Geometry Solution Manual | Reference Guide Unit 11 | Spherical Geometry
Spherical Geometry
Solutions
1. Non-Euclidean geometry is any geometry that
deviates from the postulates in Euclid’s Elements.
Non-Euclidean geometries, such as spherical
geometry, have resulted from negations of the
Parallel Postulate. In Euclidean geometry, the
Parallel Postulate states that given a line and a
point not on the line, there is exactly one line
through the given point that is parallel to the
given line. You can negate the Parallel Postulate
using this statement: given a line and a point
not on a line, there is no line that contains the
given point and is parallel to the given line. This
negation leads to a logically consistent nonEuclidean geometry called spherical geometry.
2. First, spherical geometry is derived from negating
of the Parallel Postulate. This negation leads to
the following differences between Euclidean and
spherical geometries:
(1) In Euclidean geometry, the most basic objects
are points and lines. In spherical geometry, the
most basic objects are points and great circles on
a sphere.
(2) In Euclidean geometry, any two points
define a line. In spherical geometry, any two
nonantipodal points define a great circle.
(3) In Euclidean geometry, the shortest distance
between any two points lies along a straight
line. In spherical geometry, the shortest distance
between any two points lies along a great circle.
(4) In Euclidean geometry, any two nonparallel
lines determine exactly one point. In spherical
geometry, any two great circles determine exactly
two antipodal points.
(5) In Euclidean geometry, there are parallel lines.
In spherical geometry, there are no parallel great
circles.
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3. (1) A Euclidean triangle is determined by three
noncoplanar points. A spherical triangle is
determined by three points that do not all lie on
the same great circle.
(2) The sides of a Euclidean triangle are line
segments. The sides of a spherical triangle are
arcs that lie along great circles.
(3) A Euclidean triangle’s angle measures sum
to exactly 180°. A spherical triangle’s angle
measures sum to a value between 180° and 540°.
4. The shortest distance between two points lies along
a geodesic. In Euclidean geometry, a geodesic is
a straight line. In spherical geometry, a geodesic
is a great circle. Since the earth is approximately
spherical in shape, the shortest distance between
points on the earth lies along a great circle.
Therefore, flight routes are often charted along the
great circles that two locations share.
5. In spherical geometry, a great circle is one of
the most basic objects. A great circle is a cross
section of a sphere that has been intersected by
a plane passing through the sphere’s center. It is
a circle on the surface of a sphere whose center
coincides with the sphere’s center.
6. A geodesic is the path along which the shortest
distance between two points lies. In Euclidean
geometry, a geodesic is a line. In spherical
geometry, a geodesic is a great circle.
7. A. These points determine a spherical triangle
because they do not all lie on the same
great circle.
B. These points do not determine a spherical
triangle because they all lie on the same
great circle.
C. These points do not determine a spherical
triangle because they all lie on the same
great circle.
D. These points determine a spherical triangle
because they do not all lie on the same
great circle.
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Geometry Solution Manual | Reference Guide Unit 11 | Spherical Geometry
8. No; it is impossible for great circles to be parallel to
one another. Any two great circles intersect at exactly
two antipodal points on the surface of a sphere.
9. A. 180° < angle sum < 540°
B.
C.
D.
E.
F.
180° < m∠1 + m∠2 + m∠3 < 540°
180° < 32° + 58° + m∠3 < 540°
180° < 90° + m∠3 < 540°
90° < m∠3 < 450°
90° < m∠3 < 360°
180° < angle sum < 540°
180° < m∠1 + m∠2 + m∠3 < 540°
180° < m∠1 + 90° + 43° < 540°
180° < m∠1 + 133° < 540°
47° < m∠1 < 407°
47° < m∠1 < 360°
180° < angle sum < 540°
180° < m∠1 + m∠2 + m∠3 < 540°
180° < 56° + m∠2 + 72° < 540°
180° < m∠2 + 128° < 540°
52° < m∠2 < 412°
52° < m∠2 < 360°
180° < angle sum < 540°
180° < m∠1 + m∠2 + m∠3 < 540°
180° < 92° + 67° + m∠3 < 540°
180° < 159° + m∠3 < 540°
21° < m∠3 < 381°
21° < m∠3 < 360°
180° < angle sum < 540°
180° < m∠1 + m∠2 + m∠3 < 540°
180° < m∠1 + 90° + 103° < 540°
180° < m∠1 + 193° < 540°
−13° < m∠1 < 347°
0° < m∠1 < 347°
180° < angle sum < 540°
180° < m∠1 + m∠2 + m∠3 < 540°
180° < 85° + m∠2 + 113° < 540°
180° < m∠2 + 198° < 540°
−18° < m∠2 < 342°
0° < m∠2 < 342°
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Copying or distributing without K12’s written consent is prohibited.
10. A. 360° < angle sum < 720°
360° < m∠5 + m∠6 + m∠7 + m∠8 < 720°
360° < 92° + 79°+ 89° + m∠8 < 720°
360° < 260° + m∠8 < 720°
100° < m∠8 < 460°
100° < m∠8 < 360°
B. 360° < angle sum < 720°
360° < m∠5 + m∠6 + m∠7 + m∠8 < 720°
360° < 101° + 82° + m∠7 + 94° < 720°
360° < 277° + m∠7 < 720°
83° < m∠7 < 443°
83° < m∠7 < 360°
C. 360° < angle sum < 720°
360° < m∠5 + m∠6 + m∠7 + m∠8 < 720°
360° < 105° + m∠6 + 98° + 101° < 720°
360° < 304° + m∠6 < 720°
56° < m∠6 < 416°
56° < m∠6 < 360°
D. 360° < angle sum < 720°
360° < m∠5 + m∠6 + m∠7 + m∠8 < 720°
360° < m∠5 + 142° + 150° + 120° < 720°
360° < m∠5 + 412° < 720°
−52° < m∠5 < 308°
0° < m∠5 < 308°
11. The geodesic is the line that is the shortest
distance between two points. In spherical
geometry, the line created by the great circle is
the geodesic, which is a shorter distance than
any other line. Therefore, the great circle line is a
shorter distance than the latitudinal line.
12. Given a number of sides n, the minimum of the
sum of the interior angles is 180(n − 2)° and
the maximum of the sum of the interior angles
is 180n°. A trip with 5 flights forms a pentagon,
which has 5 sides. Therefore, for this trip, the
minimum of the sum of the interior angles is
180(n − 2)° = 180(5 − 2)° = 540°. The maximum
of the interior angles is 180n° = 180(5)° = 900°.
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