College Geometry How many possible equivalence classes of triangles might exist given some specific givenmeasures of SSS (side side side) triangle measures in these geometries?a. Euclideanb. Hyperbolicc. Taxicabd. Spherical

Answered: How many possible equivalence classes…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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College Geometry

How many possible equivalence classes of triangles might exist given some specific given
measures of SSS (side side side) triangle measures in these geometries?
a. Euclidean
b. Hyperbolic
c. Taxicab
d. Spherical

Expert Solution

Step 1

In Euclidean geometry, if three sides of of two triangles are equal in length, then they are said to be congruent by S-S-S rule.

Let T be the set of all triangles in Euclidean plane. Then the relation $≅$ congruence by S-S-S rule is an equivalence relation on T

So, in Euclidean geometry, there is one equivalence class of triangles exists given some specific measures of SSS which is $S = \{T_{i} \in T : T_{i} ≅ T_{j}\}$

If three pairs of sides of two triangles are equal in length, then they are congruent in hyperbolic plane.

Therefore, in Hyperbolic geometry, there is one equivalence class of triangles exists given some specific measures of SSS which is

$S = \{T_{i} \in T_{H} : T_{i} ≅ T_{j}\}$

Where $T_{H}$ is the set of all triangles in hyperbolic plane.

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