Question

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

Explanation:

Step1: Recall mid - point definition

If K is the mid - point of $\overline{JL}$, then $JK = KL$ by the definition of a mid - point (a mid - point of a line segment divides the segment into two congruent segments).

Step2: Use given equalities

Given $JK = MN$ and $JK = KL$, by the transitive property of equality, $KL=MN$.

Step3: Use another given equality

Given $LM = MN$ and $KL = MN$, by the transitive property of equality, $KL = LM$.

Step4: Recall congruence definition

Since $KL = LM$, then $\overline{KL}\cong\overline{LM}$ because two line segments are congruent if and only if they have the same length.

Answer:

The proof is completed as shown above, using the definition of mid - point and the transitive property of equality to show that $\overline{KL}\cong\overline{LM}$.