Question

1. given: (overline{ty}congoverline{yx},angle tvycongangle xwy) prove: (\triangle tvycong\triangle xwy)

Explanation:

Step1: Write down the given information

1. \( \overline{TY}\cong\overline{YX}\) (Given)
2. \( \angle TVY\cong\angle XWY\) (Given)

Step2: Identify vertical angles

1. \( \angle T Y V\cong\angle X Y W\) (Vertical angles are congruent)

Step3: Apply AAS (Angle - Angle - Side) congruence criterion

1. In \( \triangle TVY\) and \( \triangle XWY\), we have \( \angle TVY\cong\angle XWY\), \( \angle T Y V\cong\angle X Y W\), and \( \overline{TY}\cong\overline{YX}\)
2. \( \triangle TVY\cong\triangle XWY\) (AAS congruence criterion)

Answer:

1. \( \overline{TY}\cong\overline{YX}\); Given
2. \( \angle TVY\cong\angle XWY\); Given
3. \( \angle T Y V\cong\angle X Y W\); Vertical angles are congruent
4. In \( \triangle TVY\) and \( \triangle XWY\): \( \angle TVY\cong\angle XWY\), \( \angle T Y V\cong\angle X Y W\), \( \overline{TY}\cong\overline{YX}\); (Combined information)
5. \( \triangle TVY\cong\triangle XWY\); AAS