Question

1 \\( \angle qtr \cong \angle ptq \\) given
2 \\( m \angle ptq + m \angle qtr = 180 ^ { \circ } \\) angles forming a linear pair sum to \\( 180 ^ { \circ } \\)
3 \\( m \angle ptq + m \angle ptq = 180 ^ { \circ } \\) substitution
4 \\( m \angle ptq = 90 ^ { \circ } \\) properties of addition, subtraction, multiplication, and division
5

Explanation:

Step1: Analyze the previous steps

We know from step 4 that \(m\angle PTQ = 90^{\circ}\).

Step2: Determine the relationship between the lines

Since \(m\angle PTQ = 90^{\circ}\), by the definition of perpendicular lines (if the measure of the angle between two lines is \(90^{\circ}\), the lines are perpendicular), we can conclude that \(PS\perp QR\).

Answer:

\(PS\perp QR\) (by the definition of perpendicular lines, since \(m\angle PTQ = 90^{\circ}\))