4 Euclidean geometry is based on figures in a plane Euclidean geometry is based on figures in a plane. Non-Euclidean geometry is based on figures in a curved surface. In a non-Euclidean geometry system,the Parallel Postulate is not true. One type of non-Euclidean geometry is spherical geometry, which is the study of figures on the surface of a sphere.
5 A line in Euclidean geometry is the shortest path between two points A line in Euclidean geometry is the shortest path between two points. On a sphere, the shortest path between two points is along a great circle, so “lines” in spherical geometry are defined as great circles. In spherical geometry, there are no parallel lines. Any two lines intersect at two points.
6 The two points used to name a line cannot be exactly opposite each other on the sphere. In Example 1, AB could refer to more than one line.Caution!
7 Example 1: Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere.PQ is a linePQ is a segment.∆PQR is a triangle.
8 Check It Out! Example 1Name a line, segment, and triangle on the sphere.AD is a lineAD is a segment.∆BCD is a triangle.
9 Imagine cutting an orange in half and then cutting each half in quarters using two perpendicular cuts. Each of the resulting triangles has three right angles.
10 Example 2A: Classifying Spherical Triangles Classify the spherical triangle by its angle measures and by its side lengths.∆XYZacute isosceles triangle
11 Example 2B: Classifying Spherical Triangles Classify the spherical triangle by its angle measures and by its side lengths.∆STU on Earth has vertex S at the South Pole and vertices T and U on the equator. TU is equal to one-fourth the circumference of Earth.right equilateral triangle
12 Check It Out! Example 2Classify ∆VWX by its angle measures and by its side lengths.∆VWX is equiangular and equilateral.
13 The area of a spherical triangle is part of the surface area of the sphere. For the piece of orange on page 726, the area is of the surface area of the orange, or If you know the radius of a sphere and the measure of each angle, you can find the area of the triangle.
15 Example 3A: Finding the Area of Spherical Triangles Find the area of each spherical triangle. Round to the nearest tenth, if necessary.A ≈ 62.8 cm2
16 Example 3B: Finding the Area of Spherical Triangles Find the area of the spherical triangle. Round to the nearest tenth, if necessary.∆QRS on Earth with mQ = 78°, mR = 92°, and mS = 45°.A ≈ 9,574,506.9 mi2
17 Check It Out! Example 3Find the area of ∆KLM on a sphere with diameter 20 ft, where mK = 90°, mL = 90°, and mM = 30°. Round to the nearest tenth.A ≈ 52.4 ft2