Answer the following underlined questions on the answer sheet. Many of the answers are evident by the diagrams given in the problem (which should guide your lab), but you should confirm the findings by drawing your own figures on the hyperbolic geometry software. When the problem is divided into parts “a” and “b”, the “a” part asks about what we currently know in Euclidean (Plane) Geometry. You should be able to answer the “a” part without investigation.
1) What is the shortest path between two points in the Euclidean Plane?
2) Euclid’s first postulate states that for every point \(P\) and for every point \(Q\) where \(P \neq Q\), a unique line passes through \(P\) and \(Q\).
Create a new disk (File-New). Draw two points \(P\) and \(Q\) on the disk. Draw a line that passes through these two points. Label two points on the line. Try to see if you can draw a different line through these two points. Does Euclid’s first postulate hold in hyperbolic geometry?
1) What is the shortest path between two points in the Euclidean Plane?
2) Euclid’s first postulate states that for every point \(P\) and for every point \(Q\) where \(P \neq Q\), a unique line passes through \(P\) and \(Q\).
Create a new disk (File-New). Draw two points \(P\) and \(Q\) on the disk. Draw a line that passes through these two points. Label two points on the line. Try to see if you can draw a different line through these two points. Does Euclid’s first postulate hold in hyperbolic geometry?
3) Consider 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?
4a) Draw two lines on the Euclidean plane. In how many ways do these lines intersect?
4b) Create a new disk (File-New). Draw two lines on the hyperbolic plane. In how many points do the lines intersect?
5a) Two lines are defined as being parallel if they have no points in common. Given a line \(\ell\) on the Euclidean plane and a point \(A\) that is not on \(\ell\). How many possible lines can you construct through \(A\) that is parallel to \(\ell\)?
5b) Create a new disk (File-New). Draw a line \(m\) on the hyperbolic plane. Mark a point \(P\) not on \(m\). Draw a line through \(P\) that is parallel to \(m\). How many possible lines can you construct? Does the Parallel Postulate hold on the hyperbolic plane?
6) How would you re-word Euclid’s parallel postulate so that it is true for the hyperbolic plane?
7) Lines in Euclidean geometry are of infinite length. Can the same be said of lines in the hyperbolic plane? Explain.
7) Lines in Euclidean geometry are of infinite length. Can the same be said of lines in the hyperbolic plane? Explain.
8) Euclid’s third postulate states that a circle can be drawn with any center and any radius.Create a new disk (File-New). Draw a number of circles with centers located at different points in the hyperbolic plane. What appears to happen to the circle as the center gets nearer the edge of the disk? 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.
9) Vertical angles on the Euclidean plane are congruent.
Create a new disk (File-New). 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?
Create a new disk (File-New). 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?
10) On Euclidean plane, given a line \(\ell\) and a point \(A\) not on the line, only one perpendicular can be drawn to line \(\ell\) through point \(A\).
Create a new disk (File-New). Draw a line on the hyperbolic plane. Locate a point \(P\) not on the line. Can you construct a perpendicular from the point to the line? If so, how many perpendiculars can you construct?
Measure the angle at the point of intersection to confirm that the angle is a right angle.
Create a new disk (File-New). Draw a line on the hyperbolic plane. Locate a point \(P\) not on the line. Can you construct a perpendicular from the point to the line? If so, how many perpendiculars can you construct?
Measure the angle at the point of intersection to confirm that the angle is a right angle.
11) On Euclidean plane, given a pair of parallel lines and a transversal, each pair of
corresponding angles are congruent.
corresponding angles are congruent.
Create a new disk (File-New). Draw a pair of parallel lines and a transversal on the hyperbolic plane. Measure the pairs of corresponding angles. Is the corresponding angles postulate valid on the hyperbolic plane?
12) On Euclidean plane, 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.
Create a new disk (File-New). Draw two parallel lines \(\ell\) and \(m\) on the hyperbolic plane. At a point on \(\ell\) draw a perpendicular transversal. Is the Perpendicular Transversal Theorem valid on the hyperbolic plane?
Create a new disk (File-New). Draw two parallel lines \(\ell\) and \(m\) on the hyperbolic plane. At a point on \(\ell\) draw a perpendicular transversal. Is the Perpendicular Transversal Theorem valid on the hyperbolic plane?
13) On Euclidean plane, if two lines are parallel to the same line, then they are parallel to each other.
Create a new disk (File-New). 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?
14) On Euclidean plane, if two lines are perpendicular to the same line, then they are parallel to each other.
Create a new disk (File-New). 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?
Create a new disk (File-New). 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?