Lesson 10.8 Spherical Geometry pp. 451-453

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.

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.

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.

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

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

Homework pp. 452-453

►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

►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

►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

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

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

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

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

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

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

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

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

►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?

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

■ 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.

■ 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.

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

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