Question

$\angle srt \cong \angle puq$, $\angle pqu \cong \angle rts$, and $\overline{tu} \cong \overline{qr}$. complete the proof that $\overline{rs} \cong \overline{pu}$.

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

	statement	reason
2	$\angle pqu \cong \angle rts$
3	$\overline{tu} \cong \overline{qr}$
4	$qu = tu + qt$
5	$rt = qr + qt$
6	$qu = qr + qt$
	$rt = qu$
	$\triangle pqu \cong \triangle str$
9	$\overline{rs} \cong \overline{pu}$

Explanation:

Step1: Identify given congruences

$\angle SRT \cong \angle PUQ$, $\angle PQU \cong \angle RTS$, $\overline{TU} \cong \overline{QR}$ (Given)

Step2: Segment addition postulate

$QU = TU + QT$, $RT = QR + QT$ (Segment Addition Postulate)

Step3: Substitute TU with QR

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

Step4: Prove QU ≅ RT

$RT = QU$ (Transitive Property, from Step2 and Step3)

Step5: ASA congruence for triangles

$\triangle PQU \cong \triangle STR$ (ASA: $\angle PUQ \cong \angle SRT$, $\overline{QU} \cong \overline{RT}$, $\angle PQU \cong \angle RTS$)

Step6: Corresponding parts congruence

$\overline{RS} \cong \overline{PU}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

$\overline{RS} \cong \overline{PU}$ (proven via ASA congruence of $\triangle PQU$ and $\triangle STR$ and CPCTC)