Introduction Geometry construction tools can also be used to create perpendicular and parallel lines. While performing each construction, it is important to remember that the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but in constructions this is not allowed. You can adjust the opening of your compass to verify that lengths are equal. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts Perpendicular Lines and Bisectors Perpendicular lines are two lines that intersect at a right angle (90˚). A perpendicular line can be constructed through the midpoint of a segment. This line is called the perpendicular bisector of the line segment. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued It is impossible to create a perpendicular bisector of a line, since a line goes on infinitely in both directions, but similar methods can be used to construct a line perpendicular to a given line. It is possible to construct a perpendicular line through a point on the given line as well as through a point not on a given line. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Bisector of a Line Segment Using a Compass To construct a perpendicular bisector of , put the sharp point of your compass on endpoint A. Open the compass wider than half the distance of . Make a large arc intersecting . Without changing your compass setting, put the sharp point of the compass on endpoint B. Make a second large arc. It is important that the arcs intersect each other. (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Use your straightedge to connect the points of intersection of the arcs. Label the new line . Do not erase any of your markings. is perpendicular to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Bisector of a Line Segment Using Patty Paper Use a straightedge to construct onto patty paper. Fold the patty paper so point A meets point B. Be sure to crease the paper. Unfold the patty paper. Use your straightedge to mark the creased line. Label the new line . is perpendicular to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Line Through a Point on the Given Line Using a Compass To construct a perpendicular line through the point, A, on a line, put the sharp point of your compass on point A. The opening of the compass does not matter, but try to choose a setting that isn’t so large or so small that it’s difficult to make markings. Make an arc on either side of point A on the line. Label the points of intersection C and D. Place the sharp point of the compass on point C. Open the compass so it extends beyond point A. (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Create an arc on either side of the line. Without changing your compass setting, put the sharp point of the compass on endpoint D. Make a large arc on either side of the line. It is important that the arcs intersect each other. Use your straightedge to connect the points of intersection of the arcs. Label the new line . Do not erase any of your markings. is perpendicular to line through point A. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Line Through a Point on the Given Line Using Patty Paper Use a straightedge to construct a line, , on the patty paper. Label a point on the line A. Fold the patty paper so the line folds onto itself through point A. Be sure to crease the paper. Unfold the patty paper. Use your straightedge to mark the creased line. Label the new line . Line is perpendicular to line through point A. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Line Through a Point Not on the Given Line Using a Compass To construct a perpendicular line through the point, G, not on the given line , put the sharp point of your compass on point G. Open the compass until it extends farther than the given line. Make a large arc that intersects the given line in exactly two places. Label the points of intersection C and D. (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Without changing your compass setting, put the sharp point of the compass on point C. Make a second arc below the given line. Without changing your compass setting, put the sharp point of the compass on point D. Make a third arc below the given line. The third arc must intersect the second arc. Label the point of intersection E. Use your straightedge to connect points G and E. Label the new line . Do not erase any of your markings. Line is perpendicular to line through point G. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Perpendicular Line Through a Point Not on the Given Line Using Patty Paper Use a straightedge to construct a line, , on the patty paper. Label a point not on the line, G. Fold the patty paper so the line folds onto itself through point G. Be sure to crease the paper. Unfold the patty paper. Use your straightedge to mark the creased line. Label the new line . Line is perpendicular to line through point G. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Parallel Lines Parallel lines are lines that either do not share any points and never intersect, or share all points. Any two points on one parallel line are equidistant from the other line. There are many ways to construct parallel lines. One method is to construct two lines that are both perpendicular to the same given line. 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Parallel Line Using a Compass To construct a parallel line through a point, A, not on the given line , first construct a line perpendicular to . Put the sharp point of your compass on point A. Open the compass until it extends farther than line Make a large arc that intersects the given line in exactly two places. Label the points of intersection C and D. (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Without changing your compass setting, put the sharp point of the compass on point C. Make a second arc below the given line. Without changing your compass setting, put the sharp point of the compass on point D. Make a third arc below the given line. The third arc must intersect the second arc. Label the point of intersection E. Use your straightedge to connect points A and E. Label the new line . Line is perpendicular to line . (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Construct a second line perpendicular to line . Put the sharp point of your compass on point A. Open the compass until it extends farther than line . Make a large arc that intersects line in exactly two places. Label the points of intersection F and G. Without changing your compass setting, put the sharp point of the compass on point F. Make a second arc to the right of line . (continued) 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Without changing your compass setting, put the sharp point of the compass on point G. Make a third arc to the right of line . The third arc must intersect the second arc. Label the point of intersection H. Use your straightedge to connect points A and H. Label the new line . Do not erase any of your markings. Line is perpendicular to line . Line is parallel to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Key Concepts, continued Constructing a Parallel Line Using Patty Paper Use a straightedge to construct line on the patty paper. Label a point not on the line A. Fold the patty paper so the line folds onto itself through point A. Be sure to crease the paper. Unfold the patty paper. Fold the new line onto itself through point A. Use your straightedge to mark the second creased line. Label the new line . Line is parallel to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Common Errors/Misconceptions inappropriately changing the compass setting moving the patty paper before completing the construction not creating large enough arcs to find the point of intersection attempting to measure lengths and angles with rulers and protractors 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice Example 3 Use a compass and a straightedge to construct a line perpendicular to line through point B that is not on the line. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued Draw line with point B not on the line. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued Make a large arc that intersects line . Put the sharp point of your compass on point B. Open the compass until it extends farther than line . Make a large arc that intersects the given line in exactly two places. Label the points of intersection F and G, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued Make a set of arcs above line . Without changing your compass setting, put the sharp point of the compass on point F. Make a second arc above the given line, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued Without changing your compass setting, put the sharp point of the compass on point G. Make a third arc above the given line. The third arc must intersect the second arc. Label the point of intersection H, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued Draw the perpendicular line. Use your straightedge to connect points B and H. Label the new line , as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued ✔ Do not erase any of your markings. Line is perpendicular to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 3, continued http://www.walch.com/ei/00137 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice Example 4 Use a compass and a straightedge to construct a line parallel to line through point C that is not on the line. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Draw line with point C not on the line. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Construct a line perpendicular to line through point C. Make a large arc that intersects line . Put the sharp point of your compass on point C. Open the compass until it extends farther than line . Make a large arc that intersects the given line in exactly two places. Label the points of intersection J and K, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Make a set of arcs below line . Without changing your compass setting, put the sharp point of the compass on point J. Make a second arc below the given line. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Without changing your compass setting, put the sharp point of the compass on point K. Make a third arc below the given line. Label the point of intersection R. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Draw the perpendicular line. Use your straightedge to connect points C and R. Label the new line . Do not erase any of your markings. Line is perpendicular to line . 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Construct a second line perpendicular to line . Put the sharp point of your compass on point C. Make a large arc that intersects line on either side of point C. Label the points of intersection X and Y, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Make a set of arcs to the right of line . Put the sharp point of your compass on point X. Open the compass so that it extends beyond point C. Make an arc to the right of line , as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Without changing your compass setting, put the sharp point of the compass on point Y. Make another arc to the right of line . Label the point of intersection S, as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Draw the perpendicular line. Use your straightedge to connect points C and S. Label the new line , as shown on the next slide. 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued Do not erase any of your markings. Line is perpendicular to line . Line is parallel to line . ✔ 1.2.3: Constructing Perpendicular and Parallel Lines

Guided Practice: Example 4, continued http://www.walch.com/ei/00138 1.2.3: Constructing Perpendicular and Parallel Lines