Question

∠uxy and ∠vux are supplementary. complete the proof that $overleftrightarrow{wy}paralleloverleftrightarrow{tv}$.
statement\treason
1 ∠uxy and ∠vux are supplementary\tangles forming a linear pair sum to 180°
2 m∠uxy + m∠vux = 180°\tdefinition of supplementary angles
3 m∠suv + m∠vux = 180°\t
4 m∠uxy + m∠vux = m∠suv + m∠vux\t
5 m∠uxy = m∠suv\t
6 $overleftrightarrow{wy}paralleloverleftrightarrow{tv}$\tgiven

Explanation:

Step1: Recall supplementary - angle property

By the definition of supplementary angles, if two angles are supplementary, the sum of their measures is 180°. Given that $\angle UXY$ and $\angle VUX$ are supplementary, we have $m\angle UXY + m\angle VUX=180^{\circ}$.

Step2: Identify another linear - pair

$\angle SUV$ and $\angle VUX$ form a linear pair. Since angles forming a linear pair sum to 180°, we have $m\angle SUV + m\angle VUX = 180^{\circ}$.

Step3: Use the transitive property of equality

Since $m\angle UXY + m\angle VUX=180^{\circ}$ and $m\angle SUV + m\angle VUX = 180^{\circ}$, by the transitive property of equality (if $a + c=b + c$, then $a = b$), we can subtract $m\angle VUX$ from both sides of the equations. So, $m\angle UXY=m\angle SUV$.

Step4: Apply the corresponding - angles postulate

$\angle UXY$ and $\angle SUV$ are corresponding angles. If corresponding angles are congruent, then the two lines are parallel. So, $\overleftrightarrow{WY}\parallel\overleftrightarrow{TV}$.

Answer:

The reasons for statements 3 - 6 are:

1. Angles forming a linear pair sum to 180°
2. Transitive property of equality
3. Subtraction property of equality
4. Corresponding angles postulate (If corresponding angles are congruent, then the lines are parallel)