Question

use the diagram and the given and prove statements. given: m ⊥ p and n ⊥ p prove: m || n statements 1. m ⊥ p and n ⊥ p 2. m∠1 = 3. m || n reasons 1. given 8. 9. 10.

Explanation:

Step1: Recall perpendicular - angle property

Since \(m\perp p\) and \(n\perp p\), by the definition of perpendicular lines, \(\angle1 = 90^{\circ}\) and \(\angle2=90^{\circ}\). So \(m\angle1 = 90^{\circ}\) and \(m\angle2 = 90^{\circ}\).

Step2: Use corresponding - angle postulate

If two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel. Here, \(\angle1\) and \(\angle2\) are corresponding angles. Since \(m\angle1=m\angle2 = 90^{\circ}\), we can conclude that \(m\parallel n\).
For the statements and reasons:

• Statement 2: Since \(m\perp p\) and \(n\perp p\), the measure of the angles formed by the perpendicular lines and the transversal \(p\) are right - angles. So \(m\angle1 = 90^{\circ}\) and \(m\angle2 = 90^{\circ}\).
• Reason 9: Definition of perpendicular lines (if a line is perpendicular to another line, the angles formed are \(90^{\circ}\)).
• Reason 10: Corresponding angles postulate (if corresponding angles are congruent, the two lines are parallel).

Answer:

1. \(90^{\circ}\)
2. \(90^{\circ}\)
3. Definition of perpendicular lines
4. Corresponding angles postulate