Chapter 16 — Answers
Selected answers for Chapter 16: Non-Euclidean Geometries, transcribed from the book’s appendix. They are meant for checking your work, not for copying — the book prints answers to selected exercises only.
Back to Chapter 16 · Teacher’s answers: Chapter 16 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF
Lesson 1 (pages 692–695)
1. At the center of the sphere.
2. A radius of the sphere.
3. Two hemispheres.
7. Euclidean geometry.
10. Both geometries.
16. 186°.
18. ∠A = 72°, ∠B = ∠C = 60°; ∠A + ∠B + ∠C = 192°.
22. Yes.
25. Arcs of great circles.
30. Both curves cannot correspond to lines.
31. One.
32. No.
40. No. The sum of the angles of △ABC is greater than 180°.
45. It has three right angles rather than one.
46. It is four times as long.
Lesson 2 (pages 698–700)
1. Its legs.
2. The base angles.
3. The summit.
5. WY and MO.
6. ∠Y and ∠O.
7. Two points determine a line.
9. All right angles are equal.
14. The Ruler Postulate.
16. It is a birectangular quadrilateral whose legs are equal.
17. The summit angles of a Saccheri quadrilateral are equal.
20. An exterior angle of a triangle is greater than either remote interior angle.
23. A Saccheri quadrilateral.
25. The fact that ∠D > ∠C.
26. If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order.
29. CD.
31. DM = CM because △ADM ≅ △BCM (SAS).
33. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.
38. DA.
40. The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them.
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42. In a plane, two lines perpendicular to a third line are parallel.
44. The opposite angles of a parallelogram are equal.
Lesson 3 (pages 704–706)
1. The summit angles of a Saccheri quadrilateral are acute in Lobachevskian geometry.
3. Substitution.
4. A line segment has exactly one midpoint.
6. The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them.
7. A quadrilateral that has two sides perpendicular to a third side is birectangular.
9. The Addition Theorem of Inequality.
14. A birectangular quadrilateral whose legs are equal is a Saccheri quadrilateral.
15. The summit of a Saccheri quadrilateral is longer than its base in Lobachevskian geometry.
20. Through a point not on a line, there is more than one line parallel to the line.
28. Birectangular.
34. In Riemannian geometry, the summit of a Saccheri quadrilateral is shorter than its base.
37. They are all half as long.
42. They are congruent squares.
Lesson 4 (pages 709–712)
2. ∠D and ∠E.
4. A Saccheri quadrilateral.
5. The summit angles of a Saccheri quadrilateral are acute in Lobachevskian geometry.
8. Substitution.
10. The sum of the angles of a triangle is less than 180° in Lobachevskian geometry.
14. Corresponding angles of similar triangles are equal.
17. The angles in a linear pair are supplementary.
21. No. The sums of the angles of the two triangles may not be equal.
26. ∠A + ∠B + ∠C > 180°.
27. Riemannian geometry.
31. ∠1 = ∠A + ∠B.
35. △ADE ∼ △ABC (AA).
41. ∠F is a right angle.
42. CFDE is a birectangular quadrilateral (in two different ways).
43. ∠FDE is acute.
46. DE > 2BC.
Review (pages 714–717)
1. A great circle.
2. False.
4. True (because antipodal points are considered to be identical).
7. AD.
12. 360°.
16. ∠ABC = 90°; ∠BCD = 120°; ∠CDA = 90°; ∠DAB = 120°.
17. The quadrilateral is a parallelogram.
19. It is a rhombus.
23. No.
26. Saccheri quadrilaterals.
27. Acute.
32. Substitution.
34. A midsegment of a triangle is less than half as long as the third side.
37. Two points determine a line.
40. The sum of the angles of a triangle is less than 180°.
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