Question

given: p ⊥ q and q || r
prove: p ⊥ r

Explanation:

Step1: Definition of perpendicular lines

If \(p\perp q\), then \(\angle1 = 90^{\circ}\) (by the definition of perpendicular lines, which form right - angles).

Step2: Corresponding angles postulate

Since \(q\parallel r\), \(\angle1\) and \(\angle2\) are corresponding angles. So, \(\angle1=\angle2\) (by the corresponding angles postulate: if two parallel lines are cut by a transversal, then corresponding angles are congruent).

Step3: Substitution

Since \(\angle1 = 90^{\circ}\) and \(\angle1=\angle2\), then \(\angle2 = 90^{\circ}\) (by substitution property).

Step4: Definition of perpendicular lines

If \(\angle2 = 90^{\circ}\), then \(p\perp r\) (by the definition of perpendicular lines: if two lines intersect to form a right - angle, then the lines are perpendicular).

Answer:

Statements	Reasons
\(\angle1 = 90^{\circ}\)	definition of perpendicular lines
\(\angle1=\angle2\)	corresponding angles postulate (\(q\parallel r\))
\(\angle2 = 90^{\circ}\)	substitution (\(\angle1 = 90^{\circ}\) and \(\angle1=\angle2\))
\(p\perp r\)	definition of perpendicular lines (\(\angle2 = 90^{\circ}\))