Lesson 9.4 Proving Circle Conjectures
Case 1: As a group, go through the flowchart proof of Case 1, one box at a time.
What does each statement mean? How does it relate to the given diagram? How
does the reason below the box support the statement? How do the arrows connect
the flow of ideas? Discuss the logic of the proof with your group members. 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 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.
Case 2: The circle’s center is outside the inscribed angle.
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
1
̂
Show: m∠ABC = 𝑚𝐴𝐶
2
Flowchart Proof of Case 2
As a group, go through the flowchart proof of Case 2 and fill in the missing
reasons. Then work together to create a flowchart proof for Case 3, similar to the
proof of Case 2.
Case 3: The circle’s center is inside the inscribed
angle. 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
1
̂
Show: m∠ABC = 𝑚𝐴𝐶
2
Once we have proved all three cases, we have proved the Inscribed Angle
Conjecture. You can now accept it as true to write proofs of other conjectures in
the exercises.