Question

w is the midpoint of \\(\overline{uv}\\) and \\(\overline{uv} \perp \overline{tw}\\). complete the proof that \\(\angle vtw \cong \angle utw\\).
triangle diagram with vertices t, v, u and midpoint w on uv, and segment tw drawn

statement	reason
2. \\(\overline{uv} \perp \overline{tw}\\)
3. \\(\angle twu \cong \angle twv\\)	reflexive property of congruence
4. \\(\overline{uw} \cong \overline{vw}\\)
5. \\(\overline{tw} \cong \overline{tw}\\)	sas
6. \\(\triangle tuw \cong \triangle twv\\)	cpctc
7. \\(\angle vtw \cong \angle utw\\)

Explanation:

Step1: Use the definition of midpoint

Since \(W\) is the midpoint of \(\overline{UV}\), by the definition of midpoint, \(UW = VW\), so \(\overline{UW}\cong\overline{VW}\) (Statement 3: \(\overline{UW}\cong\overline{VW}\), Reason: Definition of midpoint).

Step2: Use the reflexive property

For \(\overline{TW}\), by the reflexive property of congruence, \(\overline{TW}\cong\overline{TW}\) (Statement 4: \(\overline{TW}\cong\overline{TW}\), Reason: Reflexive Property of Congruence).

Step3: Use the right - angle property

Given \(UV\perp TW\), so \(\angle T WU=\angle TWV = 90^{\circ}\), then \(\angle TWU\cong\angle TWV\) (Statement 2: \(\angle TWU\cong\angle TWV\), Reason: \(UV\perp TW\) (given, and right - angles are congruent)).

Step4: Use the SAS (Side - Angle - Side) criterion

In \(\triangle TUW\) and \(\triangle TVW\), we have \(\overline{UW}\cong\overline{VW}\), \(\angle TWU\cong\angle TWV\), \(\overline{TW}\cong\overline{TW}\). By the SAS criterion for triangle congruence, \(\triangle TUW\cong\triangle TVW\) (Statement 5: \(\triangle TUW\cong\triangle TVW\), Reason: \(SAS\)).

Step5: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TUW\cong\triangle TVW\), then \(\angle VTW\cong\angle UTW\) (Statement 6: \(\angle VTW\cong\angle UTW\), Reason: \(CPCTC\)).

Answer:

1. \(W\) is the midpoint of \(\overline{UV}\) - Given (assumed from the problem setup as the starting point for the proof).
2. \(\angle TWU\cong\angle TWV\) - \(UV\perp TW\) (right - angles are congruent).
3. \(\overline{UW}\cong\overline{VW}\) - Definition of midpoint.
4. \(\overline{TW}\cong\overline{TW}\) - Reflexive Property of Congruence.
5. \(\triangle TUW\cong\triangle TVW\) - \(SAS\).
6. \(\angle VTW\cong\angle UTW\) - \(CPCTC\).
7. \(\angle VTW\cong\angle UTW\) - \(CPCTC\) (this might be a duplicate numbering in the original table, but the key is the application of \(CPCTC\) to get the final angle congruence).