Question

∠svw and ∠tsv are supplementary. complete the proof that uw || rt. 1. ∠svw and ∠tsv are supplementary. given 2. m∠svw + m∠tsv = 180°. definition of supplementary angles 3. m∠qst + m∠tsv = 180°. angles forming a linear pair sum to 180° 4. m∠svw + m∠tsv = m∠qst + m∠tsv. angles forming a linear pair sum to 180° 5. m∠svw = m∠qst. properties of addition, subtraction, multiplication, and division 6. uw || rt. definition of congruence, reflexive property of congruence, reflexive property of equality

Explanation:

Step1: Recall angle - addition and supplementary - angle concepts

We know that $\angle SVW$ and $\angle TSV$ are supplementary, so $m\angle SVW + m\angle TSV=180^{\circ}$ (by the definition of supplementary angles).

Step2: Use given angle - equality

We are given some relationship between the angles such that $m\angle SVW + m\angle TSV=m\angle QST + m\angle TSV$. By the subtraction property of equality, if $a + c=b + c$, then $a = b$. So, $m\angle SVW=m\angle QST$.

Step3: Apply parallel - line criteria

If corresponding angles (in this case, $\angle SVW$ and $\angle QST$) are equal, then the lines are parallel. So, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ (by the converse of corresponding - angles postulate).

Answer:

The proof is completed by using the definition of supplementary angles, subtraction property of equality, and the converse of corresponding - angles postulate to show that $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.