Question

prove: ru = ts
ru = ts
ru ≅ ts
△rts ≅ △tru
rt ≅ rt
ru || ts
statements
reasons
definition of congruent segments
cpctc
asa
reflexive property of congruence
given
∠rts ≅ ∠srt and ∠srt ≅ ∠utr because of the alternating interior angles theorem
∠rts ≅ ∠tru and ∠srt ≅ ∠utr because of the alternating interior angles theorem
∠rts ≅ ∠rtu and ∠str ≅ ∠utr because of the alternating interior angles theorem
∠rts ≅ ∠tru and ∠str ≅ ∠utr because of the alternating interior angles theorem

Explanation:

Step1: Analyze the given information

We are given that $\overline{RU}\parallel\overline{TS}$ and need to prove $\overline{RU}=\overline{TS}$. Also, we know some congruence - related statements and reasons in the table.

Step2: Identify the key elements

Since $\overline{RU}\parallel\overline{TS}$, by the alternate - interior angles theorem, $\angle RTS\cong\angle TRU$ (alternate interior angles formed by parallel lines $\overline{RU}$ and $\overline{TS}$ and transversal $\overline{RT}$).

Step3: Use congruence postulate

We have $\overline{RT}\cong\overline{RT}$ (reflexive property of congruence), $\angle RTS\cong\angle TRU$ (from step 2), and assume some other congruent parts (not shown in full question but from the context of congruent - triangle proof). By the ASA (Angle - Side - Angle) congruence postulate, $\triangle RTS\cong\triangle TRU$.

Step4: Use CPCTC

Since $\triangle RTS\cong\triangle TRU$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, $\overline{RU}=\overline{TS}$.

Answer:

The correct reason for the missing part in the proof (to show the congruence of the two triangles) is the alternate - interior angles theorem which gives $\angle RTS\cong\angle TRU$ because $\overline{RU}\parallel\overline{TS}$. And the overall proof uses ASA to prove $\triangle RTS\cong\triangle TRU$ and then CPCTC to get $\overline{RU}=\overline{TS}$. The key angle - congruence relation based on parallel lines is $\angle RTS\cong\angle TRU$ due to the alternate - interior angles theorem.