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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 . You name a line segment by its endpoints, such as A line segment (in spherical geometry) is an arc of a great circle. It is the shortest ¯¯¯¯¯ distance between two points. AC .

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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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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?

Add the angle measures. 110 + 97 + 108 = 315

Add the angle measures. 47 + 90 + 90 = 227

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 know the sum of the measures C 63, 63, 54 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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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-9 Comparing Spherical and Euclidean Geometry

Problem 4

TEKS Process Standard (1)(G)

Comparing Spherical and Euclidean Geometries

A In Euclidean geometry, how many lines are perpendicular to a line ℓ through a point P that does not lie on the line?

There is exactly one line through point P that is perpendicular to line ℓ.

B The statement in part (A) is the Perpendicular Postulate in Euclidean geometry. Is the Perpendicular Postulate true in spherical geometry? Explain.

In spherical geometry, a line is a great circle. There are many lines that contain point P that are perpendicular to line ℓ. So the Euclidean Perpendicular Postulate is not true in spherical geometry. Plan How can a model help you?

You can use a globe to help you visualize the points, lines, and planes of spherical geometry.

C Two airplanes travel around the globe along great circles that meet at right angles. How many times do the flight paths intersect?

The flight paths of the airplanes are perpendicular lines in spherical geometry. The lines intersect twice, at point A and at point B, as shown in the figure.

PRACTICE and APPLICATION EXERCISES

1. Classify each set of measures as the angle measures of a triangle in Euclidean geometry, a triangle in spherical geometry, or neither. Scan page for a Virtual a. 28, 28, 118 Nerd™ tutorial video.

b. 68, 45, 67

c. 55, 103, 61

For Exercises 2–4, explain how each property of spherical geometry compares to what is true in Euclidean geometry.

2. There are pairs of points on a sphere through which you can draw more than one line. For additional

3. Two equiangular triangles can have different angle measures. support when completing your 4. A triangle can have two obtuse angles. homework, go to 5. Determine whether each of the following properties of Euclidean geometry is true in spherical geometry. If the property is PearsonTEXAS.com . not true, draw a counterexample.

a. Two lines that are perpendicular to the same line do not intersect.

b. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

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c. There can be at most one right angle in a triangle.

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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

a. The figure at the right appears to show parallel lines on a sphere. Explain why this is not the case.

b. Explain why a piece of the upper circle in the figure is not a line segment.

7. Explain Mathematical Ideas (1)(G) In Euclidean geometry vertical angles are congruent. Does this seem to be true in spherical geometry? Explain.

8. The lines of latitude and longitude are shown on the globe.

a. Which lines of longitude are lines in spherical geometry? Explain.

b. Which lines of latitude are lines in spherical geometry? Explain.

c. How many lines perpendicular to the equator can you draw from the North Pole?

d. Explain why lines of latitude are not parallel lines in spherical geometry.

9. Describe a line segment in spherical geometry. How does it differ from a line segment in Euclidean geometry?

10. Explain why the Euclidean Parallel Postulate does not hold in spherical geometry.

11. Create Representations to Communicate Mathematical Ideas (1)(E) In spherical geometry how many perpendicular line segments can you draw from a point to a line? Draw a sketch to support your answer.

TEXAS Test Practice

12. Which CANNOT be the measures of the angles of a triangle in spherical geometry?

A. 150

B. 185

C. 245

D. 270

13. Two airplanes follow flight paths around the globe along different great circles. How many times will the flight paths intersect?

F. 0

G. 1

H. 2

J. infinitely many

14. Which of the following statements is true in spherical geometry?

A. A triangle can have at most one right angle.

B. Through any two points there is exactly one line.

C. Two perpendicular lines form eight 90° angles.

D. All equilateral triangles have three angles measuring 60.

15. Compare and contrast triangles in Euclidean geometry and spherical geometry.

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