Exercises for Chapter Two
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He is unworthy of the name of man who
is ignorant of the fact that the diagonal of a square is incommensurable with
its side.
—
Plato (429–347 B.C.)
Exercise 2.1. Identify the error or errors in the proof that all triangles are isosceles.
Exercise 2.2. Identify the error or errors in the proof of the Rusty Compass Theorem.
Exercise 2.3. For each model (Euclidean, Taxicab, Max-distance, Missing-Strip, and Poincaré Half-plane), find the distance between points P(–1, 2) and Q(3, 4).
Exercise 2.4. Show the Taxicab distance satisfies the definition of distance.
Exercise 2.5. Show the Max-distance distance satisfies the definition of distance.
Exercise 2.6. Show the Missing Strip distance satisfies the definition of distance.
Exercise 2.7. Show the Poincaré Half-plane distance satisfies the definition of distance.
Exercise 2.8. (a) Show the Hamming distance satisfies the definition of distance. (b) Does the Discrete Plane distance satisfy the definition of distance? Justify.
Exercise 2.9. Define a distance function for the Modified Riemann Sphere.
Exercise 2.10. Sketch and describe a circle for each model.
Exercise 2.11.
Find the axioms
from a high school geometry book that correspond to SMSG Postulates 1 and 5.
Exercise
2.12. Prove Theorem 2.2.
Exercise
2.13. Show the Missing Strip Plane satisfies (a) SMSG Postulate 1 and
(b) SMSG Postulate 5.
Exercise 2.14. Show the Poincaré Half-plane satisfies (a) SMSG
Postulate 1 and (b) SMSG Postulate 5.
Exercise
2.15. Does a Discrete plane satisfy (a) SMSG Postulate 1? Justify. (b) SMSG Postulate 5? Justify.
Exercise
2.16. Do the Riemann Sphere and Modified Riemann Sphere satisfy (a) SMSG
Postulate 1? Justify. (b) SMSG Postulate 5? Justify.
Exercise 2.17. Find the axioms from a high school geometry book that
correspond to SMSG Postulates 2, 3, and 4.
Exercise
2.18. How do the SMSG Postulates 3 and 4 relate to
"real-world" applications?
Exercise
2.19. For each model (Euclidean, Taxicab, Max-distance, Missing Strip,
and Poincaré Half-plane) find a ruler where f(P) = 0 and f(Q) > 0 for (a) P(3,
4) and Q(3, 7); and (b) P(–1, 3) and Q(1, 2).
Exercise
2.20. Complete the proof that the Ruler Placement Postulate is not
independent, Theorem 2.5.
Exercise
2.21. Show the stated model satisfies SMSG Postulate 3, the Ruler
Postulate, for (a) Taxicab Plane; (b)
Max-Distance Plane; (c) Missing Strip Plane; and (d)
Poincaré Half-Plane.
Exercise
2.22. Does the Modified Riemann Sphere satisfy SMSG Postulate 3, the
Ruler Postulate? Explain.
Exercise
2.23. Explain why collinear is necessary in the definition of
betweenness. (Hint. Look for an example
in either the Taxicab or Max-distance plane where the distance condition is
satisfied, but the point would not be on the line.)
Exercise
2.24. Prove a segment has a unique midpoint.
Exercise
2.25. Find the midpoint of the segment AB for each model (Euclidean, Taxicab, Max-distance, Missing Strip,
and Poincaré Half-Plane) where (a) A(1, 1) and B(1, 5); and (b) A(–1,
1) and B(3, 2). (Show the work using the standard ruler
for each model.)
Exercise
2.26. Find the ray AB for
each model (Cartesian, Missing Strip, and Poincaré Half-Plane) where (a) A(–3,
1) and B(–3, 7); and (b) A(–1, 5) and B(3, 1).
Exercise
2.27. An equivalence relation, ~, is a relation on a set that satisfies
each of the following: (i) a ~ a (reflexive property) (ii) If a ~ b, then b ~ a. (symmetric
property) (iii) If a ~ b
and b ~
c, then a ~ c. (transitive property).
Prove that is an equivalence relation for the set of all
segments.
Exercise 2.28.
Is the Plane Separation Postulate discussed or mentioned
in the high school geometry book you are using as a reference?
Exercise 2.29.
Is the Plane Separation Postulate independent of the other axioms? Explain.
Exercise
2.30. Find an analytic example that shows the Missing Strip plane does
not satisfy Pasch's Postulate.
Exercise
2.31. Prove Pasch's Postulate.
Exercise 2.32. Find the axioms
from a high school book that correspond to SMSG Postulates 11, 12, and 13.
Exercise
2.33. Find the measures of the three angles determined by the points
A(1, 1), B(1, 2) and C(2, 1) where the points are in the (a) Euclidean Plane;
and (b) Poincaré Half-plane.
Also, find the sum of the measures of the
angles of the triangles.
Exercise
2.34. Find the angle bisector of ,
if A(0, 5), B(0, 3), and
where the points are in the (a) Euclidean
Plane; and (b) Poincaré Half-plane.
Exercise
2.35. Given ,
,
and
.
Prove or disprove
.
Exercise
2.36. Prove or disprove that all right angles are congruent.
Exercise
2.37. Prove or disprove that an angle has a unique bisector.
Exercise
2.38.
(a) Prove that given a line and a
point on the line, there is a line perpendicular to the given line and point on
the line.
(b) Prove the existence of two lines perpendicular to each other.
Exercise
2.39. Prove Theorem 2.7.
Exercise 2.40. Prove congruence of angles is an equivalence relation on the set of all angles.
Exercise 2.41. Find the axiom or theorem from a high school book that corresponds to the Supplement Postulate. Is it an axiom or theorem in the high school book? If it is a theorem, how was it proven?
Exercise
2.42. Prove or disprove. If two angles are supplementary, then they
form a linear pair.
Exercise
2.43. Justify each numbered step and fill in any gaps in the following
proof that the Supplement Postulate is not independent of the other axioms.
We need to show that given a linear
pair of angles that they are supplementary. Assume and
form a linear pair of angles. We must show
that
.
We do this by showing that
and
both lead to contradictions.
Case 1. Suppose .
1. There is a unique ray BE with E and C on the same side
of line AB and with .
2. .
3. Then .
4. Thus .
5. Also, .
6. Then .
7. Thus ,
which is not possible.
Case 2. Suppose .
8.
9. There is a unique ray BF with F and C on the same side
of line AB and with .
10. .
11. .
12. Thus .
13. Hence .
14. Also, .
15. Then .
16. Thus ,
which is not possible.
Thus, by Cases 1 and 2, we must have .
Exercise
2.44. Prove the Vertical Angle Theorem.
Exercise
2.45. Prove that two angles supplement to the same angle are congruent.
Exercise
2.46. (a) Prove that if two congruent adjacent angles form a linear pair,
then they are right angles.
(b) Prove that the four angles formed by two perpendicular lines are right
angles.
Exercise
2.47. Prove that a segment has a unique perpendicular bisector.
Exercise 2.48.
Find the axiom from a high school book that corresponds
to the SAS Postulate.
Exercise
2.49. Show the SAS Postulate is not satisfied by the (a) Taxicab plane;
and (b) Max-distance plane. Thus showing independence.
Exercise
2.50. Prove or disprove. If quadrilateral ABCD is such that segment BD
and segment AC intersect at a point M and M is the midpoint of both segment BD and segment AC, then
segment AB is congruent to segment CD.
Exercise
2.51. Prove or disprove. If quadrilateral ABCD has segment CD congruent
to segment CB and ray CA is the bisector of ,
then segment AB is congruent to
segment AD.
Exercise
2.52. Prove or disprove. If is a quadrilateral and
,
then
.
Exercise
2.53. State and prove each theorem.
(a) SSS Theorem; and (b) ASA
Theorem.
Exercise
2.54. Prove that all points equidistant from two points A and B are on the perpendicular bisector of segment AB.
Exercise 2.55.
For the Poincaré
Half-plane, find all lines parallel to the given line through the given point.
(a)
(2, 1) and
(b)
(2, 1) and
Exercise
2.56. Justify each step and fill in any gaps in the proof of the
Exterior Angle Theorem.
Exercise
2.57. Prove Theorem 2.12. (Hint.
Use SAS Postulate to prove the existence of a perpendicular line and the
Exterior Angle Theorem to prove uniqueness.)
Exercise
2.58. Prove Theorem 2.13.
Exercise
2.59. (a) Prove that given a line and a
point not on the line, there is a line parallel to the given line and passing
through the point
not on the line.
(b)
Prove Theorem 2.14.
Exercise
2.60. Justify each step and fill in any gaps in the proof of the
Theorem 2.15.
Exercise
2.61. State and prove the AAS Theorem for congruent triangles.
Exercise 2.62.
Prove Theorem
2.16.
Exercise
2.63. Prove Theorem 2.19.
Exercise
2.64. Prove Theorem 2.20.
Exercises 2.65.
Show the Poincaré
Half-plane does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic
geometry software to construct an example.
(b) Find an analytic example.
Exercises
2.66. Show the Poincaré
Half-plane does not satisfy
Exercise
2.67. (a) Prove five of the propositions below using the Euclidean
Parallel Postulate and
Use dynamic geometry software with the
Poincaré Half-plane for the construction investigations (Geometer's Sketchpad
or NonEuclid). The script for Geometer's Sketchpad Poincaré Half-plane is in
Appendix B of the Title Page and Index. Make sure you use the Poincaré
Half-plane construction tools for lines, rays, segments, angles, circles, etc.
Do not use the Euclidean tools.
Exercise 2.68.
Construct a triangle in the Poincaré
Half-plane. (a) Find the sum of the measures of the angles. (b) Find a triangle where the sum of the
measures of the angles is less than 5.
(c) Find a triangle where the sum of the measures is more than 175.
Exercise
2.69. (a) Construct two
perpendicular lines in the Poincaré Half-plane, as illustrated on the right.
(b) Prove the two lines are perpendicular.
(c) Is this construction valid for a neutral geometry (or Euclidean
geometry)? Explain.
Exercise
2.70. Use the procedure from Exercise 2 for this exercise. (a)
Construct a Saccheri quadrilateral Poincaré Half-plane with the base and sides
all congruent. (b) May we say that in a hyperbolic world, "Nobody is a
'square'."? Explain.
Exercise
2.71. (a) Construct a
quadrilateral with three right angles in the Poincaré Half-plane. (This called a Lambert quadrilateral after
Johann Lambert (1728–1777).) (b) What
is the measure of the fourth angle? (c)
Prove or disprove that a Lambert quadrilateral is a parallelogram. (d) Is the result of part (c) valid for a
neutral geometry (or Euclidean geometry)?
Exercise
2.72. (a) Construct two parallel
lines in the Poincaré Half-plane by constructing them perpendicular to the same
line. (b) What happens with the distance between the two parallel lines?
Exercise
2.73. Investigate the question: In the Poincaré Half-plane, do two
parallel lines have a common perpendicular? If they do not always have a common
perpendicular, how are the two parallel lines related?
Exercise
2.74.
Use the models definitions, not Geometer's Sketchpad for this exercise.
Given A(0, 2), B(0, 1), C ,
and D
in the Poincaré Half-plane. (a) Show quadrilateral ABCD is a Saccheri quadrilateral.
(b) How do the lengths of summit and base compare?
Exercise 2.75. In the Riemann Sphere, what properties are true about all lines perpendicular to a given line?
Exercise 2.76. How does a Möbius strip relate to the Modified Riemann Sphere?
Exercise 2.77. Describe how is it possible to have a triangle with three right angles.
Exercise 2.78. Find an upper bound for the sum of the measures of the angles of a triangle in the Riemann Sphere.
Exercise 2.79. Use a ball to represent the Riemann Sphere, construct a Saccheri quadrilateral on the ball. Are the summit angles acute, right, or obtuse? Is the length of the summit more or less than the length of the base? (Remember the sides of the quadrilateral must be segments of great circles.)
Exercise 2.80. Discuss the questions. Is the physical world Euclidean, hyperbolic, or elliptic? How could you prove the answer to the question? Could the physical world be proved to be hyperbolic or elliptic, but not Euclidean?
