Question

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

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

Explanation:

Step1: Definition of supplementary angles

If two angles are supplementary, then the sum of their measures is \(180^{\circ}\). So, since \(\angle SRU\) and \(\angle RUV\) are supplementary, \(m\angle SRU + m\angle RUV=180^{\circ}\) (Reason: Definition of supplementary angles).

Step2: Linear - pair angles

\(\angle PRS\) and \(\angle SRU\) form a linear pair. By the linear - pair postulate, \(m\angle PRS + m\angle SRU = 180^{\circ}\) (Reason: Linear - pair postulate).

Step3: Transitive property of equality

We have \(m\angle SRU + m\angle RUV=180^{\circ}\) and \(m\angle PRS + m\angle SRU = 180^{\circ}\). So, \(m\angle SRU + m\angle RUV=m\angle PRS + m\angle SRU\) (Reason: Transitive property of equality).

Step4: Subtraction property of equality

Subtract \(m\angle SRU\) from both sides of the equation \(m\angle SRU + m\angle RUV=m\angle PRS + m\angle SRU\). We get \(m\angle RUV=m\angle PRS\) (Reason: Subtraction property of equality).

Step5: Alternate - interior angles converse

If two lines are cut by a transversal and the alternate - interior angles are congruent, then the lines are parallel. Here, \(\overleftrightarrow{TV}\) and \(\overleftrightarrow{QS}\) are cut by transversal \(\overleftrightarrow{WP}\), and \(\angle RUV\) and \(\angle PRS\) are alternate - interior angles. Since \(m\angle RUV = m\angle PRS\), then \(\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}\) (Reason: Alternate - interior angles converse).

Answer:

\(\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}\)