Question

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

Explanation:

Step1: Definition of mid - point

By the definition of a mid - point, if a point is the mid - point of a line segment, it divides the line segment into two equal parts. So if \(K\) is the mid - point of \(JL\), then \(JK = KL\), and if \(M\) is the mid - point of \(LN\), then \(LM=MN\).

Step2: Given information substitution

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

Step3: Transitive property of equality

Since \(JK = KL\) and \(JK = MN\), by the transitive property of equality, we can substitute \(JK\) in \(JK = KL\) with \(MN\) to get \(MN = KL\). Also, we know \(LM = MN\) from the mid - point definition.

Step4: Substitution

Substitute \(MN\) in \(MN = KL\) with \(LM\) (because \(LM = MN\)) to get \(LM = KL\).

Step5: Symmetric property of equality

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 line segments are congruent if they have the same length. Since \(KL = LM\), then \(\overline{KL}\cong\overline{LM}\).

Answer:

1. Definition of mid - point
2. Given
3. Substitution (using \(JK = KL\) and \(JK = MN\) and \(LM = MN\))
4. Substitution
5. Symmetric property of equality
6. Definition of congruent segments