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Exploring Inscribed Angles in Circles

Author:Colleen Shallenberger

Part 1: Explore Inscribed Angles

Recall that a CENTRAL ANGLE is an angle whose vertex is at the center of a circle. The measure of the central angle is equal to the measure of the arc that it intercepts. In the circle shown, angle CAD intercepts arc CD. Note that if point C is moved, the angle measure and the arc measure are always equal.
An INSCRIBED ANGLE is an angle whose vertex is ON the CIRCLE. In the diagram below, angle CED is an inscribed angle because the vertex is on the circle. Notice that angle CED intercepts arc CD! Answer the next question about angle CED before clicking the box to show its measure.

WITHOUT clicking to show the measure of CED, which statement do you think is true?

Select all that apply
  • A
  • B
  • C
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Now, click to show the measure of angle CED. Move point E to see what changes. Move point C to see what changes.

Describe the relationship between an inscribed angle and a central angle, IF they intercept the same arc.

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In the graph below, angle CFD is another inscribed angle that intercepts the same arc (arc CD). Which statement do you think is true? Choose your answer, then click to show the measure.

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

Part 2: Inscribed angles formed by a tangent.

Look carefully at the graph shown. Notice that angle RST is formed by a tangent line and a line that is inside the circle (it's called a chord.) The vertex of angle RST is ON the circle, which means it is, by definition, an inscribed angle.

Do you think that this type of inscribed angle will have the same relationship? Will its measure be one-half the arc that it intercepts?

Select all that apply
  • A
  • B
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Part 4: Inscribed Triangles (a triangle is inscribed in a circle if all vertices are ON the circle.)

In the graph shown, move points E and/or D until angle C is a right angle. Then, answer the question below the graph.

Fill in the blank: When an inscribed triangle is a RIGHT triangle, the hypotenuse __________________

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Part 4: Inscribed Quadrilaterals

What is true about the angles in an inscribed quadrilateral?Recall that all angles of any quadrilateral add to 180 degrees. Click the boxes to show the angles in the graph. You can look at two angles at a time (this is a hint.)

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