Question

∠uvx and ∠qsr are supplementary. complete the proof that (overleftrightarrow{uw}paralleloverleftrightarrow{rt}).
statement reason
1 ∠uvx and ∠qsr are supplementary given
2 (mangle uvx + mangle qsr=180^{circ})
3 (mangle qsr + mangle rsv = 180^{circ})
4 (mangle uvx + mangle qsr=mangle qsr + mangle rsv)
5 (mangle uvx=mangle rsv)
6 (overleftrightarrow{uw}paralleloverleftrightarrow{rt})

Explanation:

Step1: Definition of supplementary angles

If two angles are supplementary, the sum of their measures is 180°. So, since $\angle{UVX}$ and $\angle{QSR}$ are supplementary, $m\angle{UVX}+m\angle{QSR}=180^{\circ}$.

Step2: Linear - pair postulate

$\angle{QSR}$ and $\angle{RSV}$ form a linear - pair. A linear pair of angles is supplementary, so $m\angle{QSR}+m\angle{RSV}=180^{\circ}$.

Step3: Transitive property of equality

Since $m\angle{UVX}+m\angle{QSR}=180^{\circ}$ and $m\angle{QSR}+m\angle{RSV}=180^{\circ}$, we can say $m\angle{UVX}+m\angle{QSR}=m\angle{QSR}+m\angle{RSV}$ by the transitive property of equality.

Step4: Subtraction property of equality

Subtract $m\angle{QSR}$ from both sides of the equation $m\angle{UVX}+m\angle{QSR}=m\angle{QSR}+m\angle{RSV}$. We get $m\angle{UVX}=m\angle{RSV}$.

Step5: Corresponding - angles postulate

$\angle{UVX}$ and $\angle{RSV}$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.

Answer:

1. Definition of supplementary angles
2. Linear - pair postulate
3. Transitive property of equality
4. Subtraction property of equality
5. Corresponding - angles postulate (If corresponding angles are congruent, then the lines are parallel)