1. The problem involves constructing a line parallel to a given line at a specified distance using geometric methods. 2. The key property used is that angles on the opposite side of a transversal intersecting parallel lines are equal. 3. Steps for construction: 1. Draw line AB. 2. Mark point C on AB and draw an arc intersecting AB at points X and Y. 3. With Y as center, draw an arc intersecting the previous arc at Q. 4. With O as center and same radius, draw another arc intersecting the previous arc at P. 5. With P and Q as centers, draw arcs intersecting at Z. 6. Connect C to Z and extend it; mark point D on CZ such that CD equals the given distance. 7. Angle ∠DCB is 90°, ensuring perpendicularity. 8. Cut line CZ at D equal to the given distance. 9. Repeat steps 1 to 6 with D as center to draw line DE. 10. Line DE is parallel to AB by construction. 4. Applying this to specific problems: Problem 1: - Draw line l. - Draw perpendicular at any point on l. - Mark point X on perpendicular 4 cm from l. - Draw line through X parallel to l. Problem 2: - Draw segment PQ = 7 cm. - Construct parallel line 5 cm above PQ. Problem 3: - Draw segment PQ = 4.5 cm below. - Check if lines are parallel by verifying equal corresponding angles or equal distance. Problem 4: - Draw segment AB = 5.6 cm. - Construct parallel line 6.5 cm from AB. Problem 5: - Draw segment AB = 6.8 cm. - Take point P outside AB. - Using ruler and compass, draw line through P parallel to AB. Problem 6: - Draw triangle ABC. - Through A, draw line parallel to BC. 5. The unitary method is used to find required distances or units in constructions. 6. The construction ensures that the parallel line is at the exact given distance and maintains the properties of parallelism. Final answer: Using compass and ruler, parallel lines can be constructed at a given distance by replicating angles and distances as per the steps above, ensuring the lines are parallel and the distance between them is as specified.