Question

∠prs and ∠vuw are supplementary. complete the proof that qs || tv. statement: 1. ∠prs and ∠vuw are supplementary 2. m∠prs + m∠vuw = 180° 3. m∠prs + m∠vuw = 180° 4. m∠prs + m∠sru = m∠prs + m∠vuw 5. m∠vuw = m∠sru 6. qs || tv reason: given definition of supplementary angles angles forming a linear pair sum to 180° properties of addition, subtraction, multiplication, and division reflexive property of congruence reflexive property of equality substitution transitive property of congruence

Explanation:

Step1: Given information

We are given that $\angle PRS$ and $\angle VUW$ are supplementary, so $m\angle PRS + m\angle VUW=180^{\circ}$ (by the definition of supplementary angles).

Step2: Analyze linear - pair angles

We know that $\angle PRS$ and $\angle SRU$ form a linear - pair. So $m\angle PRS + m\angle SRU = 180^{\circ}$ (angles forming a linear pair sum to $180^{\circ}$).

Step3: Use the transitive property

Since $m\angle PRS + m\angle VUW=180^{\circ}$ and $m\angle PRS + m\angle SRU = 180^{\circ}$, by the transitive property of equality, we have $m\angle VUW=m\angle SRU$.

Step4: Conclude parallel lines

If two lines are cut by a transversal such that a pair of corresponding angles are congruent, then the lines are parallel. Here, $\angle VUW$ and $\angle SRU$ are corresponding angles. So, $\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$ (by the converse of the corresponding - angles postulate).

Answer:

The reasons for each step in order are: Given, Definition of supplementary angles, Angles forming a linear pair sum to $180^{\circ}$, Transitive property of equality, Converse of the corresponding - angles postulate. And the proof shows that $\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$.