Final answer:
The Pythagorean Inequality Theorem for obtuse triangles states that in a triangle with an obtuse angle, the sum of the squares of the lengths of the two shorter sides is less than the square of the length of the longest side.
Step-by-step explanation:
The Pythagorean Inequality Theorem for obtuse triangles states that in a triangle with an obtuse angle (greater than 90 degrees), the sum of the squares of the lengths of the two shorter sides is less than the square of the length of the longest side. Mathematically, it can be written as a² + b² < c², where a, b, and c are the lengths of the triangle's sides and c is the length of the longest side (hypotenuse).
For example, if a triangle has side lengths of 4, 5, and 8, with 8 being the longest side, we can use the Pythagorean Inequality Theorem to check if it is an obtuse triangle. Calculating, we have 4² + 5² = 41, which is less than 8² = 64. Therefore, the triangle is an obtuse triangle.
It is important to note that this theorem only applies to obtuse triangles, which are triangles with one angle greater than 90 degrees. In right triangles, which have a 90-degree angle, the Pythagorean theorem applies.