Obtuse Angle
An obtuse angle is an angle that measures greater than 90 degrees and less than 180 degrees.
Formula
90 < \text{angle} < 180^\circ
Definition
An obtuse angle is bigger than a right angle but smaller than a straight line, measuring strictly between $90^\circ$ and $180^\circ$ ($\pi/2 < \theta < \pi$ radians); obtuse angles look wide and open, like a reclining chair leaning back past $90^\circ$. A triangle can contain at most one obtuse angle, since the three angles must sum to $180^\circ$, and the angle $C$ opposite side $c$ is obtuse if and only if $c^2 > a^2 + b^2$, equivalently $\cos C < 0$; in vector terms, $u$ and $v$ form an obtuse angle exactly when $u \cdot v < 0$.
Example
A $120^\circ$ angle is obtuse, as is the larger angle between clock hands at 10 o'clock. In a triangle with angles $30$, $50$, and $100$ degrees, the $100^\circ$ angle is obtuse, making it an obtuse triangle; the supplement of an obtuse $110^\circ$ angle is $70^\circ$ (acute). For sides $3, 5, 7$: since $49 > 9+25=34$, the angle opposite the side of length $7$ is obtuse.
Key Insight
The word "obtuse" means dull or blunt, the opposite of the sharp, pointed look of acute angles. In any obtuse triangle, the side opposite the obtuse angle is the longest side, and the law of cosines gives a value greater than $a^2+b^2$ precisely because $\cos C < 0$ when $C$ is obtuse. The sign of the dot product immediately classifies an angle between vectors as acute (positive), right (zero), or obtuse (negative), an algebraic test that replaces angle computation in many proofs and is fundamental in optimization and machine learning, such as checking gradient directions.