Is it possible to build triangle with all the same sides but angles that are not congruent?
$\endgroup$2 Answers
$\begingroup$If you mean "two triangles who have the same set of side lengths", then the law of cosines shows that angles are determined by the three side lengths, so it is not possible.
If you really mean "all sides are the same length, does that mean the angles are congruent", then the law of cosines again determines the angles, and by symmetry all three angles have to be the same.
(All of this is assuming Euclidean geometry, btw.)
$\endgroup$ $\begingroup$No, if the triangle has all three sides equal, it is equilateral. The law of cosines (for example) then shows each angle is $\frac \pi 3$. You can also prove it from Euclidean geometry without trig.
$\endgroup$
