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Given: [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], where [tex]\( a \neq 0 \)[/tex].

Prove: [tex]\( RSTU \)[/tex] is a square.

\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{Statements} & \multicolumn{1}{|c|}{Reasons} \\
\hline
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] & 1. Given \\
\hline
2. [tex]\( RS = a \)[/tex] units & 2. ? \\
\hline
3. [tex]\( ST = a \)[/tex] units & 3. Distance formula \\
\hline
4. [tex]\( \overline{RS} \cong \overline{ST} \)[/tex] & 4. ? \\
\hline
5. [tex]\( RSTU \)[/tex] is a square & 5. ? \\
\hline
\end{tabular}

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

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



Answer :

GinnyAnswer
Let's carefully go through the statements and identify the corresponding reasons to prove that the rectangle RSTU is a square.

### Given:
- RSTU 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].

We need to prove that RSTU is a square, and we have the statements that need proper justification:

1. RSTU 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].
- Reason: [tex]\( \text{given} \)[/tex]

2. [tex]\( RS = a \)[/tex] units
- Reason: [tex]\( \text{distance formula} \)[/tex]

3. [tex]\( ST = a \)[/tex] units
- Reason: [tex]\( \text{distance formula} \)[/tex]

4. [tex]\( \overline{RS} \cong \overline{ST} \)[/tex]
- Reason: [tex]\( \text{definition of congruence} \)[/tex]

5. RSTU is a square
- Reason: [tex]\( \text{if two consecutive sides of a rectangle are congruent, then it's a square} \)[/tex]

Now, let's provide a detailed step-by-step solution:

### Steps and Reasons:
1. Given: RSTU 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].
- Reason: Given

2. Calculate the Length of [tex]\( \overline{RS} \)[/tex]:
[tex]\[ \overline{RS} = \text{distance between } R(0,0) \text{ and } S(0,a) = a \text{ units} \][/tex]
- Reason: Distance Formula

3. Calculate the Length of [tex]\( \overline{ST} \)[/tex]:
[tex]\[ \overline{ST} = \text{distance between } S(0,a) \text{ and } T(a,a) = a \text{ units} \][/tex]
- Reason: Distance Formula

4. Conclude that [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{ST} \)[/tex] are Congruent:
[tex]\[ \overline{RS} \cong \overline{ST} \][/tex]
- Reason: Definition of Congruence (two segments are congruent if their lengths are equal)

5. Conclude that RSTU is a Square:
Since [tex]\( \overline{RS} \)[/tex] and [tex]\( \overline{ST} \)[/tex] are congruent and they are consecutive sides of a rectangle RSTU, we conclude that RSTU is a square.
- Reason: If two consecutive sides of a rectangle are congruent, then it's a square

### Conclusion:
Matching these reasons with the provided choices gives us:
- Choice B: If two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence.
- Choice C: distance formula; definition of congruence, if two consecutive sides of a rectangle are congruent, then it's a square.
- Choice D: definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square.

The correct sequence of reasons is C: distance formula; definition of congruence; if two consecutive sides of a rectangle are congruent, then it's a square.

So, the correct answer is C.

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