Question

\overline{qw} \cong \overline{st}, \overline{rv} \cong \overline{ru}, \overline{tu} \cong \overline{vw}, and \overline{rs} \cong \overline{qr}. complete the proof that \delta quw \cong \delta svt.

statement | reason

1. \overline{qw} \cong \overline{st} |
2. \overline{rv} \cong \overline{ru} |
3. \overline{tu} \cong \overline{vw} |
4. \overline{rs} \cong \overline{qr} |
5. qu = qr + ru |
6. sv = rs + rv |
7. qu = rs + rv |
8. qu = sv |
9. uw = vw + uv |
10. tv = tu + uv |
11. uw = tu + uv |
12. tv = uw |
13. \delta quw \cong \delta svt |

Explanation:

Analyze the given statements

Using the Geometric Proofs knowledge point
The proof establishes \(\Delta QUW \cong \Delta SVT\) using given segment congruences and segment addition properties.

Identify reasons for statements 1 to 4

Using the Geometric Proofs knowledge point
Statements 1, 2, 3, and 4 are all given in the problem description:

• \(1. \quad \overline{QW} \cong \overline{ST}\) (Given)
• \(2. \quad \overline{RV} \cong \overline{RU}\) (Given)
• \(3. \quad \overline{TU} \cong \overline{VW}\) (Given)
• \(4. \quad \overline{RS} \cong \overline{QR}\) (Given)

Apply segment addition for statements 5, 6, 9, and 10

Using the Segment Addition Postulate knowledge point

• \(5. \quad QU = QR + RU\) (Segment Addition Postulate)
• \(6. \quad SV = RS + RV\) (Segment Addition Postulate)
• \(9. \quad UW = VW + UV\) (Segment Addition Postulate)
• \(10. \quad TV = TU + UV\) (Segment Addition Postulate)

Analyze algebraic substitutions for statements 7, 8, 11, and 12

Using the Geometric Proofs knowledge point

• \(7. \quad QU = RS + RV\) (Substitution Property of Equality, substituting \(RS\) for \(QR\) and \(RV\) for \(RU\))
• \(8. \quad QU = SV\) (Substitution Property of Equality, substituting \(SV\) for \(RS + RV\))
• \(11. \quad UW = TU + UV\) (Substitution Property of Equality, substituting \(TU\) for \(VW\))
• \(12. \quad TV = UW\) (Substitution Property of Equality, substituting \(UW\) for \(TU + UV\))

Prove triangle congruence

Using the Triangle Congruence knowledge point

• \(13. \quad \Delta QUW \cong \Delta SVT\) (SSS Congruence Theorem, using \(\overline{QW} \cong \overline{ST}\) from step 1, \(\overline{QU} \cong \overline{SV}\) from step 8, and \(\overline{UW} \cong \overline{TV}\) from step 12)

Answer:

The complete proof with statements and reasons is:

	Statement	Reason
2	\(\overline{RV} \cong \overline{RU}\)	Given
3	\(\overline{TU} \cong \overline{VW}\)	Given
4	\(\overline{RS} \cong \overline{QR}\)	Given
5	\(QU = QR + RU\)	Segment Addition Postulate
6	\(SV = RS + RV\)	Segment Addition Postulate
7	\(QU = RS + RV\)	Substitution Property of Equality
8	\(QU = SV\)	Substitution Property of Equality
9	\(UW = VW + UV\)	Segment Addition Postulate
10	\(TV = TU + UV\)	Segment Addition Postulate
11	\(UW = TU + UV\)	Substitution Property of Equality
12	\(TV = UW\)	Substitution Property of Equality
13	\(\Delta QUW \cong \Delta SVT\)	SSS (Side-Side-Side) Congruence Theorem