Question

wy || ik and ik || tv. complete the proof that m∠sxw + m∠tuz = 180°. statement reason 1 wy || ik given 2 ik || tv given 3 ∠sxw ≅ ∠ijs corresponding angles theorem 4 ∠ijs≅∠tuz corresponding angles theorem 5 ∠sxw≅∠tuz transitive property of congruence 6 definition of congruence

Explanation:

Step1: Identify corresponding - angles

Since $\overleftrightarrow{WY}\parallel\overleftrightarrow{IK}$, by the Corresponding Angles Theorem, $\angle SXW\cong\angle IJS$.

Step2: Identify another pair of corresponding - angles

Since $\overleftrightarrow{IK}\parallel\overleftrightarrow{TV}$, by the Corresponding Angles Theorem, $\angle IJS\cong\angle TUZ$.

Step3: Use the transitive property

By the Transitive Property of Congruence, if $\angle SXW\cong\angle IJS$ and $\angle IJS\cong\angle TUZ$, then $\angle SXW\cong\angle TUZ$.

Step4: Recall linear - pair relationship

We know that $\angle SXW$ and $\angle TXU$ form a linear - pair. So $m\angle SXW + m\angle TXU=180^{\circ}$. But since $\angle TXU$ and $\angle TUZ$ are vertical angles and $\angle SXW\cong\angle TUZ$, we can say that $m\angle SXW + m\angle TUZ = 180^{\circ}$. The missing statement in the proof is: $\angle SXW$ and $\angle TUZ$ are supplementary. The reason is: If two angles are congruent and one pair of them forms a linear - pair, the other pair is supplementary.

Answer:

1. $\angle SXW$ and $\angle TUZ$ are supplementary; If two angles are congruent and one pair of them forms a linear - pair, the other pair is supplementary.