Question

complete the proof that ( mangle sut + mangle wxz = 180^{circ} ).

Explanation:

Step1: Corresponding Angles Postulate

When two parallel lines are cut by a transversal, corresponding angles are congruent. Since \( \overleftrightarrow{WY}\parallel\overleftrightarrow{FH} \) and \( \overleftrightarrow{SZ} \) is the transversal, \( \angle WXZ\cong\angle FGZ \) by the Corresponding Angles Postulate.

Step2: Same - Side Interior Angles Theorem

When two parallel lines are cut by a transversal, same - side interior angles are supplementary. Since \( \overleftrightarrow{FH}\parallel\overleftrightarrow{TV} \) and \( \overleftrightarrow{SZ} \) is the transversal, \( m\angle FGZ + m\angle SUT=180^{\circ} \) by the Same - Side Interior Angles Theorem.

Step3: Substitution Property of Equality

Since \( \angle WXZ\cong\angle FGZ \) (so \( m\angle WXZ = m\angle FGZ \)), we can substitute \( m\angle FGZ \) with \( m\angle WXZ \) in the equation \( m\angle FGZ + m\angle SUT = 180^{\circ} \). By the Substitution Property of Equality, \( m\angle SUT + m\angle WXZ=180^{\circ} \).

Answer:

1. Corresponding Angles Postulate; 4. Same - Side Interior Angles Theorem; 5. Substitution Property of Equality.