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Arc and Angle Measure!

Author:Emily_Gabriel1, Suzanne Gibbons, Guillermo Bautista
Topic:Geometry
In this activity you will discover relationships in circles that involve arc and angle measures.

How do you measure arcs?

Central angles have a vertex at the center of the circle. An intercepted arc is a portion of the circumference of a circle encased by two line segments meetings at a vertex. Arcs can be measured in degrees. Drag points A, B, and C. to determine the relationship between the degree of an arc and the central angle that creates the arc.

What is the relationship between the central angle and the intercepted arc?

Inscribed Angles

An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. In the applet below, BAC is an inscribed angle with an intercepted minor arc from C to B.
Move points A, B, and C to discover the relationship between the inscribed angle and the arc that the angle intercepts. Note: that the applet above and below are two different interpretations of inscribed angles, but both are correct :)

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

If the intercepted arc is 82 degrees, what is the measurement of the inscribed angle?

Select all that apply
  • A
  • B
  • C
Check my answer (3)
In the activity below, create a circumscribed circle about the given triangle. This will create three inscribed angles. (To circumscribe means to draw a figure around another figure in such a way that the drawn figure touches the inside figure without intersecting.) (Hint: select the circle through 3 points tool)

Find the interior angle measurements for the triangle. Determine the value of the intercepted arc for the three interior angles for the triangle. [Hint: use the angle measure tool]

Angles Formed by Two Intersecting Chords

Chords are segments that have both endpoints on a circle. When two chords intersect inside of a circle, four angles are formed. At the point of intersection, two sets of congruent vertical angles are formed.
Engage with the applet below and 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.

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

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

If the intercepted arcs are 32 degrees and 54 degrees, what is the angle measure of the vertical angles?

Angles Formed Outside of Circle by Intersection

Secant lines intersect a circle at two points . The intercepted arcs are major arc DE and minor arc CB.
Use the applet below and move points D,C,B, and E to discover the relationship between the measure of the angle created by two secant lines and the measure of the intercepted arcs (major and minor arcs).

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

If the major arc is 100 degrees and the minor arc is 30 degrees, what is the measure of the angle formed by the two secants?

Exit Ticket!

What is one thing you learned today?

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