2. (18) (Hyperbolic) Use the Poincare Disk Model for Hyperbolic Geometry to demonstrate i) a triangle with angle sum less than 180^o and ii) two different lines parallel to a given line.

3. (18) (Euclidean) Prove that any tangent to a circle is perpendicular to the diameter at the point of tangency..

4. (18) (Euclidean) Prove that the measure of an angle inscribed in a circle is equal to half of the subtended arc.

5. (18) (Absolute) Use the Exterior Angle Theorem to prove that sum of two angles of a triangle is less than 180^o.

6. (4) (Spherical) Define great circle and small circle.

7. (15) (Spherical) Our Earth has a radius of 4000 miles. The plane that contains the 45th North parallel is at a distance of 2828 miles = (4000/square root of 2) from the Earth's center. (5 percent extra credit to verify this...)

i) What is the radius and circumference of the 45th North parallel (it's a small circle)?

ii) If you travelled from a point on the 45th parallel along that parallel to a point 180^o away, you would need to travel how many miles? For example, you might travel from Minneapolis, MN to a point in the Altai Mts. of Mongolia -- I don't know why you'd want to but... (The answer to this is easy -- look at answer to i...)

iii) How many miles is it between the two points mentioned in ii) along a "great circle" route? (Hint: first find the great circle that contains both of these points.)