Question

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

(image of two triangles with points u, t, q, p and s, r, t, q; table with statement and reason columns:

1. $\angle qpu \cong \angle rst$
2. $\overline{tu} \cong \overline{qr}$
3. $\angle puq \cong \angle rts$
4. $qu = tu + qt$
5. $rt = qr + qt$
6. $qu = qr + qt$
7. $rt = qu$
8. $\triangle pqu \cong \triangle srt$)

Explanation:

Step1: Identify Given Information

The problem states \( \angle QPU \cong \angle RST \), \( \overline{TU} \cong \overline{QR} \), and \( \angle PUQ \cong \angle RTS \). These are given, so for statement 1, reason is "Given".

Step2: Given for \( \overline{TU} \cong \overline{QR} \)

Statement 2 is \( \overline{TU} \cong \overline{QR} \), so reason is "Given".

Step3: Given for \( \angle PUQ \cong \angle RTS \)

Statement 3 is \( \angle PUQ \cong \angle RTS \), reason is "Given".

Step4: Segment Addition Postulate

Statement 4: \( QU = TU + QT \). This is the Segment Addition Postulate (if a point \( T \) is between \( Q \) and \( U \), then \( QU = QT + TU \)).

Step5: Segment Addition Postulate

Statement 5: \( RT = QR + QT \). By Segment Addition Postulate (point \( Q \) is between \( R \) and \( T \), so \( RT = RQ + QT \), and \( RQ = QR \) (reflexive, but here using segment addition with \( QR \)).

Step6: Substitution (from Step2)

Statement 6: \( QU = QR + QT \). Since \( TU \cong QR \), \( TU = QR \) (congruent segments have equal length), so substitute \( TU \) with \( QR \) in statement 4. Reason: "Substitution (from statement 2)".

Step7: Transitive Property of Equality

Statement 7: \( RT = QU \). From statement 5 (\( RT = QR + QT \)) and statement 6 (\( QU = QR + QT \)), so by Transitive Property (if \( a = b \) and \( b = c \), then \( a = c \)), \( RT = QU \).

Step8: AAS Congruence Criterion

Statement 8: \( \triangle PQU \cong \triangle SRT \). We have \( \angle QPU \cong \angle RST \) (statement 1), \( \angle PUQ \cong \angle RTS \) (statement 3), and \( QU \cong RT \) (statement 7, since \( RT = QU \) implies \( \overline{QU} \cong \overline{RT} \)). So by Angle - Angle - Side (AAS) Congruence Theorem, the triangles are congruent.

Answer:

1. Reason for \( \angle QPU \cong \angle RST \): Given
2. Reason for \( \overline{TU} \cong \overline{QR} \): Given
3. Reason for \( \angle PUQ \cong \angle RTS \): Given
4. Reason for \( QU = TU + QT \): Segment Addition Postulate
5. Reason for \( RT = QR + QT \): Segment Addition Postulate
6. Reason for \( QU = QR + QT \): Substitution (from statement 2)
7. Reason for \( RT = QU \): Transitive Property of Equality
8. Reason for \( \triangle PQU \cong \triangle SRT \): AAS (Angle - Angle - Side) Congruence Theorem