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For each set of three angle measures of a triangle given, does the triangle exist in Euclid geometry, spherical geometry, or neither? 3. 79°, 45°, 45° 4. 33°, 87°, 60° 5. 100°, 100°, 100°

For each set of three angle measures of a triangle given, does the triangle exist in Euclid geometry, spherical geometry, or neither? 3. 79°, 45°, 45° 4. 33°, 87°, 60° 5. 100°, 100°, 100°

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Theorem** **Spherical Geometry—Triangle Angle-Sum Theorem** The sum of the measures of the angles of a triangle is always greater than 180°. On a sphere, the lines are great circles. Since these lines are curved, the sum of the angles formed for a triangle lying on the sphere is always greater than 180°. In Euclidean geometry, the lines that form a triangle are straight and their sum always equals 180°. This is proved by the Euclidean Geometry Triangle Angle-Sum Theorem. **Problem** An airline is mapping routes for its international flights. One route leaves from Raleigh-Durham International Airport in the United States, travels to Heathrow Airport in London, then to Alexandria International Airport in Egypt, and finally returns to Raleigh-Durham. This trip forms a triangular route. Is this triangle Euclidean, spherical, or neither? *Answer:* Spherical; since the path of the airplane is curved, the angles formed will also be curved, and since the earth can be considered a sphere, the triangle is spherical. **Exercises** For each set of three angle measures of a triangle given, does the triangle exist in Euclidean geometry, spherical geometry, or neither? 3. 79°, 45°, 45° 4. 33°, 87°, 60° 5. 100°, 100°, 100° 6. 21°, 121°, 221° 7. 30°, 40°, 50° 8. 100°, 40°, 40° 9. 30°, 60°, 90° 10. 70°, 111°, 68° 11. 100°, 80°, 80° 12. Are lines of latitude examples of great circles? Are lines of longitude examples of great circles? Explain.
Transcribed Image Text:**Theorem** **Spherical Geometry—Triangle Angle-Sum Theorem** The sum of the measures of the angles of a triangle is always greater than 180°. On a sphere, the lines are great circles. Since these lines are curved, the sum of the angles formed for a triangle lying on the sphere is always greater than 180°. In Euclidean geometry, the lines that form a triangle are straight and their sum always equals 180°. This is proved by the Euclidean Geometry Triangle Angle-Sum Theorem. **Problem** An airline is mapping routes for its international flights. One route leaves from Raleigh-Durham International Airport in the United States, travels to Heathrow Airport in London, then to Alexandria International Airport in Egypt, and finally returns to Raleigh-Durham. This trip forms a triangular route. Is this triangle Euclidean, spherical, or neither? *Answer:* Spherical; since the path of the airplane is curved, the angles formed will also be curved, and since the earth can be considered a sphere, the triangle is spherical. **Exercises** For each set of three angle measures of a triangle given, does the triangle exist in Euclidean geometry, spherical geometry, or neither? 3. 79°, 45°, 45° 4. 33°, 87°, 60° 5. 100°, 100°, 100° 6. 21°, 121°, 221° 7. 30°, 40°, 50° 8. 100°, 40°, 40° 9. 30°, 60°, 90° 10. 70°, 111°, 68° 11. 100°, 80°, 80° 12. Are lines of latitude examples of great circles? Are lines of longitude examples of great circles? Explain.
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