Question

∠ruv and ∠sru are supplementary. complete the proof that (overleftrightarrow{qs}paralleloverleftrightarrow{tv}).
statement

1. ∠ruv and ∠sru are supplementary
2. (mangle ruv + mangle sru=180^{circ})
3. (mangle rqs + mangle sru = 180^{circ})
4. (mangle ruv + mangle sru=mangle rqs + mangle sru)
5. (mangle ruv=mangle rqs)
6. (overleftrightarrow{qs}paralleloverleftrightarrow{tv})

reason
1.

1. properties of addition, subtraction, multiplication, and division
2. reflexive property of congruence
3. reflexive property of equality
4. substitution
5. transitive property of congruence
6. transitive property of equality

Explanation:

Step1: Given information

$\angle RUV$ and $\angle SRU$ are supplementary is given.

Step2: Definition of supplementary angles

If two angles are supplementary, the sum of their measures is $180^{\circ}$, so $m\angle RUV + m\angle SRU=180^{\circ}$.

Step3: Angle - angle relationship

We assume that there is some relationship such that $m\angle RQS + m\angle SRU = 180^{\circ}$ (this might be from linear - pair or other angle - addition facts in the figure).

Step4: Substitution property

Since $m\angle RUV + m\angle SRU=180^{\circ}$ and $m\angle RQS + m\angle SRU = 180^{\circ}$, by the substitution property, $m\angle RUV + m\angle SRU=m\angle RQS + m\angle SRU$.

Step5: Subtraction property of equality

Subtract $m\angle SRU$ from both sides of the equation $m\angle RUV + m\angle SRU=m\angle RQS + m\angle SRU$. We get $m\angle RUV=m\angle RQS$.

Step6: Alternate - interior angles converse

If $m\angle RUV=m\angle RQS$, then by the alternate - interior angles converse (if alternate - interior angles are congruent, then the two lines are parallel), $\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$.

Answer:

1. Given
2. Definition of supplementary angles
3. Angle - addition or linear - pair property (assumed from figure)
4. Substitution property
5. Subtraction property of equality
6. Alternate - interior angles converse