Angle

Understanding Check

Calculate the answer to the problem and write it down. Then click on the blue and yellow words to see the correct answer.

1. The degree measure of an angle that is 1/3 of a circle is: Click for answer.360 / 3 = 120
2. The degree measure of an angle that is 1/17 of a circle is: Click for answer.360 / 17 ≈ 21.17

One reason the measure of radians is so useful has to do with Euler's famous equation that relates exponentiation with trigonometry using complex numbers: e^iθ = cos(θ) + i·sin(θ). This equation only works if the angle θ is measured in radians.

Understanding Check

Calculate the answer to the problem and write it down. Then click on the blue and yellow words to see the correct answer.

1. The radian measure of an angle that is 1/3 of a circle is: Click for answer.2·π / 3 ≈ 2.09
2. The radian measure of an angle that is 1/17 of a circle is: Click for answer.2·π / 17 ≈ 0.37

For convenience in discussions of angles and trigonometry, angles are divided into classes.^{[3 page 10]} The class an angle belongs to is determined by its measure. Table 1 shows the classes of angles and their measures. Click on the blue point in manipulative 3 and drag it to change the figure. Manipulative 3 also shows the classes of angles and their measures.

Table 1: Classes of Angles

Ex.	Angle Measure		Class
	Degrees	Radians
	0 < θ < 90	0 < θ < π/2	Acute angle
	θ = 90	θ = π/2	Right angle
	90 < θ < 180	π/2 < θ < π	Obtuse angle
	θ = 180	θ = π	Straight angle
	180 < θ < 360	π < θ < 2·π	Reflex angle
	θ = 360	θ = 2·π	Full angle

Discovery

1. All inscribed angles are greater than what measure?
   Click for Answer All inscribed angles are greater than zero. Move the endpoints of the legs of the angle close together, but away from the vertex. The closer the endpoints are to each other, they closer the angle is to zero.
2. All inscribed angles are smaller than what measure?
   Click for Answer All inscribed angles are smaller than a straight angle. Move the endpoints of the legs of the angle close to each side of the vertex, but on opposite sides of the vertex. The closer the endpoints are to the vertex, they closer the angle is to a straight angle.
3. How does the measure of the inscribed angle change when only the vertex is moved without moving across one of the endpoints?
   Click for Answer. If the endpoints of the inscribed angle are left alone and the vertex of the inscribed angle is moved, the measure of the angle does not change.
4. How does the measure of the inscribed angle change when the vertex is moved across one of the endpoints?
   Click for Answer. If the vertex is moved across one of the endpoints, the previous angle and the new angle are complementary. The angles add up to a straight angle.
5. If an angle is inscribed in the diameter of a circle, what is the measure of the inscribed angle?
   Click for Answer. The measure of an angle inscribed in the diameter of a circle is a right angle.

A central angle of a circle is an angle with the vertex at the center of the circle and the other two points on the circumference of the circle.^{[3 page 41]} Click on the blue points in manipulative 6 and drag them to change the figure.

For an inscribed angle and a central angle with the same endpoints, the measure of the inscribed angle is half the measure of the central angle. Click on the blue points in manipulative 7 and drag them to change the figure.

An angle of rotation is the amount of rotation about a center of rotation. Click on the blue points in manipulative 8 and drag them to change the figure.

Supplementary Angles

The angle addition postulate states that adjacent angles can be added together to form a larger angle. This is a postulate or axiom, meaning it is accepted as true without proof. The exact mathematical definition of the Angle Addition Postulate is:

Given noncollinear points A, B, C and a point D in the interior of ∠BAC, m∠BAD + m∠DAC = m∠BAC.