Given: [tex]$RSTU$[/tex] is a rectangle with vertices [tex]$R(0,0)$[/tex], [tex]$S(0, a)$[/tex], [tex]$T(a, a)$[/tex], and [tex]$U(a, 0)$[/tex], where [tex]$a \neq 0$[/tex].

Prove: [tex]$RSTU$[/tex] is a square.

\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{\textbf{Statements}} & \multicolumn{1}{c|}{\textbf{Reasons}} \\
\hline 1. [tex]$RSTU$[/tex] is a rectangle with vertices [tex]$R (0,0)$[/tex], [tex]$S (0, a)$[/tex], [tex]$T (a , a)$[/tex], and [tex]$U (a , 0)$[/tex]. & 1. Given \\
\hline 2. [tex]$RS = a$[/tex] units & 2. Distance formula \\
\hline 3. [tex]$ST = a$[/tex] units & 3. Distance formula \\
\hline 4. [tex]$\overline{RS} \cong \overline{ST}$[/tex] & 4. Definition of congruence \\
\hline 5. [tex]$RSTU$[/tex] is a square. & 5. If two consecutive sides of a rectangle are congruent, then it is a square \\
\hline
\end{tabular}

What is the correct order of reasons that complete the proof?

A. Distance formula; if two consecutive sides of a rectangle are congruent, then it's a square; definition of congruence

B. Distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square

C. Definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square

D. If two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence

To solve this problem, we need to prove that RSTU is a square and determine the correct order of reasons to complete the proof. Here is the detailed step-by-step solution:

Given:
- RSTU is a rectangle with vertices [tex]$ R (0,0), S (0, a ), T ( a , a )$[/tex], and [tex]$ U ( a , 0) $[/tex], where [tex]$ a \neq 0 $[/tex].

To Prove:
- RSTU is a square.

Proof:
1. Statement 1: [tex]$ RSTU $[/tex] is a rectangle with vertices [tex]$ R (0,0), S (0, a ), T ( a , a )$[/tex], and [tex]$ U ( a , 0) $[/tex].
- Reason 1: given

2. Statement 2: [tex]$ RS = a $[/tex] units
- Reason 2: distance formula

To find the length of [tex]$ RS $[/tex]:
[tex]\[ RS = \sqrt{(0-0)^2 + (a-0)^2} = \sqrt{0 + a^2} = \sqrt{a^2} = a \text{ units} \][/tex]

3. Statement 3: [tex]$ ST = a $[/tex] units
- Reason 3: distance formula

To find the length of [tex]$ ST $[/tex]:
[tex]\[ ST = \sqrt{(a-0)^2 + (a-a)^2} = \sqrt{a^2 + 0} = \sqrt{a^2} = a \text{ units} \][/tex]

4. Statement 4: [tex]$ \overline{RS} \cong \overline{ST} $[/tex]
- Reason 4: definition of congruence (two segments are congruent if they have the same length)

5. Statement 5: [tex]$ RSTU $[/tex] is a square.
- Reason 5: if two consecutive sides of a rectangle are congruent, then it's a square

Since [tex]$ RSTU $[/tex] is given as a rectangle and we have proven that two consecutive sides [tex]$ RS $[/tex] and [tex]$ ST $[/tex] are congruent ([tex]$ RS = ST = a $[/tex]), it follows that [tex]$ RSTU $[/tex] is a square.

Conclusion:
The correct order of reasons that complete the proof is:
B. distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square