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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].

Prove: RSTU is a square.

\begin{tabular}{|l|l|}
\hline
\multicolumn{1}{|c|}{Statements} & \multicolumn{1}{c|}{Reasons} \\
\hline
1. RSTU 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. 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. RSTU is a square. & 5. if two consecutive sides of a rectangle are congruent, then it's 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. definition of congruence; distance formula; if two consecutive sides of a rectangle are congruent, then it's a square
C. distance formula; definition of congruence; 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



Answer :

GinnyAnswer
To complete the proof that [tex]\( RSTU \)[/tex] is a square, we need to provide the correct order of reasons for the given statements. Let's go through the solution step by step:

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

2. [tex]\( RS = a \)[/tex] units:
- Reason: Distance formula. We use the distance formula to calculate the distance between points [tex]\( R(0,0) \)[/tex] and [tex]\( S(0,a) \)[/tex]:
[tex]\[ RS = \sqrt{(0-0)^2 + (a-0)^2} = \sqrt{a^2} = a \][/tex]

3. [tex]\( ST = a \)[/tex] units:
- Reason: Distance formula. We use the distance formula to calculate the distance between points [tex]\( S(0,a) \)[/tex] and [tex]\( T(a,a) \)[/tex]:
[tex]\[ ST = \sqrt{(a-0)^2 + (a-a)^2} = \sqrt{a^2} = a \][/tex]

4. [tex]\( \overline{RS} \cong \overline{ST} \)[/tex]:
- Reason: Definition of congruence. Since [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are both equal to [tex]\( a \)[/tex], they are congruent. Congruence of line segments is defined by having equal lengths.

5. RSTU is a square:
- Reason: If two consecutive sides of a rectangle are congruent, then it's a square. We have shown that [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex], which are consecutive sides of the rectangle RSTU, are congruent. This means that RSTU must be a square.

From these steps, the correct order of reasons is:

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

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