As a further doubt about the validity of the parallel postulate (see problem 27), one can appear to prove directly that the sum of the angles of any triangle is equal to 180°. Take any triangle and divide it arbitrarily into two triangles. The internal angles of the figure are a, b, c, d, e, and f. Let us designate as x the sum of the angles of a triangle-what we are seeking to determine.

We have a + b + f = x = c + d + e and a + b + c + d + e + f = 2x. But e and f, being adjacent and supplementary, have the sum 180°. Therefore:

a + b + c + d + 180° = 2x

x + 180° = 2x

x = 180°

Where is the error?

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Tags: angles, free puzzles, geometric puzzles, mathematics, puzzles

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