mravinjakag

Answered

2022-06-27

Prove: If the sum of the angles of a triangle is a constant n, then $n=180$ and thus the geometry is Euclidean

Answer & Explanation

Angelo Murray

Expert

2022-06-28Added 23 answers

Step 1
Here:
1) $\angle A+\angle B+\angle C=n$ (constant)
Step 2
Two parallel lines ${l}_{1}$ and ${l}_{2}$ , such that ${l}_{1}$ passes through A and ${l}_{2}$ passes through side BC.
From the property of straight line:
$\mathrm{\angle }1+\mathrm{\angle }A+\mathrm{\angle }2={180}^{\circ }\dots \dots \left(2\right)$
Also,
From the property of alternate interior angles:
$\left\{\begin{array}{l}\mathrm{\angle }1=\mathrm{\angle }B\\ \mathrm{\angle }2=\mathrm{\angle }C\end{array}\dots \dots \left(3\right)$
Step 3
Using (2) and (3):
$\angle A+\angle B+\angle C=180°$
Hence, from (1):
$n=180$
The above figure is a 2-dimensional plane figure with sum of all angles 180.

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