Triangles - Inequalities in a Triangle

Angle-Side Relationship (Longer Side): If two sides of a triangle are unequal, the angle opposite to the longer side is greater. For example, in a triangle where side $AC$ is clearly longer than side $AB$, the angle facing $AC$ (which is $\angle B$) will be larger than the angle facing $AB$ (which is $\angle C$).

Side-Angle Relationship (Greater Angle): In any triangle, the side opposite to the larger angle is longer. Visually, if you widen the opening of an angle in a triangle, the side connecting the two rays of that angle must grow longer to close the triangle.

The Triangle Inequality Theorem: The sum of any two sides of a triangle is always greater than the third side. If you have three segments of lengths $a$, $b$, and $c$, they can only form a closed triangle if $a + b > c$, $b + c > a$, and $a + c > b$. If the sum of two sides were equal to the third, the 'triangle' would collapse into a straight line.

The Difference Property: The difference between any two sides of a triangle is always less than the third side. Mathematically, for sides $a$, $b$, and $c$, $|a - b| < c$. This ensures that the two sides are long enough to actually meet when connected to the third side.

Perpendicular Distance: From a point not on a given line, the perpendicular line segment drawn to the line is the shortest of all line segments that can be drawn. This is because any other segment forms a right-angled triangle where the perpendicular is one side and the other segment is the hypotenuse; since the hypotenuse is opposite the $90^\circ$ angle (the largest angle), it must be the longest side.

Side Length Range: For any triangle with known sides $a$ and $b$, the length of the third side $x$ must fall within the range $|a - b| < x < a + b$. Visually, this represents the third side being longer than the 'gap' between the two known sides but shorter than the two sides laid out in a straight line.