Question

$\overline{tu} \cong \overline{sv}$ and $\overline{st} \cong \overline{uv}$. complete the proof that $\overline{st} \parallel \overline{uv}$.
(image of a quadrilateral with vertices s, t, u, v and intersection point w of diagonals. then a table with statements and reasons:

1. $\overline{tu} \cong \overline{sv}$
2. $\overline{st} \cong \overline{uv}$
3. $\overline{su} \cong \overline{su}$
4. $\triangle suv \cong \triangle ust$
5. $\angle suv \cong \angle tsu$
6. $\overline{st} \parallel \overline{uv}$)

Explanation:

Step1: Identify Given Information

The problem states \(\overline{TU} \cong \overline{SV}\) and \(\overline{ST} \cong \overline{UV}\). These are given, so for statement 1, the reason is "Given". For statement 2, the reason is also "Given".

Step2: Reflexive Property of Congruence

For statement 3, \(\overline{SU} \cong \overline{SU}\) because a segment is congruent to itself. This is the Reflexive Property of Congruence.

Step3: SSS Congruence Criterion

For statement 4, we have three sides: \(\overline{TU} \cong \overline{SV}\), \(\overline{ST} \cong \overline{UV}\), and \(\overline{SU} \cong \overline{SU}\). So by the Side - Side - Side (SSS) Congruence Criterion, \(\triangle SUV \cong \triangle UST\).

Step4: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

For statement 5, since \(\triangle SUV \cong \triangle UST\), their corresponding angles \(\angle SUV\) and \(\angle TSU\) are congruent. This is by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Step5: Alternate Interior Angles Theorem

For statement 6, \(\angle SUV\) and \(\angle TSU\) are alternate interior angles formed by the transversal \(\overline{SU}\) intersecting \(\overline{ST}\) and \(\overline{UV}\). If alternate interior angles are congruent, then the lines are parallel. So by the Alternate Interior Angles Theorem, \(\overline{ST} \parallel \overline{UV}\).

Answer:

1. Reason for \(\overline{TU} \cong \overline{SV}\): Given
2. Reason for \(\overline{ST} \cong \overline{UV}\): Given
3. Reason for \(\overline{SU} \cong \overline{SU}\): Reflexive Property of Congruence
4. Reason for \(\triangle SUV \cong \triangle UST\): SSS Congruence Criterion
5. Reason for \(\angle SUV \cong \angle TSU\): CPCTC
6. Reason for \(\overline{ST} \parallel \overline{UV}\): Alternate Interior Angles Theorem