the measure of each angle. $$11(10)=110$$ One angle is 110°. $$7(10)=70$$ The other angle is 70°.

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