Arc and Angle Measure

Topic:
Geometry
In this activity you will discover relationships in circles that involve arc and angle measures.

Inscribed Angles

Activity Directions: Circumscribe circles about each of the given triangles to create inscribed angles.
Inscribed angles are formed by two chords. The vertex of an inscribed angle is on the circle. Activity Directions: Move points A, B, and C to discover the relationship between the inscribed angle and the arc that the angle intercepts.

1. What is the relationship between an inscribed angle and the arc that the inscribed angle intercepts?

Circumscribed Angles

Activity Directions: Inscribe circles inside each of the given triangles below to create circumscribed angles.
A circumscribed angle is outside of the circle and is formed by lines that are tangent to the circle. Lines that are tangent to a circle intersect the circle at exactly one point. Activity Directions: Move points A, B, and C below to discover the relationship between the measure of the circumscribed angle and the measures of the arcs that it intercepts.

2. What is the relationship between the circumscribed angle and the arcs that it intercepts?

3. What are the measures of AFD and AHD? Explain your reasoning.

Angles formed by Secant Lines

Secant lines intersect a circle at two points. Activity Directions: Move points D, C, B, and E to discover the relationship between the measure of the angle created by two secant lines and the measures of the intercepted arcs.

4. What is the relationship between the measure of the angle formed by two secant lines and the measures of the intercepted arcs?

Angles Formed by Intersecting Chords

Chords are segments that have both endpoints on a circle. Activity Directions: Move points D, C, B, and E to discover the relationship between the measure of the angles formed by intersecting chords and the measures of their intercepted arcs.

5. What is the relationship between the measures of angles formed by chords and the measures of their intercepted arcs?

Summarize

6. How do you find the angle measure of an angle inside of a circle if you know the measures of the intercepted arcs? 7. How do you find the angle measure of an angle outside of a circle if you know the measures of the intercepted arcs?