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1. [tex]$RSTU$[/tex] is a rectangle with vertices [tex]$K (0,0), S (0, a), T (a, a)$[/tex], and [tex]$U (a, 0)$[/tex]. & 1. given \\
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2. [tex]$RS = a$[/tex] units & 2. ? \\
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3. [tex]$ST = a$[/tex] units & 3. distance formula \\
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4. [tex]$\overline{KS} \cong \overline{ST}$[/tex] & 4. ? \\
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5. [tex]$KSTU$[/tex] is a square. & 5. ? \\
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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. if two consecutive sides of a rectangle are congruent, then it's a square; distance formula; definition of congruence

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

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

Sagot :

GinnyAnsweridn
Let's solve this step by step, filling in the missing reasons to complete the proof.

First, let's analyze the information given:

1. [tex]\( RSTU \)[/tex] is a rectangle with vertices [tex]\( K (0,0), S (0, a), T (a, a), \)[/tex] and [tex]\( U (a, 0) \)[/tex].

Reason: given

Next, let's look at the second row of the table:

2. [tex]\( RS = a \)[/tex] units

To determine this, you are using the distances between points [tex]\( R \)[/tex] and [tex]\( S \)[/tex]. This distance can be calculated using the distance formula.

Reason: distance formula

Now, let's move to the third row:

3. [tex]\( ST = a \)[/tex] units

This has already been given with the distance formula as its reason.

Reason: distance formula

Next, look at the fourth row:

4. [tex]\( \overline{KS} \cong \overline{ST} \)[/tex]

This means that the segment [tex]\( \overline{KS} \)[/tex] is congruent to the segment [tex]\( \overline{ST} \)[/tex]. This follows from the definition of congruence since both segments are equal in length, specifically both are [tex]\( a \)[/tex] units.

Reason: definition of congruence

Finally, row five:

5. [tex]\( KSTU \)[/tex] is a square

If two consecutive sides of a rectangle are congruent, then the rectangle must be a square. In this case, because [tex]\( \overline{KS} \)[/tex] is congruent to [tex]\( \overline{ST} \)[/tex], [tex]\( RSTU \)[/tex] meets the criteria for being a square.

Reason: if two consecutive sides of a rectangle are congruent, then it's a square

Putting it all together, the completed table should look like this:

[tex]\[ \begin{array}{|l|l|} \hline 1. \text{\(RSTU\) is a rectangle with vertices \(K (0,0), S (0, a ), T ( a , a ), \text{ and } U ( a , 0)\).} } & 1. \text{given} \\ \hline 2. \text{\(RS = a\) units} & 2. \text{distance formula} \\ \hline 3. \text{\(ST = a\) units} & 3. \text{distance formula} \\ \hline 4. \text{\(\overline{KS} \cong \overline{ST}\)} & 4. \text{definition of congruence} \\ \hline 5. \text{\(KSTU \text{ is a square}\)} & 5. \text{if two consecutive sides of a rectangle are congruent, then it's a square} \\ \hline \end{array} \][/tex]

Thus, the correct order of reasons that complete the proof 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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