Question

complete the proof that $overleftrightarrow{rt}perpoverleftrightarrow{su}$. statement reason 1 $angle rvscongangle svt$ given 2 $mangle rvs + mangle svt=180^{circ}$ 3 $mangle rvs + mangle rvs = 180^{circ}$ substitution 4 $mangle rvs = 90^{circ}$ properties of addition, subtraction, multiplication, and division 5 $overleftrightarrow{rt}perpoverleftrightarrow{su}$ definition of perpendicular lines

Explanation:

Step1: Identify given facts

We are given $\angle RVS\cong\angle SVT$ and $m\angle RVS + m\angle SVT=180^{\circ}$.

Step2: Substitute equal - angles

Since $\angle RVS\cong\angle SVT$, we can substitute $m\angle SVT$ with $m\angle RVS$ in the equation $m\angle RVS + m\angle SVT = 180^{\circ}$, getting $m\angle RVS+m\angle RVS = 180^{\circ}$.

Step3: Solve for $m\angle RVS$

Combining like - terms in $m\angle RVS+m\angle RVS = 180^{\circ}$ gives $2m\angle RVS=180^{\circ}$. Then, dividing both sides by 2 using the division property of equality, we have $m\angle RVS = 90^{\circ}$.

Step4: Use perpendicular - lines definition

By the definition of perpendicular lines (if the angle formed by two lines is $90^{\circ}$, the lines are perpendicular), since the angle between $\overleftrightarrow{RT}$ and $\overleftrightarrow{SU}$ at point $V$ (i.e., $\angle RVS$) is $90^{\circ}$, we can conclude that $\overleftrightarrow{RT}\perp\overleftrightarrow{SU}$.

Answer:

The proof is completed as shown above, with the reasons for each step being: 1. Given; 2. Given; 3. Substitution; 4. Properties of addition, subtraction, multiplication, and division; 5. Definition of perpendicular lines.