= Triangle
{title2=$\triangle ABC$}
{wiki}
A <triangle> is the <convex hull> of three noncollinear <points>: the three vertices must not lie on one <line>. Its boundary consists of the three joining <line segments>; depending on context, "<triangle>" can mean this boundary or the filled region it encloses. Here a <triangle>'s <area> refers to the filled region. The noncollinearity assumption excludes a collapsed shape with zero <area>, for which ordinary interior <angles> are not defined.
For example, the positions $(0,0)$, $(4,0)$ and $(0,3)$ form a <right triangle> with a <right angle> at the origin. Taking the side of <length> $4$ as its base gives perpendicular height $3$, so its <area> is $4\cdot3/2=6$. The height is measured at a <right angle> to the chosen base; it is not generally the <length> of another side. In this example it happens to be a side because the <triangle> is a <right triangle>.
By contrast, side <lengths> $3,4,6$ define a valid <triangle>: each side is shorter than the sum of the other two, as required by the strict <triangle inequality>. But $3^2+4^2=25\ne36=6^2$, so this is not a <right triangle> and the <Pythagorean theorem> cannot calculate its longest side from the other two. The <law of cosines> instead gives the opposite interior <angle> $C$ through
$$
\cos C=\frac{3^2+4^2-6^2}{2\cdot3\cdot4}=-\frac{11}{24},\qquad C\approx117.28^\circ.
$$
This example separates the definition of a <triangle> from the additional assumption of a <right angle>.
Every <triangle> in the <Euclidean plane> has interior <angles> summing to $\pi$ <radians>, a result proved in <Euclid's Elements> I.32.https://mathcs.clarku.edu/~djoyce/elements/bookI/propI32.html[Euclid, Elements I.32]{ref} It lets two measured <angles> determine the third, and underlies the angle-angle criterion for <similar triangles>. The qualification "<Euclidean plane>" matters: the earlier <spherical triangle> demonstrates a different sum.
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