Geometry: Positive and Negative Angles and Arcs

Objectives

• Students will understand that if the intersection point P of two lines lies inside a circle, then the measure of the angle formed by the two secants is equal to the average of the measures of the arcs intercepted by that angle and its corresponding vertical angle.
• Students will learn that an oriented angle is defined to be the measure of rotation of a ray about a common point (the vertex).
• Students will learn that an oriented arc’s measure will be equal to the measure of its oriented central angle. The arc’s measure will be positive or negative depending on the angle’s orientation.
• Students will determine that if two lines intersect each other and also intersect a circle, then the measure of an angle of intersection of the two lines is equal to the average of the measures of the angle’s intercepted arcs.

Vocabulary

• Secant and chord
• Tangent line
• Central angle
• Intercepted arc
• Oriented angle
• Initial ray and terminal ray
• Reflex angle

About the Lesson

This lesson involves measuring oriented angles and applying these measurements to reflex angles and arcs. As a result students will:

• Observe the relationship between the measure of the angle of intersection of two lines and the measures of the intercepted arcs.
• Deduce that the angle of the intersection has a measure equal to the average of the measures of the intercepted arcs.