Parallel lines, triangle angles and the triangle inequality

The shortest way between two points is a straight line, so two sides of a triangle together must be longer than the third. A triangle with sides a, b and c exists only when a + b > c, a + c > b and b + c > a. It is enough to check that the two shorter sides add up to more than the longest side. If the sum is exactly equal, the sides lie flat on one line. If a triangle has two sides of 5 cm and 9 cm, the third side x must be greater than the difference 9 − 5 = 4 and smaller than the sum 5 + 9 = 14, that is 4 < x < 14. To construct a triangle with sides 3, 4 and 5 cm: draw AB = 5 cm; with the compass point at A draw an arc of radius 4 cm; with the point at B draw an arc of radius 3 cm; the arcs meet at C; join AC and BC.