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Display several responses where angle subtends the major arc. Ask students, “Is this relationship still true when the point is on the arc ?” To guide the discussion, label the measure of angle as and consider asking these questions:
• “If the measure of angle is degrees, what would be the measure of the central angle that traces out the entire circle except for arc ?” ( degrees.)
• “Compare that larger angle to the measure of angle . Does the inscribed angle relationship hold?” (Yes, the inscribed angle measures half of this larger angle.)
6.4 Proving the Inscribed Angle Theorem
Optional: 15 minutes
This activity is optional because it goes beyond the depth of understanding required to address the standards.
In this activity, students prove one case of the Inscribed Angle Theorem and the other cases are left as an extension. The task is presented in terms of radians to build familiarity with describing angle relationships in terms of radians.
Student Task Statement
Here is a special case of an inscribed angle where one of the chords that de nes the inscribed angle goes through the center.
The central angle measures radians, and the inscribed angle measures radians. Prove that .
Student Response
Because and are both radii of the same circle, they have the same length and triangle
is isosceles. Therefore angle must have the same measure as angle ,
radians. Because the must be radians. It
and thus .
angles in a triangle sum to or follows that its supplementary angle
radians, the measure of angle must be radians. So
Unit 7 Lesson 6: The Angle Inscribed
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