Question

complete the proof that \\(\overline{rt} \cong \overline{rs}\\).

	statement	reason
2	\\(\angle s \cong \angle t\\)
3	\\(\angle rus \cong \angle rut\\)
4	\\(\overline{ru} \cong \overline{ru}\\)
5	\\(\triangle rsu \cong \triangle rtu\\)
6	\\(\overline{rt} \cong \overline{rs}\\)

Explanation:

Step1: Reason for \(\overline{RU} \perp \overline{ST}\)

This is given (or from the diagram's right angle mark at \(U\)), so Reason: Given (or "Definition of Perpendicular Lines" as they form right angles).

Step2: Reason for \(\angle S \cong \angle T\)

From the diagram, the arcs on \(\angle S\) and \(\angle T\) indicate they are congruent, so Reason: Given (marked angles are congruent).

Step3: Reason for \(\angle RUS \cong \angle RUT\)

Since \(\overline{RU} \perp \overline{ST}\), both \(\angle RUS\) and \(\angle RUT\) are right angles. All right angles are congruent, so Reason: All right angles are congruent.

Step4: Reason for \(\overline{RU} \cong \overline{RU}\)

This is the Reflexive Property of Congruence (a segment is congruent to itself), so Reason: Reflexive Property of Congruence.

Step5: Reason for \(\triangle RSU \cong \triangle RTU\)

We have \(\angle S \cong \angle T\), \(\angle RUS \cong \angle RUT\), and \(\overline{RU} \cong \overline{RU}\), so by the AAS (Angle - Angle - Side) Congruence Theorem, \(\triangle RSU \cong \triangle RTU\). Reason: AAS Congruence Theorem.

Step6: Reason for \(\overline{RT} \cong \overline{RS}\)

Corresponding parts of congruent triangles are congruent (CPCTC). Since \(\triangle RSU \cong \triangle RTU\), their corresponding sides \(\overline{RT}\) and \(\overline{RS}\) are congruent. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Reason: Given (or Definition of Perpendicular Lines)
2. Reason: Given (marked angles are congruent)
3. Reason: All right angles are congruent
4. Reason: Reflexive Property of Congruence
5. Reason: AAS Congruence Theorem
6. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)