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## 10-Ext

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**Spherical Geometry**10-Ext Lesson Presentation Holt Geometry**Objective**Understand spherical geometry as an example of non-Euclidean geometry.**Vocabulary**non-Euclidean geometry spherical geometry**Euclidean geometry is based on figures in a plane.**Non-Euclidean geometryis 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.**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.**Caution!**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.**PQ is a line**PQ is a segment. Example 1: Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere. ∆PQR is a triangle.**AD is a line**AD is a segment. Check It Out! Example 1 Name a line, segment, and triangle on the sphere. ∆BCD is a triangle.**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.**Example 2A: Classifying Spherical Triangles**Classify the spherical triangle by its angle measures and by its side lengths. ∆XYZ acute isosceles triangle**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**Check It Out! Example 2**Classify ∆VWX by its angle measures and by its side lengths. ∆VWX is equiangular and equilateral.**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.**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**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**Check It Out! Example 3**Find 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