Question

complete the proof that m∠svw + m∠tsv = 180°. statement reason 1. (overleftrightarrow{uw}paralleloverleftrightarrow{rt}) given 2. ∠svw≅∠qst 3. m∠qst + m∠tsv = 180° angles forming a linear pair sum to 180° converse of corresponding angles theorem corresponding angles theorem definition of supplementary angles vertical angle theorem 4. m∠svw + m∠tsv = 180°

Explanation:

Step1: Identify given information

$\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ is given.

Step2: Use corresponding - angles theorem

Since $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$, by the Corresponding Angles Theorem, $\angle SVW\cong\angle QST$. So, $m\angle SVW = m\angle QST$.

Step3: Recognize linear - pair relationship

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

Step4: Substitute

Substitute $m\angle SVW$ for $m\angle QST$ in the equation $m\angle QST + m\angle TSV = 180^{\circ}$. We get $m\angle SVW + m\angle TSV = 180^{\circ}$.

Answer:

1. Corresponding Angles Theorem; 3. Angles forming a linear pair sum to $180^{\circ}$; 4. Substitution property of equality.