IM Geo.7.2 Lesson: Inscribed Angles

What do you notice? What do you wonder?
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[/img]
Use the applet to answer the questions. Do not show the angle measures until you are told to.
Name the central angle in this figure.[br]
Name the [b]inscribed angle [/b]in this figure.[br]
Move point [math]B[/math] around the circle. As you move this point, what happens to the measure of angle [math]QBC[/math]? Show the angle measures to confirm.[br]
Move points [math]C,Q,[/math] and [math]B[/math] to new positions. Record the measure of angles [math]QAC[/math] and [math]QBC[/math]. Repeat this several times.[br]
Make a conjecture about the relationship between an inscribed angle and the central angle that defines the same arc.[br]
Here is a special case of an inscribed angle where one of the chords that defines the inscribed angle goes through the center.
The central angle [math]DCF[/math] measures [math]\theta[/math] degrees, and the inscribed angle [math]DEF[/math] measures [math]\alpha[/math] degrees. Prove that [math]\alpha=\frac{1}{2}\theta[/math].[br]
The image shows a circle with chords CD, CB, ED, and EB. The highlighted arc from point C to point E measures 100 degrees. The highlighted arc from point D to point B measures 140 degrees.
[size=150]Prove that triangles [math]CFD[/math] and [math]EFB[/math] are similar.[/size][br]
Close

Information: IM Geo.7.2 Lesson: Inscribed Angles