Question

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

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem.

Step3: Definition of perpendicular lines

Since \(TW\perp UV\), \(\angle TWU = 90^{\circ}\) and \(\angle TWV=90^{\circ}\), so \(\angle TWU\cong\angle TWV\) (all right angles are congruent).

Step4: Definition of mid - point

If \(W\) is the mid - point of \(UV\), then \(UW = VW\) (by the definition of a mid - point, which divides a segment into two equal parts), so \(\overline{UW}\cong\overline{VW}\).

Step5: Reflexive property of congruence

For any segment \(a\), \(a\cong a\). So \(\overline{TW}\cong\overline{TW}\).

Step6: SAS (Side - Angle - Side) congruence criterion

In \(\triangle TUW\) and \(\triangle TVW\), we have \(\overline{UW}\cong\overline{VW}\), \(\angle TWU\cong\angle TWV\), \(\overline{TW}\cong\overline{TW}\). So \(\triangle TUW\cong\triangle TVW\) (by SAS: 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).

Step7: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TUW\cong\triangle TVW\), their corresponding angles \(\angle VTW\) and \(\angle UTW\) are congruent.

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (all 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)