We are given a triangle with sides $x$, $x + 1$, and $x + 2$, and we know that one of the angles of the triangle is 90 degrees, which means the triangle is a right-angled triangle.

Step 1: Identify the right-angle property

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras' Theorem). The hypotenuse is always the longest side. Here, the longest side is $x + 2$, so it must be the hypotenuse.

Thus, applying Pythagoras' Theorem:

$(x + 2)^2 = x^2 + (x + 1)^2$

Step 2: Expand and simplify the equation

Let's expand both sides of the equation.

Left-hand side:

$(x + 2)^2 = x^2 + 4x + 4$

Right-hand side:

$x^2 + (x + 1)^2 = x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1$

Now, equate the two sides:

$x^2 + 4x + 4 = 2x^2 + 2x + 1$

Step 3: Rearrange the terms

Move all terms to one side of the equation:

$x^2 + 4x + 4 - 2x^2 - 2x - 1 = 0$

Simplify:

$-x^2 + 2x + 3 = 0$

Multiply the whole equation by -1 to make the leading coefficient positive:

$x^2 - 2x - 3 = 0$

Step 4: Solve the quadratic equation

Factor the quadratic equation:

$(x - 3)(x + 1) = 0$

So, $x = 3$ or $x = -1$.

Step 5: Consider the positive value for side lengths

Since $x$ represents a side length, it must be positive. Therefore, we discard $x = -1$ and conclude that:

$x = 3$

Final Answer:

The value of $x$ is $3$ cm.

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Would you like to explore more about this problem or ask for additional details? Here are some related questions:

1. How do you apply the Pythagorean theorem to different types of triangles?
2. What other methods can be used to find the side lengths of a right-angled triangle besides Pythagoras' theorem?
3. How can you check if a triangle is a right-angled triangle using the side lengths?
4. Can a triangle with sides 3 cm, 4 cm, and 5 cm also satisfy the Pythagorean theorem?
5. What happens if one of the sides of a right-angled triangle is not the hypotenuse?

Tip: Always check the validity of your solutions by considering the physical meaning of the problem (e.g., side lengths must be positive).