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# How to construct a line parallel to a given line through a given point without drawing the line

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Question: The figure shows line r, points P and T on line r, and point Q not on line r. Also shown is ray PQ. Consider the partial construction of a line parallel to r through point Q. What would be the final step in the construction?

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To complete the construction of a line parallel to line r through point Q, follow these steps as the final step: 1. Place the compass at point P and draw an arc that intersects both line r and ray PQ. 2. Let the intersection points be A on line r and B on ray PQ. 3. Measure the length of segment PA using the compass, and then draw another arc from point T along the same side of line r as point Q, so that the arc crosses line r and creates a new point C. 4. Place the compass at point B and measure the length of segment AB. 5. Without changing the width of the compass, place the compass at point C and draw another arc that intersects the previously drawn arc with center T. 6. Let the intersection point be D. 7. Use a straightedge to draw a line through points Q and D. This line will be parallel to line r. This construction follows the concept that if two lines are parallel, and their corresponding angles are congruent, then their alternate interior angles are also congruent.

June 21, 2023, 10:38 a.m.

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