Question

∠tsv and ∠svw are supplementary. complete the proof that (overleftrightarrow{uw}paralleloverleftrightarrow{rt}).
statement\treason

1. ∠tsv and ∠svw are supplementary\tgiven
2. (mangle tsv + mangle svw=180^{circ})\t
3. (mangle qst + mangle tsv = 180^{circ})\t
4. (mangle tsv + mangle svw=mangle qst + mangle tsv) \ttransitive property of equality
5. (mangle svw=mangle qst) \tproperties of addition, subtraction, multiplication, and division
6. (overleftrightarrow{uw}paralleloverleftrightarrow{rt}) \tconverse of corresponding angles theorem

Explanation:

Step1: Define supplementary angles

By definition, if two angles are supplementary, the sum of their measures is 180°. So for $\angle TSV$ and $\angle SVW$ which are supplementary, we have $m\angle TSV + m\angle SVW=180^{\circ}$.

Step2: Identify linear - pair angles

$\angle QST$ and $\angle TSV$ form a linear - pair. A linear - pair of angles are supplementary, so $m\angle QST + m\angle TSV = 180^{\circ}$.

Step3: Apply transitive property

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

Step4: Simplify the equation

Using the properties of addition, subtraction, multiplication, and division (in this case, subtracting $m\angle TSV$ from both sides of the equation $m\angle TSV + m\angle SVW=m\angle QST + m\angle TSV$), we get $m\angle SVW = m\angle QST$.

Step5: Use corresponding - angles theorem

$\angle SVW$ and $\angle QST$ are corresponding angles. According to the converse of the corresponding - angles theorem, if corresponding angles are congruent, then the two lines are parallel. So, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.

Answer:

The reasons for statements 2, 3 are:

1. Definition of supplementary angles
2. Linear - pair postulate (or definition of linear - pair angles which are supplementary)