Question

3 - 4 perpendicular lines

1. write a two - column proof.

given: \\( \angle 1 \cong \angle 2, \ell \perp n \\)
prove: \\( \ell \perp p \\)

Explanation:

Step1: State the given information

Given: \(\angle1\cong\angle2\), \(\ell\perp n\)

Step2: Use the definition of perpendicular lines

Since \(\ell\perp n\), by the definition of perpendicular lines, the angle between \(\ell\) and \(n\) is \(90^{\circ}\). Let's call the angle between \(\ell\) and \(n\) as \(\angle3\), so \(\angle3 = 90^{\circ}\)

Step3: Use the converse of the alternate - interior angles theorem

Because \(\angle1\cong\angle2\), by the converse of the alternate - interior angles theorem, \(p\parallel n\)

Step4: Use the property of parallel lines and a transversal

If \(p\parallel n\) and \(\ell\) is a transversal, then the angle between \(\ell\) and \(p\) (let's call it \(\angle4\)) is congruent to the angle between \(\ell\) and \(n\) (\(\angle3\)) because of the corresponding angles postulate. So \(\angle4=\angle3\)

Step5: Substitute the value of \(\angle3\)

Since \(\angle3 = 90^{\circ}\), then \(\angle4=90^{\circ}\)

Step6: Use the definition of perpendicular lines

By the definition of perpendicular lines, if the angle between two lines is \(90^{\circ}\), the lines are perpendicular. So \(\ell\perp p\)

Answer:

Statements	Reasons
2. \(\angle3 = 90^{\circ}\)	Definition of perpendicular lines (\(\ell\perp n\))
3. \(p\parallel n\)	Converse of alternate - interior angles theorem (\(\angle1\cong\angle2\))
4. \(\angle4=\angle3\)	Corresponding angles postulate (\(p\parallel n\), \(\ell\) is a transversal)
5. \(\angle4 = 90^{\circ}\)	Substitution (\(\angle3 = 90^{\circ}\))
6. \(\ell\perp p\)	Definition of perpendicular lines (\(\angle4 = 90^{\circ}\))