Question

1. a segment exists with endpoints a and b. 2. given 3. 4. ab = ab ab equals the length of the segment with endpoints a and b. definition of congruent segments reflexive property of equality ruler postulate (post. 1.1) 4. ab ≅ ab statements reasons complete the two - column proof for the reflexive property of segment congruence (thm. 2.1).

Explanation:

Step1: State the given fact

A segment with endpoints A and B exists. This is the starting - point of our proof, so the reason is "Given".

Step2: Apply the Ruler Postulate

By the Ruler Postulate, we can measure the length of the segment AB. And the length of a segment is equal to itself, so $AB = AB$. The reason is "Ruler Postulate (Post. 1.1)".

Step3: Recall the definition of congruent segments

Congruent segments have equal lengths. Since $AB = AB$, we can say that $\overline{AB}\cong\overline{AB}$ based on the definition of congruent segments. The reason is "Definition of congruent segments".

Step4: State the reflexive property of equality

The statement $AB = AB$ also represents the reflexive property of equality. But in the context of segment congruence, we use the definition of congruent segments to go from $AB = AB$ to $\overline{AB}\cong\overline{AB}$.

Answer:

1. Statements: A segment exists with endpoints A and B. Reasons: Given
2. Statements: $AB = AB$. Reasons: Ruler Postulate (Post. 1.1)
3. Statements: $\overline{AB}\cong\overline{AB}$. Reasons: Definition of congruent segments
4. Statements: N/A (already completed the proof). Reasons: N/A