Question

w is the midpoint of \\( \overline{uv} \\) and \\( \overline{tw} \perp \overline{uv} \\). complete the proof that \\( \overline{tv} \cong \overline{tu} \\).

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of perpendicular lines

Since \(TW\perp UV\), \(\angle TWU = 90^{\circ}\) and \(\angle TWV=90^{\circ}\). By the definition of congruent angles (\(90^{\circ}=90^{\circ}\)), \(\angle TWU\cong\angle TWV\).

Step3: Definition of mid - point

If \(W\) is the mid - point of \(UV\), then by the definition of a mid - point, \(UW = VW\), so \(\overline{UW}\cong\overline{VW}\).

Step4: Reflexive property

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

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

In \(\triangle TUW\) and \(\triangle TVW\), we have \(\overline{UW}\cong\overline{VW}\) (from step 3), \(\angle TWU\cong\angle TWV\) (from step 2), and \(\overline{TW}\cong\overline{TW}\) (from step 4). So, \(\triangle TUW\cong\triangle TVW\) by the SAS (Side - Angle - Side) congruence criterion.

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

Since \(\triangle TUW\cong\triangle TVW\), their corresponding sides \(\overline{TV}\) and \(\overline{TU}\) are congruent. So, \(\overline{TV}\cong\overline{TU}\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of perpendicular lines
4. Definition of mid - point
5. Reflexive property of congruence
6. SAS (Side - Angle - Side)
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)