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6.4 Proving Circle Conjectures, Algebra and Geometry 2 1
Part 1: Introduction
Talking Points:
Yesterday, we first discovered the Inscribed Angle Conjecture: The measure of an angle inscribed in a
circle equals half the measure of its intercepted arc. We then discovered four other conjectures related to
inscribed angles.
Today, we will prove that all four of these conjectures are logical consequences of the Inscribed Angle
Conjecture.1
Part 2: Exploration
First we must prove the Inscribed Angle Conjecture itself, but how? Let’s use our reasoning strategies to
make a plan. By thinking backward, we see that a central angle gives us something to compare an
inscribed angle with. If one side of the inscribed angle is a diameter, then we can form a central angle by
adding an auxiliary line. But what if the circle’s center is not on the inscribed angle? There are three
possible cases.2
 Case 1: The circle’s center is on the inscribed angle.
o This proof uses the variables x, y, and z to represent the measures of the angles as shown in
the diagram at right.
Given: Circle O with inscribed angle ABC on diameter BC
̂
Show: m ABC = ½ m𝐴𝐶
1
2
Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press: Emeryville, CA.
Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press: Emeryville, CA.
6.4 Proving Circle Conjectures, Algebra and Geometry 2 2
The proof of Case 1 allows us now to prove the other two cases. By adding an auxiliary line, we can use
the proof of Case 1 to show that the measures of the inscribed angles that do contain the diameter are
half those of their intercepted arcs. The proof of Case 2 also requires us to accept angle addition and arc
addition, or that the measures of adjacent angles and arcs on the same circle can be added.3
 Case 2: The circle’s center is outside the inscribed angle.
o This proof uses x, y, and z to represent the measures of the angles, and p and q to represent
the measures of the arcs, as shown in the diagram at right.
Given: Circle O with inscribed angle ABC on one side of diameter
BD
̂
Show: m ABC = ½m𝐴𝐶
3
Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press: Emeryville, CA.
6.4 Proving Circle Conjectures, Algebra and Geometry 2 3
 Case 3: The circle’s center is inside the inscribed angle.
o This proof uses x, y, and z to represent the measures of the angles, and p and q to represent
the measures of the arcs, as shown in the diagram at right.
Given: Circle O with inscribed angle ABC whose sides lie on either side of diameter BD
̂
Show: m ABC = ½ m𝐴𝐶
6.4 Proving Circle Conjectures, Algebra and Geometry 2 4
Part 3: Conclusion
Complete each flowchart proof.
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2.
6.4 Proving Circle Conjectures, Algebra and Geometry 2 5
Name:______________
Part 4: Homework
Complete each flowchart proof.
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2.
Date:
Bell:
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