Question

△qrt is equilateral, \\(\overline{rs} \cong \overline{qu}\\), and \\(\angle srt \cong \angle tqu\\). complete the proof that \\(\triangle rst \cong \triangle qut\\).

statement	reason
2 \\(\overline{rs} \cong \overline{qu}\\)
3 \\(\angle srt \cong \angle tqu\\)
4 \\(\overline{qt} \cong \overline{rt}\\)
5 \\(\triangle rst \cong \triangle qut\\)

Explanation:

Step1: Reason for Statement 1

Given that \(\triangle QRT\) is equilateral, by the definition of an equilateral triangle, all its sides are equal. So \(QT = RT\) (sides of an equilateral triangle are congruent).

Step2: Reason for Statement 2

The problem states that \(\overline{RS} \cong \overline{QU}\), so this is given.

Step3: Reason for Statement 3

The problem states that \(\angle SRT \cong \angle TQU\), so this is given.

Step4: Reason for Statement 4

Since \(\triangle QRT\) is equilateral, \(QT = RT\) (all sides of an equilateral triangle are congruent), so \(\overline{QT} \cong \overline{RT}\) by the definition of congruent segments (segments with equal length are congruent).

Step5: Reason for Statement 5

We have \(\overline{RS} \cong \overline{QU}\) (Statement 2), \(\angle SRT \cong \angle TQU\) (Statement 3), and \(\overline{RT} \cong \overline{QT}\) (Statement 4). So by the Side - Angle - Side (SAS) congruence criterion, \(\triangle RST \cong \triangle QUT\) (if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent).

Answer:

1. Given (that \(\triangle QRT\) is equilateral)
2. Given (that \(\overline{RS} \cong \overline{QU}\))
3. Given (that \(\angle SRT \cong \angle TQU\))
4. Sides of an equilateral triangle are congruent (so \(\overline{QT} \cong \overline{RT}\))
5. SAS Congruence Criterion (since \(RS\cong QU\), \(\angle SRT\cong\angle TQU\), \(RT\cong QT\))