Question

∠wvx and ∠qst are supplementary. complete the proof that (overleftrightarrow{rt}paralleloverleftrightarrow{uw}).
statement

1. ∠wvx and ∠qst are supplementary
2. (mangle wvx + mangle qst=180^{circ})
3. (mangle qst + mangle tsv = 180^{circ})
4. (mangle wvx + mangle qst=mangle qst + mangle tsv)
5. (mangle wvx=mangle tsv)
6. (overleftrightarrow{rt}paralleloverleftrightarrow{uw})

reason

Explanation:

Step1: Given information

Given that $\angle WVX$ and $\angle QST$ are supplementary.

Step2: Definition of supplementary angles

If two angles are supplementary, the sum of their measures is $180^{\circ}$. So $m\angle WVX + m\angle QST=180^{\circ}$.

Step3: Linear - pair postulate

$\angle QST$ and $\angle TSV$ form a linear - pair. By the linear - pair postulate, $m\angle QST + m\angle TSV = 180^{\circ}$.

Step4: Transitive property of equality

Since $m\angle WVX + m\angle QST=180^{\circ}$ and $m\angle QST + m\angle TSV = 180^{\circ}$, by the transitive property of equality, $m\angle WVX + m\angle QST=m\angle QST + m\angle TSV$.

Step5: Subtraction property of equality

Subtract $m\angle QST$ from both sides of the equation $m\angle WVX + m\angle QST=m\angle QST + m\angle TSV$. We get $m\angle WVX=m\angle TSV$.

Step6: Alternate - interior angles converse

$\angle WVX$ and $\angle TSV$ are alternate - interior angles. If alternate - interior angles are congruent, then the lines are parallel. So $\overleftrightarrow{RT}\parallel\overleftrightarrow{UW}$.

Answer:

1. Given
2. Definition of supplementary angles
3. Linear - pair postulate
4. Transitive property of equality
5. Subtraction property of equality
6. Alternate - interior angles converse