An obtuse triangle is a triangle that has one obtuse angle (an angle greater than 90°).
Properties of obtuse triangles
- Whenever a triangle is classified as obtuse, one of its interior angles has a measure between 90 and 180 degrees.
- An obtuse triangle has only one angle greater than 90° since the sum of the angles in any triangle is 180°. If one of the angles is greater than 90°, then the sum of the other two angles must be less than 90°, so the other two angles must both be acute angles. For △DEF above, ∠D + ∠F = 60° < 90°, so ∠D and ∠F are acute angles.
- The side opposite the obtuse angle for an obtuse triangle is the longest side of the triangle. The greater the angle, the longer the side opposite it. Conversely, the longer the side, the greater the angle opposite it.
- The orthocenter is the point where all three altitudes of the triangle intersect. The orthocenter of an obtuse triangle is located outside of the triangle.
Using the Pythagorean Theorem to identify obtuse triangles
For a right triangle with hypotenuse of length c, and legs of lengths a and b, the Pythagorean Theorem states:
a2 + b2 = c2
For any triangle, if a2 + b2 < c2, where c is the longest side, the triangle is an obtuse triangle.
Is triangle △ABC an obtuse triangle?
Side AB above is the longest side. Plugging this into the Pythagorean Theorem where c = 9 and a and b = 6:
a2 + b2 = 62 + 62 = 72
c2 = 92 = 81
72 < 81
Since a2 + b2 < c2, △ABC is an obtuse triangle with obtuse angle C.