Question

complete the proof that \\( \overleftrightarrow { r t } \perp \overleftrightarrow { s u } \\).

statement	reason
2 \\( m \angle r v s + m \angle s v t = 180 ^ { \circ } \\)
3 \\( m \angle r v s + m \angle r v s = 180 ^ { \circ } \\)
4 \\( m \angle r v s = 90 ^ { \circ } \\)
5 \\( \overleftrightarrow { r t } \perp \overleftrightarrow { s u } \\)

Explanation:

Step1: Vertical angles are congruent

Vertical angles are formed when two lines intersect. $\angle RVS$ and $\angle SVT$ are vertical angles, so $\angle RVS\cong\angle SVT$ by the Vertical Angles Theorem.

Step2: Linear pair of angles sum to \(180^{\circ}\)

$\angle RVS$ and $\angle SVT$ form a linear pair. By the Linear Pair Postulate, \(m\angle RVS + m\angle SVT=180^{\circ}\)

Step3: Substitution property of equality

Since \(\angle RVS\cong\angle SVT\), we can substitute \(m\angle SVT\) with \(m\angle RVS\). So \(m\angle RVS + m\angle RVS=180^{\circ}\)

Step4: Solve the equation

Combine like - terms: \(2m\angle RVS = 180^{\circ}\). Divide both sides by 2: \(m\angle RVS=\frac{180^{\circ}}{2}=90^{\circ}\)

Step5: Definition of perpendicular lines

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

Answer:

1. Vertical Angles Theorem
2. Linear Pair Postulate
3. Substitution Property of Equality
4. Simplification (or solving \(2x = 180\) where \(x=m\angle RVS\))
5. Definition of perpendicular lines (if the angle between two lines is \(90^{\circ}\), the lines are perpendicular)