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Geometry, Student Text and Homework Helper Page 1 of 1
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-9 Comparing
Spherical and Euclidean Geometry
3-9 Comparing Spherical and Euclidean Geometry
Teks Focus
TEKS (4)(D) Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.
TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and
language as appropriate.
Additional TEKS (1)(G)
Vocabulary
Euclidean geometry – Euclidean geometry is based on Euclid's postulates. It is the geometry of flat planes, straight lines, and points.
Great circle – the intersection of a sphere and a plane that contains the center of the sphere
Line (in spherical geometry) – a great circle
Line segment (in spherical geometry) – an arc of a great circle. It is the shortest distance between two points.
Point (in spherical geometry) – A point in spherical geometry has the same meaning as in Euclidean geometry. It is a location on the surface of a sphere.
Spherical geometry – a non-Euclidean geometry in which a plane is defined as the surface of a sphere
Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated
Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
In addition to Euclidean geometry, there are other kinds of geometries, such as spherical geometry. Spherical geometry has its own postulates and theorems.
take note
Key Concept Undefined Terms in Spherical Geometry
Term Description How to Name It Diagram
A point (in spherical geometry) has the same meaning as in Euclidean geometry. You can represent a point by a dot and name it by a
It is a location on the surface of a sphere. capital letter such as A.
A great circle is the intersection of a sphere and a plane that contains the center of You name a line by two points that lie on the line,
the sphere. ← →
A line (in spherical geometry) is a great circle. such as AC .
A line segment (in spherical geometry) is an arc of a great circle. It is the shortest You name a line segment by its endpoints, such as
¯¯¯¯¯
distance between two points. AC .
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Geometry, Student Text and Homework Helper Page 1 of 2
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-9 Comparing
Spherical and Euclidean Geometry
take note
Postulate Spherical Geometry—Parallel Postulate
Through a point not on a line, there is no line parallel to the given line.
Since lines are great circles in spherical geometry, two lines always intersect. In fact, any two lines on a sphere intersect at
two points, as shown at the right.
take note
Key Concept Spherical Geometry—Triangle Angle-Sum Theorem
The sum of the measures of the angles of a triangle is always greater than 180.
A triangle in spherical geometry is formed by the intersection of three arcs that lie on different great circles.
Recall that the proof of the Triangle Angle-Sum Theorem in Euclidean geometry uses the Euclidean Parallel Postulate, which
does not hold in spherical geometry. So the Euclidean Triangle Angle-Sum Theorem is invalid in spherical geometry. Because
the sides of a triangle in spherical geometry are curved, the sum of the measures of the angles is always greater than 180.
Problem 1
TEKS Process Standard (1)(D)
Comparing Lines in Euclidean and Spherical Geometries
A In Euclidean geometry, how many lines are parallel to a line ℓ through a point P that does not lie on the line? Draw
a sketch to support your answer.
There is exactly one line through point P that is parallel to line ℓ.
B The statement in part (A) is the Parallel Postulate in Euclidean geometry. Is the Parallel Postulate true in spherical
geometry? Explain. Draw a sketch to support your answer.
In spherical geometry, a line is a great circle. Every line that contains point P will also intersect line ℓ. So there is no line
through point P that is parallel to line ℓ. Think
How can a sketch help you
answer the question?
A sketch can help you visualize
lines on a sphere and whether
they intersect.
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Geometry, Student Text and Homework Helper Page 1 of 1
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-9 Comparing
Spherical and Euclidean Geometry
Problem 2
Sums of Angle Measures of Triangles
What is the sum of the measures of the angles of the triangle?
A
Add the angle measures. 110 + 97 + 108 = 315
B
Add the angle measures. 47 + 90 + 90 = 227
C
Add the angle measures. 49 + 70 + 139 = 258
D Make a conjecture about the sum of the measures of the angles of a triangle in spherical geometry. Explain how
the results are different from the Triangle Angle-Sum Theorem in Euclidean geometry. Plan
The sum of the measures of the angles of a triangle is not a constant value and is always greater than 180. In Euclidean How do you get started?
geometry, the sum of the measures of the angles of any triangle is always equal to 180. Use what you know about the
sums of the measures of the
Problem 3 angles in a triangle in
Euclidean geometry to
compare triangles in Euclidean
Using the Spherical Geometry Triangle Angle-Sum Theorem and spherical geometries.
Three angle measures of a triangle are given. Does the triangle exist in Euclidean geometry, spherical geometry, or
neither?
A 128, 85, 30 Plan
How do you get started?
The sum of the angle measures is 243. Because the sum is greater than 180, the triangle exists in spherical geometry. To determine whether the
B 68, 52, 35 triangle exists in Euclidean
geometry, spherical geometry,
The sum of the angle measures is 155. Because the sum is less than 180, the triangle exists in neither geometry. or neither, you first need to
C 63, 63, 54 know the sum of the measures
of the angles of the triangle.
The sum of the angle measures is 180. Because the sum is equal to 180, the triangle exists in Euclidean geometry.
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