Question

1. e is the mid - point of df
2. given
3. de = ef
4. def. of midpoint
5. de + de = de+ef
6. add. prop
7. 2de = de + ef
8. simplify
9. de + ef = df
10. segment add.
11. 2de = df
12. transitive

2 given: $overline{kl}congoverline{ln}$, $overline{lm}congoverline{ln}$
prove: l is the mid - point of $overline{km}$
statements reasons

1. $overline{kl}congoverline{ln}$, $overline{lm}congoverline{ln}$ 1. given
2. kl = ln, lm = ln 2.
3. kl = lm 3.
4. l is the mid - point of $overline{km}$ 4.

3 given: $overline{pq}congoverline{tq}$, $overline{uq}congoverline{qs}$
prove: $overline{ps}congoverline{tu}$

Explanation:

Step1: Convert congruence to equality

If two line - segments are congruent, their lengths are equal. So, if $\overline{KL}\cong\overline{LN}$ and $\overline{LM}\cong\overline{LN}$, then $KL = LN$ and $LM=LN$ because of the "Def. of congruent segments".

Step2: Use transitive property

Since $KL = LN$ and $LM = LN$, by the transitive property of equality (if $a = c$ and $b = c$, then $a = b$), we have $KL=LM$.

Step3: Apply mid - point definition

A point $L$ on a line - segment $\overline{KM}$ is the mid - point if it divides $\overline{KM}$ into two equal parts. Since $KL = LM$, by the "Def. of mid - point", $L$ is the mid - point of $\overline{KM}$.

Answer:

1. Def. of congruent segments
2. Transitive property of equality
3. Def. of mid - point