Question

complete the proof that ( overleftrightarrow{qs} perp overleftrightarrow{rt} ).

Explanation:

Step1: Definition of congruent angles

If \(\angle RUS\cong\angle QUR\), then by the definition of congruent angles, their measures are equal.

Step2: Linear - pair postulate

\(\angle QUR\) and \(\angle RUS\) form a linear pair. By the linear - pair postulate, \(m\angle QUR + m\angle RUS=180^{\circ}\)

Step3: Substitution property

Since \(m\angle RUS = m\angle QUR\) (from step 1), substitute \(m\angle RUS\) with \(m\angle QUR\) in the equation \(m\angle QUR + m\angle RUS = 180^{\circ}\). So, \(m\angle QUR+m\angle QUR = 180^{\circ}\)

Step4: Solve the equation

Combine like terms: \(2m\angle QUR=180^{\circ}\). Then, using the division property of equality (\(a = b\Rightarrow a\div c=b\div c,c
eq0\)), divide both sides of the equation \(2m\angle QUR = 180^{\circ}\) by 2. We get \(m\angle QUR=\frac{180^{\circ}}{2}=90^{\circ}\)

Step5: Definition of perpendicular lines

If the measure of the angle between two lines is \(90^{\circ}\), then the lines are perpendicular. Since \(m\angle QUR = 90^{\circ}\), by the definition of perpendicular lines, \(\overleftrightarrow{QS}\perp\overleftrightarrow{RT}\)

Answer:

1. Definition of congruent angles
2. Linear - pair postulate
3. Substitution property
4. Division property of equality
5. Definition of perpendicular lines