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
