Experiments with Inscribed Angles
MINOR AND MAJOR ARC:
INTERCEPTED ARC OF AN ANGLE:
Exploratory Challenge 1
Your teacher will provide you with a straightedge, a sheet of colored paper in the shape of a trapezoid, and a sheet of plain white paper.
Exploratory Challenge 2
a. Draw several of the angles formed by connecting points 𝐴 and 𝐵 on your paper with any of the additional
points you marked as the acute angle was pushed through the points (𝐶, 𝐷, 𝐸, …). What do you notice about
the measures of these angles?
b. Draw several of the angles formed by connecting points 𝐴 and 𝐵 on your paper with any of the additional points you marked as the obtuse angle was pushed through the points from above. What do you notice about the measures of these angles?
Exploratory Challenge 3
a. Draw a point on the circle, and label it 𝐷. Create angle ∠𝐵𝐷𝐶.
b. ∠𝐵𝐷𝐶 is called an inscribed angle. Can you explain why?
c. Arc 𝐵𝐶 is called the intercepted arc. Can you explain why?
d. Carefully cut out the inscribed angle, and compare it to the angles of several of your neighbors.
e. What appears to be true about each of the angles you drew?
f. Draw another point on a second circle, and label it point 𝐸. Create ∠𝐵𝐸𝐶, and cut it out. Compare ∠𝐵𝐷𝐶 and ∠𝐵𝐸𝐶. What appears to be true about the two angles?
g. What conclusion may be drawn from this? Will all angles inscribed in the circle from these two points have the same measure?
h. Explain to your neighbor what you have just discovered.
Exploratory Challenge 4
a. In the circle below, draw the angle formed by connecting points 𝐵 and 𝐶 to the center of the circle.
b. Is ∠𝐵𝐴𝐶 an inscribed angle? Explain.
c. Is it appropriate to call this the central angle? Why or why not?
d. What is the intercepted arc?
e. Is the measure of ∠𝐵𝐴𝐶 the same as the measure of one of the inscribed angles in Exploratory Challenge 2?
All inscribed angles from the same intercepted arc have the same measure.
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