Question

∠svw and ∠tsv are supplementary. complete the proof that $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$. statement reason 1 ∠svw and ∠tsv are supplementary given 2 $mangle svw + mangle tsv=180^{circ}$ angles forming a linear pair sum to $180^{circ}$ definition of supplementary angles 3 $mangle qst + mangle tsv = 180^{circ}$ vertical angle theorem 4 $mangle svw + mangle tsv=mangle qst + mangle tsv$ properties of addition, subtraction, multiplication, and division 5 $mangle svw=mangle qst$ converse of corresponding angles theorem 6 $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$

Explanation:

Step1: Given supplementary angles

Given that $\angle SVW$ and $\angle TSV$ are supplementary.

Step2: Use supplementary - angle property

By the definition of supplementary angles, $m\angle SVW + m\angle TSV=180^{\circ}$.

Step3: Identify vertical - angle relationship

Since $\angle QST$ and $\angle SVW$ are vertical angles, by the Vertical Angle Theorem, $m\angle QST = m\angle SVW$. So $m\angle QST + m\angle TSV = 180^{\circ}$.

Step4: Equate the two sums

We have $m\angle SVW + m\angle TSV=m\angle QST + m\angle TSV$ (by substitution as $m\angle QST = m\angle SVW$).

Step5: Simplify the equation

Using the subtraction property of equality (subtract $m\angle TSV$ from both sides of the equation $m\angle SVW + m\angle TSV=m\angle QST + m\angle TSV$), we get $m\angle SVW=m\angle QST$.

Step6: Apply the converse of corresponding - angles theorem

Since $\angle SVW$ and $\angle QST$ are corresponding angles and $m\angle SVW=m\angle QST$, by the Converse of Corresponding Angles Theorem, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.

Answer:

The proof that $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ is completed as shown above.