Obtuse Triangle
An obtuse triangle has one interior angle measuring greater than 90 degrees.
Formula
a^2 + b^2 < c^2 \text{ (where } c \text{ is the longest side)}
Definition
An obtuse triangle has one angle larger than $90^\circ$; only one angle can be obtuse, since the three angles must sum to $180^\circ$ and two angles over $90^\circ$ would already exceed that. For sides $a, b, c$ with $c$ the longest, the triangle is obtuse if and only if $c^2 > a^2+b^2$ (equivalently $\cos\theta < 0$ for that angle), and both the circumcenter and orthocenter lie outside the triangle, on the far side of the longest side from the obtuse vertex.
Example
A triangle with angles $120^\circ$, $30^\circ$, and $30^\circ$ is obtuse because $120>90$; a very flat, wide triangle typically has an obtuse angle at its widest point. For sides $5, 5, 9$: $9^2=81$ versus $5^2+5^2=50$, and since $81>50$ the triangle is obtuse, with $\cos C = (25+25-81)/50 = -31/50$, giving $C\approx128^\circ$. For sides $4, 5, 8$: $c^2=64$ exceeds $a^2+b^2=41$, and $\cos C=(16+25-64)/40=-23/40$, so $C\approx125.1^\circ$.
Key Insight
A triangle can have at most one obtuse angle, since two angles over $90^\circ$ would already sum past $180^\circ$. In an obtuse triangle, the altitude from the obtuse vertex falls outside the triangle, on the extension of the base, which is why constructing the orthocenter sometimes requires extending the sides. In the moduli space of triangles, obtuse, right, and acute triangles are separated by the curve $a^2+b^2=c^2$, and obtuse triangles actually occupy a larger proportion of that space than acute ones.