Find the angle over 90°
How one angle classifies the whole shape
An obtuse triangle is significant because one wide angle changes several geometric features at once. The longest side lies opposite the obtuse angle, and some triangle centers move outside the triangle.
This type is important in classification and theorem work because many statements about heights, centers, or perpendicular constructions behave differently once the triangle becomes obtuse.
A triangle with angles 110, 40, and 30 degrees is obtuse because of its 110-degree angle. There cannot be two obtuse angles in one triangle; two angles greater than 90 would already add to more than 180.
The longest side lies opposite the obtuse angle. This is a useful check when reading a labeled triangle, but always use the actual angle or side information rather than judging only from the outline.
An obtuse triangle may be scalene or isosceles, so its wide angle does not tell you whether any sides match.
- An obtuse triangle has one angle greater than 90 degrees.
- Only one angle in a triangle can be obtuse because the interior angles sum to one hundred eighty degrees.
- The longest side lies opposite the obtuse angle.
- In an obtuse triangle, the orthocenter and circumcenter lie outside the triangle.
A triangle that changes the usual picture
Where obtuse triangles help
- Use obtuse-triangle classification when comparing angle types in triangle problems.
- Use it in center-location discussions and altitude constructions.
- Use it to decide which geometric case applies before using a theorem or proof step.
Only one wide angle is possible
Three obtuse-triangle checks
- Do not call a triangle obtuse unless one interior angle is actually greater than ninety degrees.
- Do not forget that the other two angles must then remain acute.
- Do not assume the widest-looking side is opposite the largest angle unless the figure has been read carefully.
