Elaborate 12. Discussion Picture a triangle that forms over Earth’s northern hemisphere, using the North Pole and two points on the equator. The triangle can expand to cover almost the entire hemisphere. Use the formula for area of a spherical triangle to make a conjecture about an upper bound for the sum of the angles of a spherical triangle contained in a hemisphere.
13. Essential Question Check-In Name a postulate and a theorem from Euclidean geometry that do not apply to spherical geometry.
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Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice
Fill in the blank with an appropriate word or phrase. 1.
In Euclidean geometry, the shortest distance between two points, A and B, in the plane is the length of the segment whose endpoints are A and B, namely AB. In spherical geometry, the shortest distance between two points A and B is the length of the
.
2.
In Euclidean geometry, a plane is a flat, two-dimensional surface that continues infinitely in all directions. The is the equivalent figure in spherical geometry.
3.
In Euclidean geometry, a point is a location represented by a dot. The equivalent in spherical geometry.
4.
In Euclidean geometry, given a line m and a point P not on it, exactly one line through P is parallel to m. In Spherical geometry, given a line m and a point P not on it, through P can be parallel to the given line.
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What If? An orange is cut three times so that each slice is at a right angle with the two other slices. This shows how a spherical triangle can have three right angles. If you put two flat surfaces of these shapes together so that the curved sides form one surface, the new curved shape is called a lune. Is a lune a spherical triangle? Explain.
A
In Exercises 6–8, use this sphere to answer each question. 6.
Name all lines shown on the figure. C
7.
D
Name all segments shown on the figure. B
8.
Name all triangles shown on the figure.
9.
‹ › − Points A and B are endpoints of a diameter of a great circle. Describe AB . Explain why we don’t use endpoints of a diameter of a great circle to define a line on a sphere.
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Find the area of each spherical triangle. Round to the nearest tenth, if necessary. A 100°
10. △ABC
B
106° 114°
14 cm C
11. △DEF on Earth’s surface with m∠D = 75°, m∠E = 80°, and m∠F = 30°. The average radius of Earth is about 3959 miles.
12. Find the area of △KLM on a sphere with diameter 20 ft, where m∠K = 90°, m∠L = 90°, and m∠M = 30°.
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13. Describe the curves on the basketball that are lines in spherical geometry.
14. Determine whether each of the following are Euclidean, spherical, or both. Select the correct answer for each lettered part. a. A line had infinite length. Euclidean
Spherical
Both
b. Two distinct lines intersect in two points. Euclidean c.
Spherical
Both
A line separates the plane into two parts. Euclidean
Spherical
Both
d. Two distinct lines perpendicular to the same line are parallel. Euclidean Module 18
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Geography Compare each length to the length of a great circle on earth. 15. the distance between the North Pole and the South Pole
16. the distance between the North Pole and any point on the equator
17. In Euclidean Geometry, a triangle can have a maximum of one obtuse angle, and the sum of interior angle measures is exactly 180°. a. What is the maximum number of obtuse angles in a spherical triangle?
c.
Let s represent the sum of interior angle measures of a spherical triangle. Write an inequality to express all possible values of s.
19. Suppose a lunar roving probe is sent to a planet’s 18. Geography If the area of a triangle on Earth’s moon. It roves the moon in a triangle making surface is 100,000 mi 2, what is the sum of its angle turns of 120°, 100°, and 50°. The probe computer measures? The average radius of Earth is about calculates the area of the triangle to be 1,500 3959 miles. square kilometers. What is the radius of the moon to the nearest tenth of a kilometer?
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b. What is the maximum sum of interior angle measures in a spherical triangle? (Assume all angle measures are in whole numbers.)
H.O.T. Focus on Higher Order Thinking
20. Counterexample In Euclidean geometry, if two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent. Draw a counterexample and explain how it shows that this is not true in spherical geometry.
21. Communicate Mathematical Ideas In spherical geometry, two lines intersect in two points. Some mathematicians redefined these two points as one point. With this modification, two lines in spherical geometry intersect in exactly one point. Describe a case in which two points would be considered one point.
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22. Analyze Relationships The figure shows a spherical quadrilateral ABCD. Consider the definition of a rectangle in plane geometry. Does a spherical rectangle exist? Explain.
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Lesson Performance Task If you look up at the night sky at the group of stars called the “Big Dipper,” you’ll see a pattern of stars that’s been studied for thousands of years. The fact that the pattern appears to have not changed over time suggests that the stars don’t move, but they are fixed in the same position relative to the Earth. In fact, stars move at great speeds in all directions. Why, then, do they seem to stay in the same places in the sky? To see why, look at the “celestial sphere,” the imaginary sphere we live inside of. The stars you see are scattered throughout our Milky Way galaxy. From your perspective they just appear to be attached to the celestial sphere, all of them at the same distance from us. If they’re moving so fast, why don’t you see changes in the positions in the sky over a period of time? You can find the answer by looking at the motion of a star called Alkaid, one of the seven stars that form the Big Dipper. Alkaid travels at a typical star speed of about 24,000 miles per hour. It is located about 100 light years from Earth.
THE Celestial North Pole CELESTIAL SPHERE
Earth
Celestial Equator
Celestial South Pole
1. The figure shows Alkaid at beginning position S and ending position T 1000 years later. Find ST, the distance it has traveled. Show your work.
E
S T
2. A light year is the distance that light travels in one year at a speed of 186,000 miles per second. Find ES, the distance from Earth (E) to Alkaid when it is at position S. Show your work. 3. Find the measure of ∠E. Explain your method. 1 60 of a degree. Explain why the stars don’t appear to move 4. The smallest angle the eye can detect is about __ over long periods of time.
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