Question

∠utw and ∠twx are supplementary. complete the proof that $overleftrightarrow{vx}paralleloverleftrightarrow{su}$.
statement\treason
1 ∠utw and ∠twx are supplementary\tgiven
2 m∠utw + m∠twx = 180°\tdefinition of supplementary angles
3 m∠rtu + m∠utw = 180°\tangles forming a linear pair sum to 180°
4 m∠utw + m∠twx = m∠rtu + m∠utw
5 m∠twx = m∠rtu\tproperties of addition, subtraction, multiplication, and division
6 $overleftrightarrow{vx}paralleloverleftrightarrow{su}$

Explanation:

Step1: Recall supplementary - angle definition

Given that $\angle UTW$ and $\angle TWX$ are supplementary, by the definition of supplementary angles, $m\angle UTW + m\angle TWX=180^{\circ}$.

Step2: Consider linear - pair property

Since $\angle RTU$ and $\angle UTW$ form a linear pair, $m\angle RTU + m\angle UTW = 180^{\circ}$ (angles forming a linear pair sum to $180^{\circ}$).

Step3: Set up equal - sum equation

Because $m\angle UTW + m\angle TWX=180^{\circ}$ and $m\angle RTU + m\angle UTW = 180^{\circ}$, we can write $m\angle UTW + m\angle TWX=m\angle RTU + m\angle UTW$ (transitive property of equality).

Step4: Simplify the equation

Subtract $m\angle UTW$ from both sides of the equation $m\angle UTW + m\angle TWX=m\angle RTU + m\angle UTW$. Using the properties of addition and subtraction, we get $m\angle TWX = m\angle RTU$.

Step5: Apply parallel - line theorem

$\angle TWX$ and $\angle RTU$ are corresponding angles. If corresponding angles are equal, then the lines are parallel. So, $\overleftrightarrow{VX}\parallel\overleftrightarrow{SU}$.

Answer:

The reason for statement 4 is "Transitive property of equality" and the reason for statement 6 is "If corresponding angles are equal, then the lines are parallel".