10 Ext Spherical Geometry Lesson Presentation Holt Geometry

10 -Ext Spherical. Geometry Lesson Presentation Holt Geometry

10 -Ext Spherical Geometry Objective Understand spherical geometry as an example of non-Euclidean geometry. Holt Geometry

10 -Ext Spherical Geometry Vocabulary non-Euclidean geometry spherical geometry Holt Geometry

10 -Ext Spherical Geometry 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. Holt Geometry

10 -Ext Spherical Geometry 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. Holt Geometry

10 -Ext Spherical Geometry 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. Holt Geometry

10 -Ext Spherical Geometry Example 1: Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere. PQ is a line PQ is a segment. ∆PQR is a triangle. Holt Geometry

10 -Ext Spherical Geometry Check It Out! Example 1 Name a line, segment, and triangle on the sphere. AD is a line AD is a segment. ∆BCD is a triangle. Holt Geometry

10 -Ext Spherical Geometry 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. Holt Geometry

10 -Ext Spherical Geometry Example 2 A: Classifying Spherical Triangles Classify the spherical triangle by its angle measures and by its side lengths. ∆XYZ acute isosceles triangle Holt Geometry

10 -Ext Spherical Geometry Example 2 B: 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 onefourth the circumference of Earth. right equilateral triangle Holt Geometry

10 -Ext Spherical Geometry Check It Out! Example 2 Classify ∆VWX by its angle measures and by its side lengths. ∆VWX is equiangular and equilateral. Holt Geometry

10 -Ext Spherical Geometry 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. Holt Geometry

10 -Ext Spherical Holt Geometry

10 -Ext Spherical Geometry Example 3 A: Finding the Area of Spherical Triangles Find the area of each spherical triangle. Round to the nearest tenth, if necessary. A ≈ 62. 8 cm 2 Holt Geometry

10 -Ext Spherical Geometry Example 3 B: Finding the Area of Spherical Triangles Find the area of the spherical triangle. Round to the nearest tenth, if necessary. ∆QRS on Earth with m Q = 78°, m R = 92°, and m S = 45°. A ≈ 9, 574, 506. 9 mi 2 Holt Geometry

10 -Ext Spherical Geometry Check It Out! Example 3 Find the area of ∆KLM on a sphere with diameter 20 ft, where m K = 90°, m L = 90°, and m M = 30°. Round to the nearest tenth. A ≈ 52. 4 ft 2 Holt Geometry