Question

$\overline{tu} \cong \overline{qr}$, $\angle rst \cong \angle qpu$, and $\angle puq \cong \angle rts$. complete the proof that $\triangle rst \cong \triangle qpu$.

statement	reason
2 $\angle rst \cong \angle qpu$
3 $\angle puq \cong \angle rts$
4 $qu = tu + qt$
5 $rt = qr + qt$
6 $qu = qr + qt$
7 $rt = qu$
8 $\triangle rst \cong \triangle qpu$

Explanation:

Step1: Identify given congruence

Given: $\overline{TU} \cong \overline{QR}$

Step2: Identify given angle congruence

Given: $\angle RST \cong \angle QPU$

Step3: Identify given angle congruence

Given: $\angle PUQ \cong \angle RTS$

Step4: Apply segment addition postulate

Segment Addition Postulate: $QU = TU + QT$

Step5: Apply segment addition postulate

Segment Addition Postulate: $RT = QR + QT$

Step6: Substitute congruent segments

Substitution Property: $QU = QR + QT$ (since $\overline{TU} \cong \overline{QR}$, so $TU=QR$)

Step7: Apply transitive property

Transitive Property: $RT = QU$

Step8: Apply ASA congruence rule

ASA Triangle Congruence: $\Delta RST \cong \Delta QPU$

Answer:

Statement	Reason
2. $\angle RST \cong \angle QPU$	Given
3. $\angle PUQ \cong \angle RTS$	Given
4. $QU = TU + QT$	Segment Addition Postulate
5. $RT = QR + QT$	Segment Addition Postulate
6. $QU = QR + QT$	Substitution Property (since $TU=QR$)
7. $RT = QU$	Transitive Property of Equality
8. $\Delta RST \cong \Delta QPU$	ASA Triangle Congruence Postulate