Short Answer 
A triangle is a polygon with three sides and angles, classified into types such as acute, obtuse, right-angled, equilateral, isosceles, and scalene. The Pythagorean Theorem is applied for right-angled triangles, stating that the square of the hypotenuse equals the sum of the squares of the other two sides, which also helps classify triangles based on side lengths.
Step 1: Understanding the Definition of a Triangle
A triangle is a polygon characterized by having three sides and three angles. The fundamental nature of a triangle allows it to be classified based on its angles and side lengths. Common types of triangles include:
- Acute Triangle: All angles are less than 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
- Right-Angled Triangle: One angle is exactly 90 degrees.
- Equilateral Triangle: All sides and angles are equal.
- Isosceles Triangle: Two sides are of equal length.
- Scalene Triangle: All sides and angles are different.
Step 2: Applying the Pythagorean Theorem
The Pythagorean Theorem is key for categorizing right-angled triangles. For any right-angled triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. This can be written mathematically as:
- If c is the hypotenuse and a and b are the other two sides, then: c² = a² + b²
Step 3: Classifying Triangles Based on Side Lengths
To determine whether a triangle is acute, right, or obtuse, follow these simple comparisons: For an obtuse triangle, the hypotenuse’s square is greater than the sum of the squares of the other two sides. For an acute triangle, the hypotenuse’s square is less than this sum. Example cases are:
- Triangle JKL: J² + L² = 9 + 16 = 25; K² = 36 (obtuse).
- Triangle XYZ: X² + Y² = 16 + 16 = 32; Z² = 25 (acute).