Question

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tw bisects ∠utv and tv ≅ tu. complete the proof that ∠twu ≅ ∠twv.
1 tw bisects ∠utv given
2 tv ≅ tu given
3 ∠utw ≅ ∠vtw definition of angle bisector
4 tw ≅ tw reflexive property of congruence
5 △tuw ≅ △tvw
6 ∠twu ≅ ∠twv

Explanation:

Step1: Determine the congruence criterion for triangles

We have \( \overline{TV}\cong\overline{TU}\) (given), \( \angle UTW\cong\angle VTW\) (definition of angle bisector), and \( \overline{TW}\cong\overline{TW}\) (reflexive property of congruence). This satisfies the Side - Angle - Side (SAS) congruence criterion.

Step2: Use the property of congruent triangles

If \( \triangle TUW\cong\triangle TVW\) (by SAS), then their corresponding parts are congruent. So, \( \angle TWU\) and \( \angle TWV\) are corresponding angles of congruent triangles.

Answer:

1. \(SAS\) (Side - Angle - Side)
2. Corresponding parts of congruent triangles are congruent (CPCTC)