In a triangle XYZ, which one of the following conditions is always true for the side lengths (triangle inequality)? (Assume XY, YZ, and ZX are the lengths of the three sides.)

Introduction / Context:This question tests the triangle inequality theorem, one of the most fundamental rules for triangles. A triangle is possible only if any two sides together are longer than the third side. If this is not true, the shape cannot close into a triangle; the points would lie on a straight line (degenerate) or be impossible. The theorem applies to all triangles regardless of type (scalene, isosceles, equilateral) as long as side lengths are positive.

Given Data / Assumptions:

• XYZ is a valid triangle
• XY, YZ, ZX are positive side lengths
• We need a condition that is always true

Concept / Approach:Triangle inequality requires: XY + YZ > ZX YZ + ZX > XY ZX + XY > YZ Any condition using “<” is invalid for a real triangle, and “=” represents a degenerate case (not a proper triangle).

Step-by-Step Solution: Pick the option that matches triangle inequality form “sum of two sides is greater than the third”. Option “XY + YZ > ZX” exactly matches the theorem. Therefore, it is always true for every valid triangle XYZ.

Verification / Alternative check:Example triangle with sides 3, 4, 5: 3 + 4 = 7 > 5, so the condition holds. If you try a non-triangle like 2, 3, 5, then 2 + 3 = 5 which is not greater, and it becomes a straight line, confirming why “>” is necessary.

Why Other Options Are Wrong: Any “<” option contradicts triangle inequality and implies a triangle cannot be formed. The “=” option describes a degenerate case where points are collinear, not a proper triangle.

Common Pitfalls:Using “greater than or equal to” instead of strictly greater, confusing side rules with angle rules, or accidentally interpreting expressions without the plus sign correctly.

Final Answer:The always true condition is XY + YZ > ZX.