College Geometry help How many possible equivalence classes of triangles might exist given some specific givenmeasures of SAS (side angle side) triangle measures in these geometries?a. Euclideanb. Hyperbolicс. Тахicabd. Spherical

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Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter2: Parallel Lines

Section2.4: The Angles Of A Triangle

Problem 42E

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College Geometry help

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

Expert Solution

Step 1

In Euclidean geometry, two triangle two pairs of sides of equal length and one pair of equal angle are said to be congruent by SAS law.

Let T be the set of all triangles in Euclidean plane. The relation $≅$ congruence on T is an equivalence relation

Therefore, one equivalence class of triangles exists given some specific measures of SAS which is

$S = \{T_{i} \in T : T_{i} ≅ T_{j}\}$

In hyperbolic plane, SAS congruence is a natural postulate also. Therefore, in this plane also, two triangles with two pairs of sides of equal length and one pair of equal angle are congruent.

Therefore, there is one equivalence class of triangles in hyperbolic geometry given some specific measures of SAS which is $S = \{T_{i} \in T_{H} : T_{i} ≅ T_{j}\}$ where $T_{H}$ is the set of all triangles in this plane.

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