Obtuse Angle
GeometryAn 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 less than a straight line. It measures between $90^\circ$ and $180^\circ$. Obtuse angles look wide and open, like a reclining chair leaning back past $90^\circ$.
Example
A $120^\circ$ angle is obtuse. When clock hands show 10 o'clock, the larger angle between them is obtuse. The opening angle of most open laptop screens is obtuse.
Key Insight
The word "obtuse" means dull or blunt. Obtuse angles look wide and blunt - the opposite of the sharp, pointed look of acute angles. A triangle can have at most one obtuse angle.
Definition
An obtuse angle has measure strictly between $90^\circ$ and $180^\circ$ ($90 < \theta < 180$). A triangle can contain at most one obtuse angle because the three angles must sum to $180^\circ$. The supplement of an obtuse angle is acute.
Example
In a triangle with angles $40$, $60$, and $80$ degrees - no obtuse angle. In a triangle with angles $30$, $50$, and $100$ degrees - the $100^\circ$ angle is obtuse, making it an obtuse triangle. An obtuse angle's supplement: $180 - 110 = 70$ degrees (acute).
Key Insight
In any obtuse triangle, the side opposite the obtuse angle is the longest side. The law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ gives a value greater than $a^2 + b^2$ when angle $C$ is obtuse, because $\cos C < 0$.
Definition
An obtuse angle satisfies $\pi/2 < \theta < \pi$ radians. In a triangle with sides $a, b, c$, the angle $C$ opposite side $c$ is obtuse iff $c^2 > a^2 + b^2$ (equivalently, $\cos C < 0$). In the dot product framework, vectors $u$ and $v$ form an obtuse angle iff $u \cdot v < 0$.
Example
Triangle with sides $3, 4, 7$: check largest angle (opposite side $7$): $49 > 9 + 16 = 25$, so yes, that angle is obtuse. Actually $3+4=7$ fails the triangle inequality, so adjust: sides $3, 5, 7$: $49 > 9+25=34$, so the angle opposite the side of length $7$ is obtuse.
Key Insight
The sign of the dot product immediately classifies the angle between vectors as acute (positive), right (zero), or obtuse (negative). This algebraic test replaces angle computation in many proofs and is fundamental in optimization and machine learning (e.g., gradient direction checks).