Question

△rst ≅ △rqu. complete the proof that △qtu ≅ △sut.

(image of a geometric figure with points t, s, r, q, u)

statement	reason
2 ( overline{qu} cong overline{st} )	cpctc
3 ( overline{qr} cong overline{rs} )	cpctc
4 ( overline{ru} cong overline{rt} )	cpctc
5 ( qt = qr + rt )	additive property of length
6 ( su = rs + ru )	additive property of length
7 ( qt = rs + ru )	substitution
8 ( qt = su )	transitive property of equality
9 ( overline{tu} cong overline{tu} )	reflexive property of congruence
10

Explanation:

Step1: Identify Known Congruences

We know \( \overline{QU} \cong \overline{ST} \) (from step 2), \( QT = SU \) (from step 8), and \( \overline{TU} \cong \overline{TU} \) (from step 9).

Step2: Apply SSS Congruence

To prove \( \triangle QTU \cong \triangle SUT \), we use the Side - Side - Side (SSS) congruence criterion. The sides of \( \triangle QTU \) are \( QT \), \( TU \), \( QU \) and the sides of \( \triangle SUT \) are \( SU \), \( TU \), \( ST \). We have \( QT = SU \) (step 8), \( TU = TU \) (step 9), and \( QU = ST \) (step 2). So by SSS, \( \triangle QTU \cong \triangle SUT \).

Step3: Write the Final Statement

The statement for step 10 should be \( \triangle QTU \cong \triangle SUT \) with the reason "SSS (Side - Side - Side) Congruence Postulate" (since we have three pairs of congruent sides: \( QT\cong SU \), \( TU\cong TU \), \( QU\cong ST \)).

Answer:

\(\triangle QTU \cong \triangle SUT\) (by SSS Congruence Postulate)