Question

∠utw and ∠twx are supplementary. complete the proof that $overleftrightarrow{vx}paralleloverleftrightarrow{su}$. statement reason 1 ∠utw and ∠twx are supplementary given 2 $mangle utw + mangle twx=180^{circ}$ definition of supplementary angles 3 $mangle rtu + mangle utw = 180^{circ}$ angles forming a linear pair sum to 180° 4 $mangle utw + mangle twx = mangle rtu + mangle utw$ transitive property of equality 5 $mangle twx = mangle rtu$ properties of addition, subtraction, multiplication, and division 6 $overleftrightarrow{vx}paralleloverleftrightarrow{su}$ angles forming a linear pair sum to 180° converse of corresponding angles theorem corresponding angles theorem definition of supplementary angles vertical angle theorem

Explanation:

Step1: Recall supplementary - angle definition

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

Step2: Identify linear - pair relationship

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

Step3: Apply transitive property

Since $m\angle UTW + m\angle TWX = 180^{\circ}$ and $m\angle RTU + m\angle UTW = 180^{\circ}$, by the transitive property of equality, $m\angle UTW + m\angle TWX=m\angle RTU + m\angle UTW$.

Step4: Use angle - arithmetic properties

Subtract $m\angle UTW$ from both sides of the equation $m\angle UTW + m\angle TWX=m\angle RTU + m\angle UTW$. We get $m\angle TWX = m\angle RTU$.

Step5: Apply corresponding - angles converse

$\angle TWX$ and $\angle RTU$ are corresponding angles. Since $m\angle TWX = m\angle RTU$, by the converse of the corresponding angles theorem, $\overleftrightarrow{VX}\parallel\overleftrightarrow{SU}$.

Answer:

The reason for step 6 is the Converse of Corresponding Angles Theorem.