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Lesson Menu Five-Minute Check (over Lesson 12–6) Then/Now New Vocabulary Key Concept: Lines in Plane and Spherical Geometry Example 1:Describe Sets of Points on a Sphere Example 2:Real-World Example: Identify Lines in Spherical Geometry Example 3:Compare Plane Euclidean and Spherical Geometries

Over Lesson 12–6 5-Minute Check 1 A in 2 B in 2 C in 2 D in 2 Find the surface area of the sphere. Round to the nearest tenth.

Over Lesson 12–6 5-Minute Check 2 A.1107 ft 2 B ft 2 C ft 3 D ft 3 Find the surface area of the sphere. Round to the nearest tenth.

Over Lesson 12–6 5-Minute Check 3 A cm 2 B cm 2 C cm 3 D cm 3 Find the volume of the hemisphere. Round to the nearest tenth.

Over Lesson 12–6 5-Minute Check 4 A mm 2 B mm 2 C mm 3 D mm 3 Find the volume of the hemisphere. Round to the nearest tenth.

Over Lesson 12–6 5-Minute Check 5 A ft 2 B ft 2 C ft 2 D ft 3 Find the surface area of a sphere with a diameter of 14.6 feet to the nearest tenth.

Over Lesson 12–6 5-Minute Check 6 A in 2 B in 2 C.19,704.1 in 2 D.39,408.1 in 2 A hemisphere has a great circle with circumference approximately in 2. What is the surface area of the hemisphere?

Then/Now You identified basic properties of spheres. (Lesson 12–6) Describe sets of points on a sphere. Compare and contrast Euclidean and spherical geometries.

Vocabulary Euclidean geometry spherical geometry non-Euclidean geometry

Concept

Example 1A Describe Sets of Points on a Sphere A. Name two lines containing R on sphere S. Answer: AD and BC are lines on sphere S that contain point R.

Example 1B Describe Sets of Points on a Sphere B. Name a segment containing point C on sphere S. Answer: GK

Example 1C Describe Sets of Points on a Sphere C. Name a triangle on sphere S. Answer: ΔKGH

Example 1A A. Determine which line on sphere Q does not contain point P. A.TL B.VM C.PS D.RO

Example 1B B. Determine which segment on sphere Q does not contain point N. A.LO B.MS C.NV D.SO

Example 1 A.ΔVSU B.ΔPRU C.ΔNSU D.ΔNOM C. Name a triangle on sphere Q.

Example 2 Identify Lines in Spherical Geometry SPORTS Determine whether the line h on the basketball shown is a line in spherical geometry. Explain. Answer: No; it is not a great circle. Notice that line h does not go through the poles of the sphere. Therefore line h is not a great circle and so not a line in spherical geometry.

Example 2 A.Yes, it is a line in spherical geometry. B.No, it is not a line in spherical geometry. SPORTS Determine whether the line X on the basketball shown is a line in spherical geometry.

Example 3A Compare Plane Euclidean and Spherical Geometries A. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. Answer: True; given a line, the only line on a sphere that is always the same distance from the line is the line itself. If two lines are parallel, they never intersect.

Example 3B Compare Plane Euclidean and Spherical Geometries B. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. Answer: False; two distinct lines on a sphere intersect twice. Any two distinct lines are parallel or intersect once.

Example 3A A.This holds true in spherical geometry. B.This does not hold true in spherical geometry. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. Perpendicular lines intersect at exactly one point.

Example 3B A.This holds true in spherical geometry. B.This does not hold true in spherical geometry. Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. It is possible for two circles to be tangent at exactly one point.

End of the Lesson