Sphere Presentations - One Problem Per Person
• Briefly review related content that applies to the geometry of the plane / Euclidean geometry / flat geometry.
• Discuss why the general topic is important in mathematics and the real-world.
• Summarize diverse perspectives of your sphere problem in your own words.

Problem 1 In Euclidean geometry, through a given point, only one line can be drawn parallel to a given line. Is this true on the sphere?

Problem 2 Do right triangles on the sphere satisfy the Pythagorean Theorem?

Problem 3 Can we construct a square on a sphere? Explain.

Problem 4 Can we construct every convex polyhedron on a sphere, like a soccer ball [a spherical version of a truncated icosahedron]? Are there spherical polyhedra that have no flat equivalents?

Problem 5 Is SAS (side-angle-side) always true for spherical triangles on the surface of a perfectly round beach ball? Explain.

### Suggestions

For part 2 of your presentation, searches like
"parallels are important"
or a search word related to your problem and then (with and without quotations) words like important, interesting, useful, or real-life applications. You might also pick some specific fields to see if there are applications:
Pythagorean theorem chemistry

For part 3 of your presentation, you do not need to prove an answer or completely resolve the issue on the sphere. You should look for various perspectives related to spherical geometry, and summarize those in your own words. Try different combinations of search terms related to your problem along with words like sphere, spherical, earth, spherical geometry, or double elliptic geometry. Vary your word combinations:
Spherical Polyhedron
Polyhedra on a sphere
yield very different results, and quotations can be helpful if there are too many results