Question

u is the midpoint of \\(\overline{st}\\). complete the proof that \\(\triangle rtu \cong \triangle rsu\\).

statement	reason
2 \\(\overline{rt} \cong \overline{rs}\\)	given
3 \\(\overline{su} \cong \overline{tu}\\)
4 \\(\overline{ru} \cong \overline{ru}\\)
5 \\(\triangle rtu \cong \triangle rsu\\)

Explanation:

Step1: Analyze Statement 3

Statement 3 is \(\overline{SU} \cong \overline{TU}\). Since \(U\) is the midpoint of \(\overline{ST}\) (from Statement 1), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So the reason for \(\overline{SU} \cong \overline{TU}\) is the definition of a midpoint.

Step2: Analyze Statement 4

Statement 4 is \(\overline{RU} \cong \overline{RU}\). This is the reflexive property of congruence, which states that any segment is congruent to itself. So the reason for \(\overline{RU} \cong \overline{RU}\) is the reflexive property of congruence.

Step3: Analyze Statement 5

To prove \(\triangle RTU \cong \triangle RSU\), we have:

• \(\overline{RT} \cong \overline{RS}\) (Statement 2, Given)
• \(\overline{SU} \cong \overline{TU}\) (Statement 3, Definition of Midpoint)
• \(\overline{RU} \cong \overline{RU}\) (Statement 4, Reflexive Property)

By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So the reason for \(\triangle RTU \cong \triangle RSU\) is SSS Congruence Criterion.

Answer:

• For Statement 3 (\(\overline{SU} \cong \overline{TU}\)): Reason is "Definition of Midpoint"
• For Statement 4 (\(\overline{RU} \cong \overline{RU}\)): Reason is "Reflexive Property of Congruence"
• For Statement 5 (\(\triangle RTU \cong \triangle RSU\)): Reason is "SSS Congruence Criterion"