Experiments with Inscribed Angles

New York State Common Core Math Geometry, Module 5, Lesson 4

Worksheets for Geometry, Module 5, Lesson 4

Student Outcomes

• Explore the relationship between inscribed angles and central angles and their intercepted arcs.

Experiments with Inscribed Angles

Classwork

Opening Exercise

ARC:
MINOR AND MAJOR ARC:
INSCRIBED ANGLE:
CENTRAL ANGLE:
INTERCEPTED ARC OF AN ANGLE:

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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.

• Draw two points no more than 3 inches apart in the middle of the plain white paper, and label them 𝐴 and 𝐵.
• Use the acute angle of your colored trapezoid to plot a point on the white sheet by placing the colored cutout so that the points 𝐴 and 𝐵 are on the edges of the acute angle and then plotting the position of the vertex of the angle. Label that vertex 𝐶.
• Repeat several times. Name the points 𝐷, 𝐸, ….

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?

Lesson Summary

All inscribed angles from the same intercepted arc have the same measure.

Relevant Vocabulary

• ARC: An arc is a portion of the circumference of a circle.
• MINOR AND MAJOR ARC: Let 𝐶 be a circle with center 𝑂, and let 𝐴 and 𝐵 be different points that lie on 𝐶 but are not the endpoints of the same diameter. The minor arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that are in the interior of ∠𝐴𝑂𝐵. The major arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that lie in the exterior of ∠𝐴𝑂𝐵.
• INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point.
• CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.
• INTERCEPTED ARC OF AN ANGLE: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc.

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