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. Нурerbolic c. Taxicab d. Spherical

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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
Transcribed Image Text: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=TiT:TiTj

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=TiTH:TiTj

Where TH is the set of all triangles in hyperbolic plane.

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