Question

statements reasons

1. $overline{pq}congoverline{tq},overline{uq}congoverline{qs}$ 1. given
2. $pq = tq,uq = qs$ 2.
3. $pq + qs=ps;tq + qu = tu$ 3.
4. $tq + qs = ps$ 4.
5. $tq + qs = tu$ 5.
6. $ps = tu$ 6.
7. $overline{ps}congoverline{tu}$ 7.

Explanation:

Step1: Definition of congruent segments

If two segments are congruent, their lengths are equal. So if $\overline{PQ}\cong\overline{TQ}$ and $\overline{UQ}\cong\overline{QS}$, then $PQ = TQ$ and $UQ=QS$.

Step2: Segment - addition postulate

The sum of the lengths of two adjacent segments on a line - segment is equal to the length of the entire segment. So $PQ + QS=PS$ and $TQ + QU = TU$.

Step3: Substitution property

Since $PQ = TQ$ (from step 1), we can substitute $PQ$ with $TQ$ in the equation $PQ + QS=PS$ to get $TQ + QS=PS$.

Step4: Substitution property

Since $UQ = QS$ (from step 1), we can substitute $UQ$ with $QS$ in the equation $TQ + QU = TU$ to get $TQ + QS=TU$.

Step5: Transitive property of equality

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

Step6: Definition of congruent segments

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

Answer:

1. Definition of congruent segments
2. Segment - addition postulate
3. Substitution property
4. Substitution property
5. Transitive property of equality
6. Definition of congruent segments