1. The problem asks to construct a line perpendicular to a given line segment passing through a specific point. 2. To construct a perpendicular line, we use the rule that perpendicular lines intersect at a 90° angle. 3. Draw the given line segment and mark the point through which the perpendicular line must pass. 4. Using a protractor or compass, measure a 90° angle at the point relative to the line segment. 5. Draw the line through the point at this 90° angle; this is the perpendicular line. 6. Next, construct a line parallel to the line segment passing through the point. 7. Parallel lines have the same slope and never intersect. 8. Using a compass, copy the angle the line segment makes with a horizontal reference line at the point. 9. Draw a line through the point with this copied angle; this line is parallel to the original segment. 10. To construct a rectangle, extend the perpendicular and parallel lines from the point to form right angles and equal opposite sides. 11. Draw the original line segment and the constructed perpendicular and parallel lines to form four right angles. 12. Connect the endpoints to complete the rectangle. 13. The rectangle has opposite sides equal and all angles 90°, satisfying the rectangle properties. 14. For the parallelogram, draw a new line segment and point. 15. Construct a line parallel to the new segment through the point by copying the angle as before. 16. From the endpoints of the original segment, draw lines parallel to the new segment to form a four-sided figure with opposite sides parallel. 17. Connect the vertices to complete the parallelogram. 18. The parallelogram has opposite sides parallel and equal in length. 19. For the trapezoid, draw another line segment and point. 20. Construct a line parallel to the new segment through the point. 21. From the endpoints of the original segment, draw lines to form a quadrilateral with only one pair of parallel sides. 22. Connect the vertices to complete the trapezoid. 23. For the square, draw another line segment and point. 24. Construct a line perpendicular to the segment through the point. 25. Using the length of the segment, mark equal lengths along the perpendicular and original segment from the point. 26. Construct parallel lines to these segments to form four equal sides and four right angles. 27. Connect the vertices to complete the square. 28. The square has all sides equal and all angles 90°, which justifies it as a square. 29. The method for constructing parallel lines involves copying an angle because parallel lines have equal corresponding angles. 30. The method for constructing perpendicular lines relies on the perpendicular bisector method because it ensures a 90° angle by bisecting the segment at right angles. This completes the constructions and explanations.