Question

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

Explanation:

Step1: Definition of midpoint

If \(W\) is the midpoint of \(\overline{UV}\), then \(UW = VW\) (by the definition of a midpoint).

Step2: Right - angle property

Since \(UV\perp TW\), \(\angle T W U=\angle T W V = 90^{\circ}\) (by the definition of perpendicular lines).

Step3: Common side

\(\overline{TW}\cong\overline{TW}\) (by the reflexive property of congruence, which states that any segment is congruent to itself).

Step4: SAS (Side - Angle - Side) congruence

In \(\triangle TUW\) and \(\triangle TVW\), we have \(UW = VW\) (from Step 1), \(\angle T W U=\angle T W V\) (from Step 2), and \(\overline{TW}\cong\overline{TW}\) (from Step 3). So, \(\triangle TUW\cong\triangle TVW\) by the SAS congruence criterion.

Step5: Corresponding parts of congruent triangles

Since \(\triangle TUW\cong\triangle TVW\), then \(\angle V T W=\angle U T W\) (by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem).

Answer:

1. Definition of midpoint
2. Definition of perpendicular lines
3. Reflexive property of congruence
4. \(SAS\) (Side - Angle - Side)
5. \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent)