Question

30,786 Learners found this answer helpful

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

Explanation:

Step1: Given

The problem provides that \(\angle V\cong\angle U\) as a given condition.

Step2: Definition of perpendicular lines

Since \(UW\perp TW\), by the definition of perpendicular lines, \(\angle TWV = 90^{\circ}\) and \(\angle TWU=90^{\circ}\), so \(\angle TWV\cong\angle TWU\) (right angles are congruent).

Step3: Right angles are congruent

As \(UW\perp TW\), \(\angle TWV\) and \(\angle TWU\) are right angles. By the property of right angles, \(\angle TWV\cong\angle TWU\).

Step4: Reflexive property of congruence

For any segment \(AB\), \(AB\cong AB\). So, for segment \(TW\), \(\overline{TW}\cong\overline{TW}\) by the reflexive property of congruence.

Step5: AAS (Angle - Angle - Side) congruence criterion

We have \(\angle V\cong\angle U\) (given), \(\angle TWV\cong\angle TWU\) (right angles), and \(\overline{TW}\cong\overline{TW}\) (reflexive property). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle TUW\cong\triangle TVW\).

Step6: Corresponding parts of congruent triangles are congruent (CPCTC)

Since \(\triangle TUW\cong\triangle TVW\), their corresponding parts are congruent. So, \(\overline{TU}\cong\overline{TV}\) by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Given
2. Given (since \(UW\perp TW\) is part of the figure's given information)
3. Definition of perpendicular (as \(UW\perp TW\) implies \(\angle TWV = \angle TWU = 90^{\circ}\) and right angles are congruent)
4. Reflexive property of congruence
5. AAS (Angle - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)