Question

1. kl = lm
2. l is the mid - point of km

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

statements	reasons
2. pq = tq, uq = qs	2. def of congruence
3. pq+qs = ps, tq + qu = tu	3. segment addition property
4. tq+qs = ps	4. substitution property
5. tq+qs = tu	5. substitution property
6. ps = tu	6.
7. $overline{ps}congoverline{tu}$	7.

© gina wilson (all things algebra, llc)

Explanation:

Step1: Given equal - length segments

Given $\overline{PQ}\cong\overline{TQ}$ and $\overline{UQ}\cong\overline{QS}$, which means $PQ = TQ$ and $UQ=QS$.

Step2: Segment addition

By the segment - addition postulate, $PQ + QS=PS$ and $TQ + QU = TU$.

Step3: Substitution

Since $PQ = TQ$ and $UQ = QS$, we can substitute in the equations. In the equation $PQ+QS = PS$, substituting $PQ$ with $TQ$ gives $TQ + QS=PS$. In the equation $TQ + QU = TU$, substituting $QU$ with $QS$ gives $TQ + QS=TU$.

Step4: Transitive property

Since $PS=TQ + QS$ and $TU=TQ + QS$, by the transitive property of equality, $PS = TU$.

Step5: Definition of congruent segments

If $PS = TU$, then by the definition of congruent segments, $\overline{PS}\cong\overline{TU}$.

Answer:

1. Transitive property of equality
2. Definition of congruent segments