Step-by-step guide for constructing parallel and perpendicular lines using geometric tools.

The image you uploaded contains instructions for constructing parallel and perpendicular lines using a compass and straightedge. Below, I will explain the solutions step by step for each construction method.

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1. Constructing Parallel Lines

#### Given:
- A line \( \overline{AB} \) and a point \( P \) not on the line.

#### Steps:
1. Draw a line through \( P \) intersecting \( \overline{AB} \):
- Use a straightedge to draw a line through point \( P \) that intersects \( \overline{AB} \) at some point \( C \).

2. Label the intersection point as \( C \):
- The line through \( P \) intersects \( \overline{AB} \) at \( C \).

3. Place the compass point at \( C \) and draw an arc intersecting both lines:
- Place the compass at \( C \) and draw an arc that intersects \( \overline{AB} \) at point \( D \) and the line through \( P \) at point \( E \).

4. Without changing the compass setting, place the compass point at \( P \) and draw an arc:
- Without adjusting the compass, place the compass at \( P \) and draw an arc that intersects the line through \( P \) at point \( F \).

5. Measure the distance \( DE \) with the compass:
- Set the compass to the distance between \( D \) and \( E \).

6. Using this new compass setting, place the compass point at \( F \) and draw an arc intersecting the previous arc:
- Place the compass at \( F \) and draw an arc that intersects the arc drawn in step 4 at point \( G \).

7. Draw a line through \( P \) and \( G \):
- Use a straightedge to draw a line through points \( P \) and \( G \). This line is parallel to \( \overline{AB} \).

#### Explanation:
This construction uses the principle of corresponding angles. By transferring the angle \( \angle DCE \) (formed by \( \overline{AB} \) and the transversal) to the same position at point \( P \), we ensure that the new line is parallel to \( \overline{AB} \).

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2. Constructing a Perpendicular Line at a Given Point on the Line

#### Given:
- A line \( \overline{AB} \) and a point \( P \) on the line.

#### Steps:
1. Set the compass to any radius greater than half the distance from \( P \) to either end of the line segment:
- Choose a radius larger than half the distance from \( P \) to either endpoint of the line segment.

2. With the compass point at \( P \), draw an arc intersecting the line at two points:
- Place the compass at \( P \) and draw an arc that intersects \( \overline{AB} \) at points \( Q \) and \( R \).

3. Without changing the compass setting, place the compass point at \( Q \) and draw an arc above \( P \):
- Place the compass at \( Q \) and draw an arc above \( P \).

4. Without changing the compass setting, place the compass point at \( R \) and draw another arc intersecting the first arc:
- Place the compass at \( R \) and draw an arc that intersects the arc drawn in step 3 at point \( S \).

5. Draw a line through \( P \) and \( S \):
- Use a straightedge to draw a line through points \( P \) and \( S \). This line is perpendicular to \( \overline{AB} \).

#### Explanation:
This construction uses the property of circles and perpendicular bisectors. By drawing arcs from points \( Q \) and \( R \) with the same radius, the intersection point \( S \) lies directly above or below \( P \), forming a right angle with \( \overline{AB} \).

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3. Constructing a Perpendicular Line at a Given Point NOT on the Line

#### Given:
- A line \( \overline{AB} \) and a point \( P \) not on the line.

#### Steps:
1. Set the compass to any radius greater than the distance from \( P \) to the line:
- Choose a radius larger than the perpendicular distance from \( P \) to \( \overline{AB} \).

2. With the compass point at \( P \), draw an arc intersecting the line at two points:
- Place the compass at \( P \) and draw an arc that intersects \( \overline{AB} \) at points \( Q \) and \( R \).

3. Without changing the compass setting, place the compass point at \( Q \) and draw an arc above and below the line:
- Place the compass at \( Q \) and draw arcs above and below \( \overline{AB} \).

4. Without changing the compass setting, place the compass point at \( R \) and draw another arc intersecting the first arcs:
- Place the compass at \( R \) and draw arcs that intersect the arcs drawn in step 3 at points \( S \) (above) and \( T \) (below).

5. Draw a line through \( S \) and \( T \):
- Use a straightedge to draw a line through points \( S \) and \( T \). This line is perpendicular to \( \overline{AB} \) and passes through \( P \).

#### Explanation:
This construction is similar to the previous one but involves finding the perpendicular from a point not on the line. By creating arcs from points \( Q \) and \( R \) and finding their intersections, we locate points that form a perpendicular line passing through \( P \).

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Final Answer:

The constructions are explained step by step using geometric principles of angles, arcs, and perpendicular bisectors. Each method ensures accuracy in drawing parallel and perpendicular lines using only a compass and straightedge.

Boxed Final Answer:
\[
\boxed{\text{See detailed steps above for each construction method.}}
\]