Points, Lines, and Triangles in Hyperbolic Geometry

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Points, Lines, and Triangles in Hyperbolic Geometry.

The following postulates will be examined: 1. There exists a unique line through any two points. 2. If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two. 3. If two lines intersect then their intersection is exactly one point. 4. A line can be extended infinitely. 5. A circle can be drawn with any center and any radius. 6. The Parallel Postulate: If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. 7. The Perpendicular Postulate: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. 8. Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 9. Corresponding Angles Converse: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 10. SAS Congruence Postulate: If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

The following theorems will be explored: 1. Vertical Angles Theorem: Vertical angles are congruent. 2. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3. Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 4. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. 5. Theorem: If two lines are parallel to the same line, then they are parallel to each other. 6. Theorem: If two lines are perpendicular to the same line, then they are parallel to each other. 7. Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180o. 8. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 9. Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles must also be congruent. 10. Angle-Angle Similarity Theorem: If two triangles have their corresponding angles congruent, then their corresponding sides are in proportion and they are similar. 11. Side-Side-Side (SSS) Congruence Theorem: If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. 12. Angle-Side-Angle (ASA) Congruence Theorem: If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. 13. Theorem of Pythagoras: In a right triangle, the square on the hypotenuse is equal to the sum of the squares of the legs. 14. Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite the sides are congruent. 15. Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent. 16. Equilateral Triangle Theorem: If a triangle is equilateral, then it is also equiangular.

2 Points, Lines, and Triangles in Hyperbolic Geometry

Models are useful for visualizing and exploring the properties of geometry. A number of models exist for exploring the geometric properties of the hyperbolic plane. These models do not “look like” the hyperbolic plane. The models merely serve as a means of exploring the properties of the geometry.

The Beltrami-Klein Model (or Klein Model) for Studying Hyperbolic Geometry In this model, a circle is fixed with center O and fixed radius. All points in the interior of the circle are part of the hyperbolic plane. Points on the circumference of the circle are not part of the plane itself. Lines are therefore open chords, with the endpoints of the chords on the circumference of the circle but not part of the plane.

The Poincaré Half Plane Model for Studying Hyperbolic Geometry In this model, the Euclidean plane is divided by a Euclidean line into two half planes. It is customary to choose the x-axis as the line that divides the plane. The hyperbolic plane is the plane on one side of this Euclidean line, normally the upper half of the plane where y > 0. In this model, lines are either the intersection of points lying on a line drawn vertical to the x-axis and the half plane, or points lying on the circumference of a semicircle drawn with its center on the x-axis.

m P n

Lines in the Poincaré Half Plane model

3 Angles are measured in the normal Euclidean way. The angle between two lines is equal to the Euclidean angle between the tangents drawn to the lines at their points of intersection.

A A C C

x x

Finding angle measure in the Poincaré Half Plane model

The Poincaré Disk Model for Studying Hyperbolic Geometry In this model, lines are not straight as you are used to seeing them on the Euclidean plane. Instead, lines are represented by arcs of circles that are orthogonal (perpendicular) to the circle defining the disk. In this model, the only lines that appear to be straight in the Euclidean sense are diameters of the disk. In addition, the boundary of the circle does not really exist, and distances become distorted in this model. All the points in the interior of the circle are part of the hyperbolic plane. In this plane, two points lie on a “line” if the “line” forms an arc of a circle orthogonal to C.

B m

Lines in the Poincaré model Constructing the angle between two lines in the Poincaré model.

The angle between two lines is the measure of the Euclidean angle between the tangents drawn to the lines at their points of intersection.

4 We will use the Geometry software NonEuclid to investigate Hyperbolic Geometry. Begin by downloading the software: http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html. Run the software and select File > New to reset the hyperbolic plane.

1. What is the shortest path between two points in the Euclidean plane?

2. Use the Constructions > Plot Point tool to locate two points A and B in the hyperbolic plane. Use the Constructions > Draw Line Segment tool to construct segment AB. Use the Edit > Move Point tool to drag points A and B around the hyperbolic plane. Describe line segment AB in the hyperbolic plane (using the disc model) as compared to the Euclidean line segment AB.

3. Use File > New to reset the hyperbolic plane. Construct a fixed point A. Use the Construction > Draw Line tool to construct different lines which pass through this point and another point on the disk.

a. Describe line l in the hyperbolic plane (using the disc model) as compared to the Euclidean line l. Are lines in the hyperbolic plane infinite in length?

5 b. Use the Edit > Move Point tool to move point A. Describe how the appearance of the lines changes as A moves closer to the center of the disc. Describe how the appearance of the lines changes as point A moves closer to the edge of the disc.

4. Euclid’s first postulate states that for every point P and for every point Q where P  Q, a unique line passes through P and Q. Is this postulate valid in hyperbolic geometry?

5. Locate three points A, B, and C on a line on the Euclidean plane. The Betweenness Axiom states that if A, B, and C are points on the Euclidean plane, then one and only one point is between the other two. Does the Betweenness Axiom hold on the hyperbolic plane?

A C B

P Q R

Q P R

6 6. Draw two lines on the Euclidean plane. In how many points do these lines intersect?

7. Draw two lines on the hyperbolic plane. In how many points do the lines intersect?

8. Euclid’s parallel postulate states that if l is a line and P is a point not on l, then exactly one line can be drawn through P that is parallel to l. Can you re-word this postulate so that it is true for spherical geometry? Explain.

7 9. Euclid’s second postulate states that a line segment can be extended infinitely from each side. Is this postulate valid in hyperbolic geometry? Justify your answer.

PQ = 1.94

PR = 2.56

PS = 3.18

PT = 5.33 ST = 3.38 Q R P S T

10. Euclid’s third postulate states that a circle can be drawn with any center and any radius. Use the Constructions > Draw Circle tool to construct several circles with centers located at different points in the hyperbolic plane. a. What appears to happen to the circle as the center gets nearer the edge of the disk?

b. Does this mean that the center of a circle near the edge of the disk is not located equidistant from the points on its circumference? Explain.

8 11. Draw two intersecting lines on the Euclidean plane. Use a protractor to measure the vertical angles. Confirm that the vertical angles are congruent.

C B E A D

12. Draw two lines on the hyperbolic plane. Measure the pairs of adjacent angles. Are they supplementary? Measure the vertical angles. Are the pairs of vertical angles congruent?

m1 = 70.0° R m2 = 70.0°

m3 = 110.0° 2 Q 1 3 m4 = 110.0° O 4

P S

13. Suppose we construct a line l on the Euclidean plane, and point A not on the line. How many lines can we construct passing through point A that are perpendicular to line l?

14. Draw a line on the hyperbolic plane. Locate a point P not on the line. a. Can you construct a perpendicular from the point to the line? If so, how many perpendiculars can you construct?

9 b. Measure the angle at the point of intersection to confirm that the angle is a right angle.

1 m1 = 90.0°

15. Given a pair of parallel lines and a transversal on the Euclidean plane, we know that corresponding angles are congruent.



Draw a pair of parallel lines and a transversal on the hyperbolic plane. Measure the pairs of corresponding angles and determine whether the corresponding angles postulate is valid on the hyperbolic plane.

V S 107.8 R W 84.4 P Q U

10 16. Given a pair of parallel lines and a transversal on the Euclidean plane, we know that alternate interior angles are congruent.



Draw a pair of parallel lines on the hyperbolic plane. Measure the alternate interior angles and determine whether they are congruent. T

V S 107.8 R 84.4 W P Q U

17. Given a pair of parallel lines and a transversal on the Euclidean plane, we know that same- side interior angles are supplementary.





11 Draw a pair of parallel lines on the hyperbolic plane. Measure the same-side interior angles. Are these pairs of angles supplementary?

V S 107.8 R 95.6 W P Q U

18. The Perpendicular Transversal Theorem states that if a transversal is perpendicular to one of two parallel lines on the Euclidean plane, then it is perpendicular to the other. Draw two parallel lines l and m on the hyperbolic plane. At a point on l draw a perpendicular transversal. Determine whether the above theorem is valid on the hyperbolic plane.

1 m m1 = 90.0°

m2 = 35.9° l 2

12 19. On the Euclidean plane, if two lines are parallel to the same line, then they are parallel to each other. Draw a line r on the hyperbolic plane. Through a point P not on r, draw a line s that is parallel to r. Through point Q that is not on either r or s, draw a line t that is parallel to r. Are s and t parallel?

t Q Q s s P P

t r r

Figure (a) Figure (b)

20. On the Euclidean plane, if two lines are perpendicular to the same line, then they are parallel to each other. Draw a line m on the hyperbolic plane. Locate at least two points P and Q on the line. At each point draw a perpendicular to the line. Are the two lines parallel?

m Q P R

13 21. When two lines cut by a transversal on the Euclidean plane, have congruent corresponding angles, then the two lines are parallel. Investigate whether the same is true for lines drawn on the hyperbolic plane.

1 m m1 = 90.0°

m2 = 90.0° 2 m3 = 90.0°

22. The sum of the interior angles of a triangle on the Euclidean plane is equal to 180o. Draw a triangle on the hyperbolic plane. Use the File > New tool to reset the hyperbolic plane. Use the Constructions > Draw Line Segment tool to construct a triangle. Use the Measurements > Measure Triangle tool to measure the interior angles of the triangle. Compare the sum of the angles of a hyperbolic triangle to that of a Euclidean triangle.

1 m1 = 17.7°

3 m2 = 12.9° 2 R m3 = 11.6° Q m1 + m2 + m3 = 42.2°

14 23. If we draw a triangle on the Euclidean plane, we can extend one side of the triangle to create an exterior angle. The measure of this exterior angle is equal to the sum of the measures of the two remote interior angles.

1 2 B C D

Draw a triangle on the hyperbolic plane. Use the Constructions > Draw Ray tool to extend one of the sides of the triangle. Use the Constructions > Plot Point on Object tool to construct a third point on the ray, outside of the triangle. Measure the exterior angle and compare this measure with the measure of the sum of the measure of the two non-adjacent interior angles. What can you conclude?

P=14.2° Q = 13.4° 1 2 R1= 43.4° R S R2 = 136.6° Q

15 24. On the Euclidean plane, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Draw a triangle on the hyperbolic plane. Measure the angles of the triangle. Create a second triangle with two angles in the second triangle congruent to two angles in the first. Measure the third angle of the triangle. Are the third angles congruent?

3 m1 = 25.7° m4 = 25.7° m2 = 7.0° m5 = 7.0° 2 4 m3 = 56.0° m6 = 133.0° 6 5

25. On the Euclidean plane, base angles of an isosceles triangle are congruent. Draw an isosceles triangle on the hyperbolic plane. Measure the angles at the base of the congruent sides. Are the base angles congruent?

l1 = 3.59 l1 l2 l2 = 3.59

2 1 m2 = 15.5°

m1 = 15.5°

16 26. Given a triangle with two congruent angles on the Euclidean plane, the sides opposite the congruent angles are congruent. Draw a triangle with two angles congruent on the hyperbolic plane. Measure the sides opposite the congruent and determine whether these sides are congruent.

Distance = 3.73 Distance = 3.73

2 1 m2= 17.3° m1 = 17.3°

27. On the Euclidean plane, the measure of each angle of an equilateral triangle is 600. Draw an equilateral triangle on the hyperbolic plane. a. To do this, first use the Constructions > Draw Line Segment tool to construct radius AB. b. Then use the Constructions > Draw Circle tool to construct circle A with radius AB and circle B with radius AB. c. Use the Constructions > Plot Intersection Point tool to construct points C and D where the circles intersect. d. Use the Constructions > Draw Line Segment tool to construct triangle ABC, which is equilateral. e. Use the Measurements > Measure Triangle tool to determine the measure of each angle of the equilateral triangle. f. How do your observations on the hyperbolic plane compare with those on the Euclidean plane?

17 C

B A

28. What is the formula for calculating the area of a triangle on the Euclidean plane? Investigate whether this formula is valid for calculating the area of a triangle on the hyperbolic plane.

b1 = 3.58 h1 = 2.87 0.5 b1 h1 = 5.13

b2 b3 h2 h2 = 2.40 h3 b2 = 4.05

h1 0.5 h2 b2 = 4.87

b1 h3 = 2.51 b3 = 3.94 0.5 h3 b3 = 4.95

18 29. In a right triangle on the Euclidean plane, the square on the hypotenuse is equal to the sum of the squares on the legs of the triangle. Does this Theorem of Pythagorean hold for triangles on the hyperbolic plane? a. Construct a number of right triangles on the hyperbolic plane. To do this, construct segment AB. Use the Constructions > Draw Ray at Specific Angle tool to construct a 90° angle ABC. Then construct segments BC and CA. b. Use the Measurements > Measure Triangle tool to measure the lengths of the hypotenuse and legs. Do the hyperbolic triangles satisfy the Pythagorean Theorem?

m1 = 90.0° a = 2.78 a b = 2.12 c c = 4.22 2 2 1 a + b = 12.21 2 C b c = 17.83

30. The SAS Congruence Postulate states that if two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent. Investigate whether this postulate can be accepted on the hyperbolic plane.

19 31. The SSS Congruence Theorem and the ASA Congruence Theorems are valid on the Euclidean plane. Use the NonEuclid webiste to discover whether these theorems are valid on the hyperbolic plane

F C

A B D

32. On the Euclidean plane, if three angles of one triangle are congruent to three angles of another triangle, then the corresponding sides of the triangles are in proportion and the two triangles are similar. If three angles of one triangle on the hyperbolic plane are congruent to three angles of another, what can we say about these two triangles?

A = 22.3° F B = 53.5° C C = 34.3° E D = 22.3° D E = 53.6° F = 73.9° A B