1. Let's start by defining a right-angle triangle. A right-angle triangle is a triangle in which one of the angles is exactly $90^\circ$. 2. The side opposite the right angle is called the hypotenuse, and it's the longest side in the triangle. 3. The other two sides, which form the right angle, are called the legs (or catheti). 4. To elaborate, let's say the legs are $a$ and $b$, and the hypotenuse is $c$. 5. According to the Pythagorean theorem, the relationship between these sides is given by: $$ c^2 = a^2 + b^2 $$ 6. This means that the square of the hypotenuse equals the sum of the squares of the other two sides. 7. For example, if $a = 3$ and $b = 4$, then: $$ c^2 = 3^2 + 4^2 = 9 + 16 = 25 $$ 8. Taking the square root of both sides, we get: $$ c = \sqrt{25} = 5 $$ 9. So the hypotenuse $c$ is 5 units long. 10. This right-angle triangle has sides 3, 4, and 5 units long, which is a classic Pythagorean triple. 11. This information helps in solving many geometry problems involving triangles with right angles.