Question

in the diagram, \\( \overleftrightarrow { f w } \\) and \\( \overleftrightarrow { b q } \\) intersect at point \\( x \\).
which statement would provide enough information to conclude that \\( \overleftrightarrow { f w } \\) is perpendicular to \\( \overleftrightarrow { b q } \\)?
\\( m \angle b x f = m \angle w x q \\)
\\( m \angle b x f + m \angle b x w = 180 ^ { \circ } \\)
\\( m \angle f x r = m \angle r x q \\)
\\( m \angle f x r + m \angle r x q = 90 ^ { \circ } \\)

Explanation:

Step1: Recall the definition of perpendicular lines

Perpendicular lines intersect at a right - angle (\(90^{\circ}\)). When two lines are perpendicular, the adjacent angles formed by their intersection are right angles.

Step2: Analyze each option

• Option \(m\angle BXF=m\angle WXQ\): This just shows that two vertical angles are equal. Vertical angles are always equal, and this does not imply that the lines are perpendicular.
• Option \(m\angle BXF + m\angle BXW=180^{\circ}\): This shows that two adjacent angles are supplementary (form a linear pair). This is a property of any two adjacent angles formed by two intersecting lines, not just perpendicular lines.
• Option \(m\angle FXR=m\angle RXQ\): This shows that two angles are equal. But this does not give information about the measure of the angles being \(90^{\circ}\).
• Option \(m\angle FXR + m\angle RXQ = 90^{\circ}\): If \(\overrightarrow{FW}\) and \(\overrightarrow{BQ}\) intersect at \(X\) and \(\angle FXR\) and \(\angle RXQ\) are adjacent angles formed by the intersection of \(\overrightarrow{FW}\) and \(\overrightarrow{BQ}\) (assuming \(R\) is a point such that \(\angle FXR\) and \(\angle RXQ\) are part of the angles at the intersection of \(\overrightarrow{FW}\) and \(\overrightarrow{BQ}\)), and their sum is \(90^{\circ}\), then the lines \(\overrightarrow{FW}\) and \(\overrightarrow{BQ}\) are perpendicular (since the angle between them is \(90^{\circ}\)).

Answer:

\(m\angle FXR + m\angle RXQ = 90^{\circ}\)