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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.

[tex]\[
\begin{array}{|l|l|}
\hline
\multicolumn{1}{|c|}{\text{Statements}} & \text{Reasons} \\
\hline
1. \, RSTU \text{ is a rectangle with vertices } R(0,0), S(0, a), T(a, a), \text{ and } U(a, 0) & \text{Given} \\
\hline
2. \, RS = a \text{ units} & \text{Given} \\
\hline
3. \, ST = a \text{ units} & \text{?} \\
\hline
4. \, \overline{RS} \cong \overline{ST} & \text{Distance formula} \\
\hline
5. \, RSTU \text{ is a square} & \text{?} \\
\hline
\end{array}
\][/tex]

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

Sagot :

GinnyAnsweridn
Let's carefully go through the steps given in the problem and discuss the correct order of reasons needed to complete the proof that RSTU is a square.

1. Statements: 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. Statements: [tex]\(RS = a\)[/tex] units
Reason: Given

3. Statements: [tex]\(ST = a\)[/tex] units
Reason: We need to use the distance formula to calculate the distance between points [tex]\(S\)[/tex] and [tex]\(T\)[/tex]. Therefore, the reason here should be the distance formula.

4. Statements: [tex]\(\overline{RS} \cong \overline{ST}\)[/tex]
Reason: The definition of congruence is that two segments are congruent if they have the same length. Here both [tex]\(RS\)[/tex] and [tex]\(ST\)[/tex] have lengths [tex]\(a\)[/tex].

5. Statements: RSTU is a square.
Reason: If two consecutive sides of a rectangle are congruent, then it is a square. This follows from the properties of rectangles and squares.

Order of reasons:

- 3. distance formula: This is the reason used to determine that [tex]\(ST = a\)[/tex] units.
- 4. definition of congruence: This reason is used to state that two segments are congruent if they have the same measure.
- 5. If two consecutive sides of a rectangle are congruent, then it's a square: This is used to conclude that RSTU is a square.

So, the correct order of reasons that complete the proof is:

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

This ensures that all logical steps are justified properly, proving that RSTU is a square.
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