For a given figure to be a triangle, the condition that it is a union of three…

A condition P is necessary for a conclusion Q when Q being true guarantees that P is also true (Q implies P); P is sufficient for Q when P being true guarantees that Q is also true (P implies Q). By definition, a triangle is the closed plane figure obtained when three line segments are joined end to end at their endpoints so that they enclose a region.

Every triangle's boundary is made up of exactly three line segments meeting at three vertices, so whenever a figure is a triangle, it is automatically a union of three segments — the property always follows from being a triangle, so it is a necessary condition. The reverse direction fails: three segments can be arranged so that they do not close up into a triangle at all — for example if they are placed end to end along a single straight line, or left disjoint, or made to overlap — so being a union of three segments does not by itself guarantee a triangle is formed. The property is therefore not a sufficient condition.

Checking both directions confirms this: no triangle can ever be drawn without three segments as its sides, so the necessity never fails; but three collinear segments joined end to end form a straight line rather than a triangle, so a union of three segments can exist without producing a triangle. This combination — always required, never on its own enough — is exactly a necessary but not sufficient condition.

• “a sufficient but not a necessary condition” would require that any union of three segments automatically closes into a triangle, which fails for the collinear or disjoint arrangement above.

• “both necessary and sufficient condition” additionally requires every union of three segments to be a triangle, which the same counter-arrangement rules out.

• “neither necessary nor sufficient condition” would mean a triangle could exist without being made of three segments at all, which contradicts the definition of a triangle.

Hence the condition ‘is a union of three segments’ is necessary but not sufficient for a figure to be a triangle.