Question

∠wvx and ∠qst are supplementary. complete the proof that (overleftrightarrow{rt}paralleloverleftrightarrow{uw}).
statement reason
1 ∠wvx and ∠qst are supplementary given
2 (mangle wvx + mangle qst=180^{circ}) definition of supplementary angles
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})

Explanation:

Step1: Recall supplementary - angle definition

Given that $\angle WVX$ and $\angle QST$ are supplementary, by the definition of supplementary angles, we have $m\angle WVX + m\angle QST=180^{\circ}$.

Step2: Identify linear - pair angles

$\angle QST$ and $\angle TSV$ form a linear pair. Since angles forming a linear pair sum to $180^{\circ}$, we have $m\angle QST + m\angle TSV = 180^{\circ}$.

Step3: Use transitive property of equality

Since $m\angle WVX + m\angle QST=180^{\circ}$ and $m\angle QST + m\angle TSV = 180^{\circ}$, we can set $m\angle WVX + m\angle QST=m\angle QST + m\angle TSV$ (transitive property of equality, which is part of the properties of addition).

Step4: Subtract $m\angle QST$ from both sides

Using the subtraction property of equality, if $m\angle WVX + m\angle QST=m\angle QST + m\angle TSV$, then $m\angle WVX=m\angle TSV$ (subtracting $m\angle QST$ from both sides of the equation).

Step5: Apply corresponding - angles converse

$\angle WVX$ and $\angle TSV$ are corresponding angles. By the converse of the corresponding - angles theorem, if corresponding angles are congruent, then the lines are parallel. So, since $m\angle WVX = m\angle TSV$, we can conclude that $\overleftrightarrow{RT}\parallel\overleftrightarrow{UW}$.

For step 3, the correct reason is "Transitive property of equality (from properties of addition)".

Answer:

1. Angles forming a linear pair sum to $180^{\circ}$
2. Transitive property of equality (from properties of addition)
3. Subtraction property of equality
4. Converse of Corresponding Angles Theorem