Construct 4 pairs of parallel lines in different orientations.
Step 1: Draw a line m and a transversal l intersecting m at point A.
Step 2: Mark the angle between m and l.
Step 3: Choose a point B on l through which the parallel line should pass.
Step 4: Copy the angle at A to point B, creating the same corresponding angle.
Step 5: The line through B making this angle with l will be parallel to m.
Step 6: Repeat this process with different orientations of the initial line and transversal to create parallel lines in various directions.
Why it works: Parallel lines can be constructed by copying angles. When a transversal intersects two lines, if the corresponding angles are equal, the lines are parallel. By copying the angle that line m makes with transversal l at a new point B on the transversal, we create a line through B that is parallel to m. This method uses only ruler and compass, relying on the angle-copying technique.
Explanation
The chapter explains that parallel lines can be constructed by copying angles. When a transversal intersects two lines, if the corresponding angles are equal, the lines are parallel. By copying the angle that line m makes with transversal l at a new point B on the transversal, we create a line through B that is parallel to m. This method uses only ruler and compass, relying on the angle-copying technique.
Solution Steps
Draw a line m and a transversal l intersecting m at point A.
Mark the angle between m and l.
Choose a point B on l through which the parallel line should pass.
Copy the angle at A to point B, creating the same corresponding angle.
The line through B making this angle with l will be parallel to m.
Repeat this process with different orientations of the initial line and transversal to create parallel lines in various directions.