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1. If two adjacent side lengths are integer numbers, then with the concluded angle between them of 30°, 45°, 135° and/or 150°
the opposite side can not have integer length due to the cosine theorem.
It can not even have the length expressed by a rational number.
It follows from the cosine theorem (as I just said) and from the fact that
cos(30°) = , cos(45°) = ), cos(135°) = and cos(150°) =
are irrational numbers.
So, it kills all the opportunities for angles 30°, 45°, 135° and/or 150°.
2. Regarding the angles 60° and 120°, such triangles also are impossible, since their area
(which is half the product of two adjacent side lengths by sin(60°) or sin(120°))
also can not be integer, because sin(60°) = sin(120°)) = is an irrational number.
Answer. The number of triangles under the question is 0 (zero, ZERO).
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Good problem for the day of April, 1.
April, 1 is an International Fools' day
For further info see this Wikipedia article
https://en.wikipedia.org/wiki/April_Fools%27_Day