The Triangle and its Properties - Sum of the Lengths of Two Sides of a Triangle

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides is always strictly greater than the length of the third side. If this condition is not met, the three line segments cannot form a closed triangle.

To visualize this property, imagine trying to connect three sticks of lengths 2 cm, 3 cm, and 7 cm. Even if you lay the 2 cm and 3 cm sticks completely flat along the 7 cm stick, they will not meet because their combined length ($2 + 3 = 5$ cm) is less than 7 cm, leaving a gap between them.

Mathematically, for a triangle with side lengths $a$, $b$, and $c$, three conditions must simultaneously be true: $a + b > c$, $b + c > a$, and $c + a > b$. If even one of these inequalities is false, the triangle cannot exist.

The Difference Property of a triangle states that the difference between the lengths of any two sides of a triangle is always less than the length of the third side. This is a logical consequence of the sum property.

When the sum of two sides is exactly equal to the third side ($a + b = c$), the three points (vertices) will lie on a single straight line. In geometry, this is called collinear points, and they form a 'degenerate triangle' which is actually just a line segment, not a triangle.

To find the possible range for the length of an unknown third side $x$ when two sides $s_{1}$ and $s_{2}$ are known, $x$ must be greater than the positive difference of the two sides and smaller than the sum of the two sides. This creates a visual range on a number line where the third side can exist.

A practical shortcut to check if a triangle can be formed is to sum the two smallest side lengths. If their sum is greater than the longest side, then the other two conditions will automatically be satisfied, and a triangle is possible.