Name: _______________________________ Date: ______________ Class: ____________ 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. n m l 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. l 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 B B 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. C A 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 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? Q P 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. P m 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.