In basic Euclidean geometry, for two line segments AB and CD, identify the condition that must hold for AB and CD to be congruent (equal in length).

Introduction / Context:

This conceptual geometry question checks your understanding of the meaning of congruence for line segments. In Euclidean geometry, congruent segments intuitively represent line segments that can be superimposed on each other perfectly, meaning they have exactly the same length. You are asked to translate this geometric idea into an algebraic condition involving their measures.

Given Data / Assumptions:

• We have two line segments, AB and CD.
• We are working in standard Euclidean geometry.
• The question asks for the condition under which AB and CD are congruent.
• Length comparisons use the usual greater than, less than, and equal signs.

Concept / Approach:

By definition, two line segments are congruent if and only if they have the same length. When we write AB, we usually mean the length of segment AB, and similarly CD for segment CD. Therefore, the condition for congruence should express that their measures are equal, nothing more and nothing less. Inequalities such as AB greater than or less than CD describe non congruent segments.

Step-by-Step Solution:

Verification / Alternative check:

Imagine drawing two segments on graph paper and measuring them with a ruler. If they have exactly the same measurement, they can be placed on top of one another and match perfectly, which is congruence. If one segment is even slightly longer or shorter, then AB > CD or AB < CD, and they are no longer congruent. This physical interpretation supports the purely algebraic condition AB = CD.

Why Other Options Are Wrong:

The condition AB > CD means AB is strictly longer than CD, so they cannot be congruent. The condition AB < CD means AB is strictly shorter, again contradicting congruence. The option AB ≥ CD allows AB to be longer, which does not always ensure equality. The choice "None of these" is incorrect because AB = CD exactly captures the definition.

Common Pitfalls:

Some learners think that "AB ≥ CD" is acceptable because equality is included, but it also allows a non congruent case where AB is longer than CD. In precise mathematics, congruent means equal only, not greater or equal. Keeping this distinction clear is important in geometry proofs and reasoning.

Final Answer:

The necessary and sufficient condition for AB and CD to be congruent is AB = CD.