Not Every Three Lengths Can Meet A Tale of Three Intersecting Lines

Picture a tent, a tree, and a pole standing on a plain field, and someone deciding whether to walk directly between the tent and the pole, or to walk via the tree instead. The direct path is always shorter than or equal to the path through any third point -- and strictly shorter whenever the three points don't happen to fall in a straight line. Testing this with concrete lengths (10 cm, 15 cm, and 30 cm as three attempted sides of a triangle) shows it can genuinely fail to form a triangle at all: 10 plus 15 is only 25, which doesn't even reach 30, so the two shorter sides could never meet up to close the shape. This single requirement, that the sum of any two sides must exceed the third, is called the Triangle Inequality, and it decides completely whether three given lengths can ever form a real triangle.

Drawing two circles, one centred at each end of a fixed base length, with radii equal to the two remaining candidate side lengths, reveals visually exactly when a triangle is possible: if the two circles' radii add up to less than the distance between their centres, the circles never touch at all, and no triangle can close; if the radii add up to exactly the base length, the circles just barely touch at one point, giving a completely flat, zero-area 'triangle'; and only when the radii together exceed the base length do the circles genuinely cross at two points, each giving a real, proper triangle. This is the same Triangle Inequality rule, seen from a different, entirely visual angle.

A triangle can also be built knowing only two of its sides plus the angle trapped between them (constructed directly: draw one side, mark the given angle at one end using a protractor, then measure the second side's length along that new ray, and finally join up the third side) -- and separately, from just two angles plus the one side that sits between them (draw the shared side first, then mark both given angles at its two ends, extending each ray until they cross to complete the triangle). This second method has its own hidden existence condition: since the third angle must come out positive, the two given angles must add up to strictly less than 180 degrees, or the two rays drawn from either end will never cross to form a triangle at all.

Drawing a line through one vertex of a triangle, parallel to the opposite side, reveals why the three angles of ANY triangle always add up to exactly 180 degrees: the two angles this new line makes with the triangle's two other sides are each equal (by the alternate-angle rule from parallel lines) to one of the triangle's own two base angles, and all three angles (the two copied ones plus the triangle's own top angle) sit together along the newly drawn straight line, which is itself always 180 degrees. This elegant argument, connecting parallel lines to triangle angles, traces back to Euclid's Elements, a hugely influential book written around 300 BCE.

Extending one side of a triangle past a vertex creates an exterior angle, and this exterior angle always equals the sum of the triangle's two remote interior angles (the two angles NOT adjacent to it) -- a direct consequence of the angle sum property, since the exterior angle and its adjacent interior angle always form a straight line (180 degrees) together, while the interior angle sum is also 180 degrees, forcing the exterior angle to match whatever's left over once the adjacent angle is subtracted from both.