Question

given: k is the mid - point of jl, m is the mid - point of ln, jk = mn. prove: $overline{kl}congoverline{lm}$. statements: 1. k is the mid - point of jl, m is the mid - point of ln. 2. jk = kl, lm = mn. 3. jk = mn. 4. mn = kl, lm = mn. 5. lm = kl. 6. kl = lm. 7. $overline{kl}congoverline{lm}$

Explanation:

Step1: Use mid - point property

Since \(K\) is the mid - point of \(JL\), by the definition of a mid - point, \(JK = KL\). Since \(M\) is the mid - point of \(LN\), by the definition of a mid - point, \(LM=MN\).

Step2: Given information

We are given that \(JK = MN\).

Step3: Substitution

Substitute \(JK\) with \(KL\) (from step 1) and \(MN\) with \(LM\) (from step 1) in the equation \(JK = MN\). We get \(MN = KL\) and \(LM = MN\).

Step4: Transitive property

From \(MN = KL\) and \(LM = MN\), by the transitive property of equality (\(a = b\) and \(b = c\) implies \(a = c\)), we have \(LM = KL\).

Step5: Symmetric property

The symmetric property of equality states that if \(a = b\), then \(b = a\). So if \(LM = KL\), then \(KL = LM\).

Step6: Definition of congruent segments

Two segments are congruent if they have the same length. Since \(KL = LM\), we can say \(\overline{KL}\cong\overline{LM}\).

Answer:

The proof is complete as shown above to prove \(\overline{KL}\cong\overline{LM}\).