Question

given: k is the midpoint of jl, m is the midpoint of ln, jk = mn
prove: \\(\overline{kl} \cong \overline{lm}\\)

statements	reasons
2. jk = kl, lm = mn	2. definition of midpoint
3. jk = mn	3.
4. mn = kl, lm = mn	4.
5. lm = kl	5.
6. kl = lm	6.
7. \\(\overline{kl} \cong \overline{lm}\\)	7.

Explanation:

Step1: Reason for Statement 3

Statement 3 is \( JK = MN \), which is given in the problem. So the reason is "Given".

Step2: Reason for Statement 4

From Step 2, we have \( JK = KL \) and \( LM = MN \), and from Step 3, \( JK = MN \). By substitution (replacing \( JK \) with \( MN \) in \( JK = KL \)), we get \( MN = KL \), and we already know \( LM = MN \) from Step 2. So the reason is "Substitution Property (from Steps 2 and 3)".

Step3: Reason for Statement 5

From Step 4, we have \( MN = KL \) and \( LM = MN \). By the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \); here \( LM = MN \) and \( MN = KL \), so \( LM = KL \)), the reason is "Transitive Property of Equality".

Step4: Reason for Statement 6

Statement 6 is \( KL = LM \), which is just the symmetric property of equality (if \( a = b \), then \( b = a \); from \( LM = KL \), we get \( KL = LM \)). So the reason is "Symmetric Property of Equality".

Step5: Reason for Statement 7

To prove \( \overline{KL} \cong \overline{LM} \), we use the definition of congruent segments: if two segments have equal length, then they are congruent. Since \( KL = LM \) (from Step 6), the reason is "Definition of Congruent Segments (if \( KL = LM \), then \( \overline{KL} \cong \overline{LM} \))".

Answer:

1. Given
2. Definition of Midpoint
3. Given
4. Substitution Property (from Steps 2 and 3)
5. Transitive Property of Equality
6. Symmetric Property of Equality
7. Definition of Congruent Segments