1. Pearson Unit 1 Topic 3: Parallel and Perpendicular Lines 3-9: Comparing Spherical and Euclidean Geometry Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(4)(D) Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.
(1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

3. Euclidean geometry is based on Euclid’s postulates
Euclidean geometry is based on Euclid’s postulates. It is the geometry of flat planes, straight lines, and points.
Spherical geometry is a non-Euclidean geometry in which a plane is defined as the surface of a sphere.

4. The undefined terms in Euclidean geometry are point, line, and plane.
The undefined terms in Spherical geometry are point, great circle, line, and line segment.

5. A great circle is the intersection of a sphere and a plane that contains the center of the sphere.
On a sphere the shortest path between two points is along a great circle.
A line in spherical geometry is a great circle.
Pilots usually fly along great
circles because a great circle
is the shortest route between
two points on Earth.

6. A point (in spherical geometry) has the same meaning as in Euclidean geometry. It is a location on the surface of a sphere.
A line segment (in spherical geometry) is an arc of a great circle. It is the shortest distance between two points.
A line segment in Euclidean geometry is the shortest path between two points.

10. The two points used to name a line cannot be exactly opposite each other on the sphere.
Caution!

12. Example: 1 lines segments triangles
Name a line, a segment, and a triangle on the sphere.
Possible Answers:
lines
segments
triangles

13. Example: 2
Classify each spherical triangle by its angle measures and by its side lengths.
ΔABC
ΔABC is an obtuse scalene triangle.

14. Example: 3
Classify each spherical triangle by its angle measures and by its side lengths.
ΔVWX
ΔVWX is an equiangular and
equilateral triangle.

15. Example: 4
Three angle measures of a triangle are given. Does the triangle exist in Euclidean geometry, spherical geometry, or neither?
A. 128, 85, 30
B. 68, 52, 35
C. 63, 63, 54
Answer: A. spherical geometry; sum > 180
Answer: B. neither geometry; sum < 180
Answer: A. Euclidean geometry; sum = 180

16. Example: 5
A. Which lines of longitude are lines in spherical geometry? Explain. Prime Meridian and International Date LIne
B. Which lines of latitude are lines in spherical geometry? Explain. Equator
C. How many lines perpendicular to the equator can you draw from the North Pole? Infinite
D. Explain why lines of latitude are not parallel lines in spherical geometry. There are no parallel lines in spherical geometry, and only the equator is a great circle.

17. Example: 6 Find the area of each spherical triangle.
Round to the nearest tenth, if necessary.
ΔABC
478.9 cm3.

18. EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW.
ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

19. Example: 7 A Find the area of each spherical triangle.
Round to the nearest tenth, if necessary.
Δ DEF on Earth’s surface with m∠D = 75°, m∠E = 80°, and m∠F = 30°.
(Hint: average radius of Earth = 3959 miles) A