Geometry Problem Solver What is an obtuse angle?
An obtuse angle is any angle that measures more than 90° but less than 180°. It is wider than a square corner and still short of a straight line. 91° and 179° both qualify; 90° and 180° do not.
How it compares to the other angles
Angles get their names from where they fall between 0° and 360°. Obtuse is one of five bands, and the boundaries are exact: an angle of exactly 90° is right, not obtuse, and exactly 180° is straight, not obtuse.
How to spot one without measuring
Put the corner of a sheet of paper into the angle, lining one edge up with one arm. If the other arm of the angle falls outside the paper's corner, the angle opens past 90°, so it is obtuse. If the arm disappears under the paper, the angle is acute.
This works because the corner of any cut sheet is a right angle. It is the same trick as a protractor, with worse resolution and no equipment.
Where obtuse angles show up in problems
A triangle can have at most one obtuse angle. Since the three angles add to 180°, a second angle over 90° would already overshoot on its own. A triangle with one is called an obtuse triangle.
In a parallelogram that is not a rectangle, two of the four angles are obtuse and the other two are acute, and each obtuse angle is supplementary to the acute one next to it.
The obtuse angle is also where the law of cosines earns its keep: with an angle over 90°, its cosine is negative, which is what makes the side opposite it longer than the Pythagorean theorem alone would predict.
The five angle names, by measure
| Name | Measure | Looks like |
|---|---|---|
| Acute | More than 0°, less than 90° | A narrow wedge |
| Right | Exactly 90° | A square corner |
| Obtuse | More than 90°, less than 180° | An open corner, not flat |
| Straight | Exactly 180° | A straight line |
| Reflex | More than 180°, less than 360° | The larger way around |
Questions
Is 90° an obtuse angle?
No. 90° is a right angle exactly. Obtuse starts above 90°, so the smallest whole-degree obtuse angle is 91°.
Can a triangle have two obtuse angles?
No. Two angles over 90° would sum to more than 180° by themselves, leaving nothing for the third angle. Every triangle has at most one.
What is the difference between an obtuse and a reflex angle?
An obtuse angle stops short of a straight line, so it is under 180°. A reflex angle goes past the straight line, between 180° and 360°. If you measure the wide side of a sharp corner, you are reading a reflex angle.
Is an obtuse angle always inside a shape?
No. Any two rays sharing an endpoint make an angle, shape or no shape. In polygons, though, an interior angle over 180° makes the shape concave, and that angle is reflex rather than obtuse.
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
