Question

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Explanation:

Analyze the given information

We are given:

1. \(\angle TUW \cong \angle UTV\) (Given)
2. \(\overline{TU} \parallel \overline{VW}\) (Given)
3. \(\overline{TV} \cong \overline{UW}\) (Given)

We need to complete the reasons for the missing steps in the proof that \(\Delta TVW \cong \Delta UWV\).

Analyze parallel line relationships

Statement 4: \(\angle UTV \cong \angle TVW\)
Statement 5: \(\angle UWV \cong \angle TUW\)

Since \(\overline{TU} \parallel \overline{VW}\), the transversal line \(\overline{TV}\) creates alternate interior angles \(\angle UTV\) and \(\angle TVW\).
Similarly, the transversal line \(\overline{UW}\) creates alternate interior angles \(\angle UWV\) and \(\angle TUW\).
Therefore, the reason for both Statement 4 and Statement 5 is the Alternate Interior Angles Theorem.

Analyze reflexive property

Statement 8: \(\overline{VW} \cong \overline{VW}\) (or \(\overline{VW} \cong \overline{WV}\))

Any segment is congruent to itself.
Therefore, the reason for Statement 8 is the Reflexive Property of Congruence.

Summarize the missing reasons

• Reason 4: Alternate Interior Angles Theorem
• Reason 5: Alternate Interior Angles Theorem
• Reason 8: Reflexive Property of Congruence

Answer:

To complete the proof, the missing reasons are:

• Reason 4: Alternate Interior Angles Theorem
• Reason 5: Alternate Interior Angles Theorem
• Reason 8: Reflexive Property of Congruence