Question

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$$ \overleftrightarrow { t w } $$ bisects $$ \angle u t v $$. complete the proof that $$ \overline { v w } \cong \overline { u w } $$.

1 $$ \overleftrightarrow { t w } $$ bisects $$ \angle u t v $$
2 $$ \overline { t v } \cong \overline { t u } $$
3 $$ \angle u t w \cong \angle v t w $$
4 $$ \overline { t w } \cong \overline { t w } $$
5 $$ \triangle t u w \cong \triangle t v w $$
6 $$ \overline { v w } \cong \overline { u w } $$

Explanation:

Step1: Given

The problem statement provides this information.

Step2: Given

From the diagram (marked segments).

Step3: Definition of angle bisector

Since \(\overleftrightarrow{TW}\) bisects \(\angle UTV\), by definition \(\angle UTW\cong\angle VTW\).

Step4: Reflexive property of congruence

Any segment is congruent to itself.

Step5: SAS (Side - Angle - Side) congruence criterion

We have \(\overline{TV}\cong\overline{TU}\) (Step 2), \(\angle UTW\cong\angle VTW\) (Step 3), and \(\overline{TW}\cong\overline{TW}\) (Step 4). So, \(\triangle TUW\cong\triangle TVW\) by SAS.

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TUW\cong\triangle TVW\), their corresponding sides \(\overline{VW}\) and \(\overline{UW}\) are congruent.

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)