Question

$$ \overline { v w } \cong \overline { t u } $$ and $$ \overline { t u } \parallel \overline { v w } $$. complete the proof that $$ \triangle t v w \cong \triangle v t u $$.

statement	reason
2 $$ \overline { t u } \parallel \overline { v w } $$
3 $$ \angle t v w \cong \angle u t v $$
4 $$ \overline { t v } \cong \overline { t v } $$
5 $$ \triangle t v w \cong \triangle v t u $$

Explanation:

Step1: Given

Statement 1: \(\overline{VW}\cong\overline{TU}\) is given in the problem.

Step2: Given

Statement 2: \(\overline{TU}\parallel\overline{VW}\) is given in the problem.

Step3: Alternate - interior angles

Since \(\overline{TU}\parallel\overline{VW}\) and \(\overline{TV}\) is a transversal, by the alternate - interior angles theorem, \(\angle TVW\cong\angle UTV\).

Step4: Reflexive property

For any segment \(\overline{a}\), \(\overline{a}\cong\overline{a}\). So, \(\overline{TV}\cong\overline{TV}\) by the reflexive property of congruence.

Step5: SAS (Side - Angle - Side)

We have two sides and the included angle congruent. In \(\triangle TVW\) and \(\triangle VTU\), \(\overline{VW}\cong\overline{TU}\) (side), \(\angle TVW\cong\angle UTV\) (angle), \(\overline{TV}\cong\overline{TV}\) (side). So, \(\triangle TVW\cong\triangle VTU\) by the SAS (Side - Angle - Side) congruence criterion.

Answer:

1. Given
2. Given
3. Alternate - interior angles theorem
4. Reflexive property of congruence
5. SAS (Side - Angle - Side) congruence criterion