One result of the Parallel Postulate in Euclidean geometry is the Triangle Angle-Sum Theorem. The spherical geometry Parallel Postulate gives a very different result.

Activity 2

Hold a string taut between any two points on a sphere. The string forms a “segment” that is part of a great circle. Connect three such segments to form a triangle on the sphere.

Below are examples of triangles on a sphere.

4. What is the sum of the angle measures in the first triangle? The second triangle? The third triangle?

5. How are these results different from the Triangle Angle-Sum Theorem in Euclidean geometry? Explain.

Exercises

For Exercises 6 and 7, draw a sketch to illustrate each property of spherical geometry. Explain how each property compares to what is true in Euclidean geometry.

6. There are pairs of points on a sphere through which you can draw more than one line.
7. Two equiangular triangles can have different angle measures.
8. For each of the following properties of Euclidean geometry, draw a counterexample to show that the property is not true in spherical geometry.
   1. Two lines that are perpendicular to the same line do not intersect.
   2. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
9. 1. The figure below appears to show parallel lines on a sphere. Explain why this is not the case.
   2. Explain why a piece of the top circle in the figure is not a line segment. (Hint: What must be true of line segments in spherical geometry?)

   

10. In Euclidean geometry, vertical angles are congruent. Does this seem to be true in spherical geometry? Explain. Make figures on a globe, ball, or balloon to support your answer.

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