Question

w is the midpoint of \\( \overline{uv} \\). complete the proof that \\( \triangle tvw \cong \triangle tuw \\).

statement	reason
2	\\( \overline{tw} \perp \overline{uv} \\)	given
3	\\( \angle twu \cong \angle twv \\)	all right angles are congruent
4	\\( \overline{uw} \cong \overline{vw} \\)	definition of midpoint
5	\\( \overline{tw} \cong \overline{tw} \\)	reflexive property of congruence
6	\\( \triangle tvw \cong \triangle tuw \\)	aas, asa, cpctc, definition of congruence, sas, etc.

Explanation:

Step1: Analyze the given information

We have two triangles \( \triangle TVW\) and \( \triangle TUW\). We know \( \overline{UW}\cong\overline{VW}\) (from the definition of mid - point), \( \angle TWU\cong\angle TWV\) (right angles are congruent), and \( \overline{TW}\cong\overline{TW}\) (reflexive property).

Step2: Recall the congruence criteria

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \( \triangle TVW\) and \( \triangle TUW\), we have \( \overline{UW}\cong\overline{VW}\), \( \angle TWV\cong\angle TWU\), and \( \overline{TW}\cong\overline{TW}\). The angle \( \angle TWV\) and \( \angle TWU\) is the included angle between \( \overline{UW}\) and \( \overline{TW}\) in \( \triangle TUW\), and between \( \overline{VW}\) and \( \overline{TW}\) in \( \triangle TVW\).

Answer:

SAS