Question

1 ↔sv bisects ∠tvx given
2 ↔sv bisects ∠usw given
3 ∠wvx ≅ ∠tvu vertical angle theorem
4 ∠svx ≅ ∠svt definition of angle bisector
5 ∠vsw ≅ ∠usv definition of angle bisector
6 m∠svw = m∠svx + m∠wvx additive property of angle measure
7 m∠svu = m∠svt + m∠tvu additive property of angle measure
8 m∠svw = m∠svt + m∠tvu substitution
9 m∠svu = m∠svw transitive property of equality
10 ◼{sv} ≅ ◼{sv}
11 △svw ≅ △svu asa
12 ◼{su} ≅ ◼{sw} cpctc

Explanation:

Step1: Analyze the given information

We are given that \( \overleftrightarrow{SV}\) bisects \( \angle TVX\) and \( \overleftrightarrow{SV}\) bisects \( \angle USW\), and we use geometric theorems like the Vertical Angle Theorem, Definition of angle bisector, Additive Property of Angle Measure, Substitution, Transitive Property of Equality, and congruence postulates (ASA) and CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Step2: Verify each step

• Step 1 - 5: Use the given (bisector information) and Vertical Angle Theorem.
• Step 6 - 7: Apply the Additive Property of Angle Measure (\(m\angle A=m\angle B + m\angle C\) if \( \angle A=\angle B+\angle C\)).
• Step 8: Substitute \(m\angle WVX\) with \(m\angle TVU\) (from Step 3) in the equation from Step 6.
• Step 9: Use the Transitive Property of Equality (\(a = b\) and \(b = c\) implies \(a = c\)) since \(m\angle SVW=m\angle SVT + m\angle TVU\) (Step 8) and \(m\angle SVU=m\angle SVT + m\angle TVU\) (Step 7).
• Step 10: \( \overline{SV}\cong\overline{SV}\) by the Reflexive Property of Congruence.
• Step 11: For \( \triangle SVW\) and \( \triangle SVU\), we have \( \angle VSW\cong\angle USV\) (Step 5), \( \overline{SV}\cong\overline{SV}\) (Step 10), and \(m\angle SVU = m\angle SVW\) (Step 9), so by ASA (Angle - Side - Angle) congruence postulate \( \triangle SVW\cong\triangle SVU\).
• Step 12: By CPCTC, since \( \triangle SVW\cong\triangle SVU\), their corresponding sides \( \overline{SU}\) and \( \overline{SW}\) are congruent.

Answer:

The proof is correct as it systematically uses geometric definitions (angle bisector), theorems (Vertical Angle Theorem), properties (Additive Property of Angle Measure, Transitive Property of Equality, Reflexive Property of Congruence), congruence postulates (ASA), and CPCTC to show \( \overline{SU}\cong\overline{SW}\).