Question

∠qur ≅ ∠rus. complete the proof that (overleftrightarrow{rt}perpoverleftrightarrow{qs}).
statement\treason
1 ∠qur ≅ ∠rus\tgiven
2 (mangle qur + mangle rus=180^{circ})\tangles forming a linear pair sum to (180^{circ})
3 (mangle qur + mangle qur = 180^{circ})\t
4 (mangle qur = 90^{circ})\tproperties of addition, subtraction, multiplication, and division
5 (overleftrightarrow{rt}perpoverleftrightarrow{qs})\tdefinition of perpendicular lines

Explanation:

Step1: Substitute equal - angle measure

Since $\angle QUR\cong\angle RUS$, we can substitute $m\angle RUS$ with $m\angle QUR$ in the equation $m\angle QUR + m\angle RUS=180^{\circ}$, getting $m\angle QUR + m\angle QUR = 180^{\circ}$.

Step2: Solve for $m\angle QUR$

Combining like - terms, we have $2m\angle QUR = 180^{\circ}$. Then, dividing both sides of the equation $2m\angle QUR = 180^{\circ}$ by 2 using the division property of equality, we get $m\angle QUR=\frac{180^{\circ}}{2}=90^{\circ}$.

Step3: Use perpendicular - line definition

Since $\overrightarrow{RT}$ and $\overrightarrow{QS}$ intersect at $U$ and $m\angle QUR = 90^{\circ}$, by the definition of perpendicular lines (two lines are perpendicular if the angle formed by their intersection is a right angle, i.e., $90^{\circ}$), we can conclude that $\overrightarrow{RT}\perp\overrightarrow{QS}$.

Answer:

The reasons for step 3 is "Substitution property of equality" and the reason for step 4 is "Division property of equality".