Question

w is the midpoint of \\( \overline { u v } \\). complete the proof that \\( \angle u \cong \angle v \\).
1\\( w \\) is the midpoint of \\( \overline { u v } \\)given
2\\( \overline { t w } \perp \overline { u v } \\)given
3\\( \angle t w u \cong \angle t w v \\)
4\\( \overline { u w } \cong \overline { v w } \\)
5\\( \overline { t w } \cong \overline { t w } \\)
6\\( \triangle t u w \cong \triangle t v w \\)
7\\( \angle u \cong \angle v \\)

Explanation:

Step1: Definition of midpoint

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

Step2: Right angle property

Given \(TW\perp\overline{UV}\), then \(\angle TWU = 90^{\circ}\) and \(\angle TWV=90^{\circ}\). So, \(\angle TWU\cong\angle TWV\) (right angles are congruent).

Step3: Reflexive property of congruence

For segment \(\overline{TW}\), by the reflexive property of congruence (a segment is congruent to itself), \(\overline{TW}\cong\overline{TW}\).

Step4: SAS (Side - Angle - Side) congruence criterion

In \(\triangle T UW\) and \(\triangle T VW\), we have \(\overline{UW}\cong\overline{VW}\) (from step 1), \(\angle TWU\cong\angle TWV\) (from step 2), and \(\overline{TW}\cong\overline{TW}\) (from step 3). So, \(\triangle T UW\cong\triangle T VW\) by the SAS (Side - Angle - Side) congruence criterion.

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

Since \(\triangle T UW\cong\triangle T VW\), then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle U\cong\angle V\).

Answer:

1. Right angles are congruent; 4. Definition of mid - point; 5. Reflexive property of congruence; 6. SAS (Side - Angle - Side); 7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)