- a) Give a list of steps with a carefully labeled diagram on how to use a compass and straightedge to find the bisector of a line segment on the plane. Your list of steps should include a proof that the construction process does in fact bisect the line segment. b) Does your method work equally well on the sphere? Explain why or why not.
- a) Give a list of steps with a carefully labeled diagram on how to use a compass and straightedge to find the bisector of any planar angle. Your list of steps should include a proof that the construction process does in fact bisect the angle. b) Does your method work equally well on the sphere? Explain why or why not.
- Given a small triangle and its three interior angles, derive the formula for the area of a small triangle. Include in your derivation the following: A brief explanation for how you thought about adding up the various areas to get the initial formula A brief explanation for how to conceptualize the derivation of the formula for the area of a lune Appropriate illustrations to help clarify your explanations and derivations
- Use your formula for the area of a small triangle on sphere to argue that the sum of the interior angles for a triangle on the sphere must be greater than 180 degrees and cannot equal 180.
- Your derivation in number 3 was for a small triangle. Does this formula also work for the “shark fin” triangle? How about the “anti-triangle”? If yes, show how it applies. If no, explain why not.
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