Question

∠vux and ∠uxy are supplementary. complete the proof that (overleftrightarrow{wy}paralleloverleftrightarrow{tv}).
statement\treason
1 ∠vux and ∠uxy are supplementary
2 (mangle vux + mangle uxy=180^{circ})
3 (mangle suv + mangle vux = 180^{circ})
4 (mangle vux + mangle uxy=mangle suv + mangle vux)
5 (mangle uxy = mangle suv)
6 (overleftrightarrow{wy}paralleloverleftrightarrow{tv})

Explanation:

Step1: Definition of supplementary angles

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

Step2: Linear - pair postulate

$\angle SUV$ and $\angle VUX$ form a linear - pair. By the linear - pair postulate, the sum of the measures of two angles in a linear pair is 180°. So, $m\angle SUV + m\angle VUX = 180^{\circ}$.

Step3: Transitive property of equality

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

Step4: Subtraction property of equality

Subtract $m\angle VUX$ from both sides of the equation $m\angle VUX + m\angle UXY=m\angle SUV + m\angle VUX$. We get $m\angle UXY = m\angle SUV$.

Step5: Alternate - interior angles converse

If alternate - interior angles are congruent, then the lines are parallel. $\angle UXY$ and $\angle SUV$ are alternate - interior angles. Since $m\angle UXY = m\angle SUV$, $\overleftrightarrow{WY}\parallel\overleftrightarrow{TV}$ by the alternate - interior angles converse.

Answer:

1. Given
2. Definition of supplementary angles
3. Linear - pair postulate
4. Transitive property of equality
5. Subtraction property of equality
6. Alternate - interior angles converse