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Math  /  Geometry

QuestionTWundefined\overleftrightarrow{T W} bisects ∠UWY\angle U W Y and ∠X≅∠V\angle X \cong \angle V. Complete the proof that △TVW≅△TXW\triangle T V W \cong \triangle T X W. \begin{tabular}{|c|c|c|c|c|} \hline & Statement & & Reason & \\ \hline 1 & TWundefined\overleftrightarrow{T W} bisects ∠UWY\angle U W Y & & Glven & \\ \hline 2 & ∠X≅∠V\angle X \cong \angle V & & Given & \\ \hline 3 & ∠XWY≅∠UWV\angle X W Y \cong \angle U W V & & Vertical Angle Theorem & \\ \hline 4 & ∠TWY≅∠TWU\angle T W Y \cong \angle T W U & & Definition of angle bisector & \\ \hline 5 & m∠TWX=m∠TWY+m∠XWYm \angle T W X=m \angle T W Y+m \angle X W Y & & Additive Property of Angle Measure & \\ \hline 6 & m∠TWV=m∠TWU+m∠UWVm \angle T W V=m \angle T W U+m \angle U W V & & | & - \\ \hline 7 & m∠TWX=m∠TWU+m∠UWVm \angle T W X=m \angle T W U+m \angle U W V & + & Substitution & \\ \hline 8 & m∠TWV=m∠TWXm \angle T W V=m \angle T W X & & Transitive Property of Equality & \\ \hline 9 & TW‾≅TW‾\overline{T W} \cong \overline{T W} & & Reflexive Property of Congruence & \\ \hline 10 & △TVW≅△TXW\triangle T V W \cong \triangle T X W & & & . \\ \hline \end{tabular}

Studdy Solution
Use the properties of angle bisectors and congruence to complete the proof:
- From the given information, ∠XWY≅∠UWV\angle X W Y \cong \angle U W V by the Vertical Angle Theorem. - ∠TWY≅∠TWU\angle T W Y \cong \angle T W U by the definition of angle bisector. - Using the Additive Property of Angle Measure, we have: m \angle T W X = m \angle T W Y + m \angle X W Y \] m \angle T W V = m \angle T W U + m \angle U W V \] - By substitution, since ∠XWY≅∠UWV\angle X W Y \cong \angle U W V and ∠TWY≅∠TWU\angle T W Y \cong \angle T W U, we have: $ m \angle T W X = m \angle T W U + m \angle U W V \] - By the Transitive Property of Equality, \(m \angle T W V = m \angle T W X\). - The line \(\overline{T W} \cong \overline{T W}\) by the Reflexive Property of Congruence.
Finally, we can conclude that △TVW≅△TXW\triangle T V W \cong \triangle T X W by the Angle-Side-Angle (ASA) Congruence Postulate, as we have two pairs of congruent angles and the included side TW‾\overline{T W} congruent.
The completed proof shows that △TVW≅△TXW\triangle T V W \cong \triangle T X W.

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