Instruction 1-1

Pre-Test		Post-Test
Q&A	Radian Measurement	Q&A

Measuring Angles | Areas of Triangles | Summary

In geometry, we classified all straight lines in three groups: segments, rays, and lines. We studied all these figures in detail in the Geometry lessons.  We will use rays to define how lines become angles. We will then begin discussing ways to measure angles. This lesson covers the following topics:

• Rays
• The Measurement of an Angle
• Measuring Scales of Angles
• The Relationship between Radian and Degree Measures
• Angle of Elevation

Rays
Recall that a ray, or half-line, is a portion of a line that begins at a point (initial point) and extends indefinitely. In the ray shown below, A is the starting point, or the initial point and B is the extension point. The arrow at B indicates that the ray extends in that direction forever.

When two rays share one initial point, they form a figure that is called an angle.
For example, in the figure below, the rays AM and AN share the initial point A. The result is a figure that is called angle MAN:

An angle has two main parts: vertex and sides. The vertex of an angle is the point where the initial points of two rays overlap. Each ray is a side of an angle. For example, in the figure above, A is the vertex and AM and AN are the sides of the angle.

TIP: An angle is denoted by the label of its vertex or by the labels of the vertex and the endpoints of its sides. These labels usually follow the sign “v “. For example, the angle above can be denoted by vA, vMAN, or vNAM.

The Measure of an Angle

The measurement of an angle is the magnitude, or degree, of openness between its sides. Such a quantity is shown in the two different angles below:

The sides of both angles look similar. Remember, though, the length of each side has no effect on the measures of angles. That is, two equal angles can have different side lengths.

For example, these two angles are equal, although one side is longer than the other side.

Measuring Scales of Angles

figure 1.1

Divide the circumference of a circle into 360 equal arcs, like points A, B, C, and D in figure 1.1.

Join each point to the center point, labled O in figure 1.1. Each two consecutive points form an arc intercepted by a central angle. For example, the arc AB, BC, or CD is intercepted by the angle AOB, BOC, or COD, respectively. The measure of each of such angles is one degree. Thus, the one-degree angle measurement is of the circumference of any circle. Therefore, if the openness of an angle that is, for example,  32 times the openness of one of these angles, then its measure is 32 degrees [1/360 x 32/360 = 32/360]. The degree unit has partial units, as well. A one-degree angle is equal to 60 minutes. Each minute is equal to a 60-second angle measurement. That is, if the measurement of an angle is 360 minutes, then it is 8 degrees because .

Figure 1.2

The Relationship between Radian and Degree Measures

If we measure an angle in degrees by D and in radian by R. Then the following formula is true:

Practice 1. Convert 120˚ (˚ is the symbol for degree) to radian.

Solution. Replace D = 120 in the formula

A Real Life Application 1. Angle of Elevation

If we stand in front of a building or a tower, the imaginary line that connects our eyes to the top edge of it is called the line of sight. For example, in the figure below, the top point of the tower is A. The observer is located at point B. The imaginary line AB is the line of sight.

Any line of sight forms an angle with the horizontal plane. Such an angle is called angle of elevation. In the diagram above, ABC is the angle of elevation. The concept of the angle of elevation is used frequently in astronomy and geography.

Video Instruction
*Availability of You Tube video links may vary. eTAP has no control of these materials.

• Angles Definition, Supplementary Angles and Complementary Angles [9:55]

• Radians and Degrees [9:55]

• Unit Circle [9:07]

• Measuring With a Protractor [5:59]

for Students, Parents and Teachers

Now let's do Practice Exercise 1-1 (top).

Next Page: Areas of Triangles (top)