1. Lesson 10.8
Spherical Geometry pp

2. Objectives:
1. To identify the concepts of spherical geometry as an example of non-Euclidean geometry.
2. To stress the importance of Scripture as the only absolute source of truth.

3. An example of non-Euclidean geometry is Riemannian geometry
An example of non-Euclidean geometry is Riemannian geometry. Riemannian geometry is also called spherical geometry because it is represented with a sphere.

4. In spherical geometry great circles can be thought of as lines and the spherical surface as the plane. Since all great circles intersect there are no parallel lines.

5. In spherical geometry, a circle can be considered a line if it has the same radius as the given sphere.

6. In spherical geometry, the sum of the measures of the angles of a triangle is never 180º.

7. Homework pp

8. ►A. Exercises
1. Imagine walking south along a longitudinal line from the North Pole and turning east at the equator. What angle did you turn? N
equator S

9. ►A. Exercises
2. Which of the following are “lines” in spherical geometry: the equator, tropic of Cancer, latitudinal circles, longitudinal circles, prime
meridian, tropic
of Capricorn,
Arctic Circle? N
equator S

10. ►A. Exercises
3. Minneapolis is at 45°N latitude. The circle of latitude at 45°N is parallel to the equator. Why do we say that there are no parallels in the
model of
Riemannian
geometry? N
equator S

11. ►B. Exercises
4. Does the “plane” of the earth contain at least three noncollinear points?

12. ►B. Exercises
5. Does every “line” contain at least two points?

13. ►B. Exercises
6. Are every pair of points on a “line”?

14. ►B. Exercises
7. Do two intersecting “lines” intersect in exactly one point?

15. ►B. Exercises
8. Do every pair of points determine
exactly one “line”?

16. ►B. Exercises
9. How does the sum of the measures of the angles in PST compare to 180°? T P S

17. ►B. Exercises
10. Sketch a triangle with three right angles (an equilateral right triangle). T P S

18. ►C. Exercises Answer the questions using Riemannian geometry.
11. Why are trapezoids and parallelograms impossible?

19. ►C. Exercises Answer the questions using Riemannian geometry.
13. How does the measure of an exterior angle of a triangle compare to the sum of the measures of the remote interior angles?

20. ■ Cumulative Review
14. A regular heptagon and a regular octagon are inscribed in congruent circles. Which polygonal region has more area?

21. ■ Cumulative Review
15. Is the following argument valid? Sound? What type of argument is it?
All lizards are reptiles.
All salamanders are lizards.
Therefore, all salamanders are reptiles.

22. ■ Cumulative Review
16. Sketch and label the altitude, perpendicular bisector, angle bisector, and median to a side in a triangle. They must be different lines and intersect the same side.

23. ■ Cumulative Review
17. Draw an illustration for the Angle Addition Postulate and explain it. One angle must be 47°.

24. ■ Cumulative Review
18. What definition acts as a Segment Addition Postulate?