Question

∠utw and ∠twx are supplementary. complete the proof that su ║ vx.
statement

1. ∠utw and ∠twx are supplementary
2. m∠utw + m∠twx = 180°
3. m∠rtu + m∠utw = 180°
4. m∠utw + m∠twx = m∠rtu + m∠utw
5. m∠twx = m∠rtu

reason
given
definition of supplementary angles
angles forming a linear - pair sum to 180°
transitive property of equality
properties of addition, subtraction, multiplication, and division

Explanation:

Step1: Recall the given information

We are given that $\angle UTW$ and $\angle TWX$ are supplementary, so $m\angle UTW + m\angle TWX=180^{\circ}$ (by the definition of supplementary angles). Also, we know that angles forming a linear - pair sum to $180^{\circ}$, and $m\angle RTU + m\angle UTW = 180^{\circ}$.

Step2: Apply the transitive property of equality

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

Step3: Use the subtraction property of equality

Subtract $m\angle UTW$ from both sides of the equation $m\angle UTW + m\angle TWX=m\angle RTU + m\angle UTW$. We get $m\angle TWX=m\angle RTU$ (by the properties of addition, subtraction).

Step4: Conclude parallel - lines

If $\angle TWX$ and $\angle RTU$ are equal (alternate interior angles), then $\overrightarrow{SU}\parallel\overrightarrow{VX}$ (alternate interior angles are equal, then the lines are parallel).

Answer:

The proof is completed by using the definition of supplementary angles, the property of linear - pair angles, the transitive property of equality, and the subtraction property of equality to show that alternate interior angles are equal, thus proving $\overrightarrow{SU}\parallel\overrightarrow{VX}$.