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Holt Geometry
10-Ext Spherical Geometry10-Ext Spherical Geometry
Holt Geometry
Lesson PresentationLesson Presentation
Holt Geometry
10-Ext Spherical Geometry
Understand spherical geometry as an example of non-Euclidean geometry.
Objective
Holt Geometry
10-Ext Spherical Geometry
non-Euclidean geometryspherical geometry
Vocabulary
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
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!
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 2A: 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 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
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 Geometry
Holt Geometry
10-Ext Spherical Geometry
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
Holt Geometry
10-Ext Spherical Geometry
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
Holt Geometry
10-Ext Spherical Geometry
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
Check It Out! Example 3