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Constructing Parallel Lines Videos: Constructing Parallel Lines: http://www.teachertube.com/video/constructing-parallel-lines-with-a-safety-com-228901 Given a line and a point, construct a line through the point, parallel to the given line using Corresponding Angles (CA) 1. Begin with point P and 2. Draw an arbitrary line 3. Center the compass at 4. Set the compass radius to line k. through point P, intersecting point Q and draw* an arc the distance between the two line k. Call the intersection intersecting both lines. intersection points of the first point Q. Now the task is to Without changing the radius arc. Now center the compass construct an angle with of the compass, center it at at the point where the second vertex P, congruent to the point P and draw another arc intersects line PQ. Mark angle of intersection. arc. the arc intersection point R. 5. Label the new line PR 6. Box a statement to show (or cursive l) and the two 4 angles (example: 2 and 4). l the two angles are congruent, corresponding 2 2 4, CA, l ǁ k angles (CA), so line l is parallel to line k. *Note: these pair of angles can go in any of the four directions (in pairs) from the two vertices Q and P Given a line and a point, construct a line through the point, parallel to the given line using Alternate Exterior Angles 1. Begin with point P and 2. Draw an transversal line 3. Center the compass at 4. Span the compass radius line k. through point P, intersecting point Q and draw* an arc to the distance between the line k. Label the intersection intersecting both lines. two intersection points of the point Q. Now construct an Without changing the radius first arc. Now center the angle with vertex P, of the compass, center it at compass at the point where congruent to one of the two point P and draw another the second arc intersects exterior angles (1 or 2) arc 180⁰ in the opposite transversal line PQ. Mark the at Q. direction (make sure they arc intersection point R. both pass thru the transversal). R P • P • Q Q • 1 5. Label the new line PR k 2 R P (or cursive l) and the two l Q • 1 2 Q • 1 2 k • 1 2 6. Box a statement to show the two angles are 4 P • angles (example: 2 and 4). P • congruent, Alternate 2 4, AEA, l ǁ k Exterior Angles (AEA), so k line l is parallel to line k. *Note: these pair of angles can be either the obtuse or acute angles (in pairs) from the two vertices Q and P k Given a line and a point, construct a line through the point, parallel to the given line using Alternate Interior Angles 1. Begin with point P and line 2. Draw a transversal 3. Center the compass at 4. Span the compass radius to k. (Make sure the point is line through point P, point Q and draw* an arc the distance between the two farther away than is was in intersecting line k. intersecting both lines. intersection points of the first prior examples) Label the intersection Without changing the radius arc. Now center the compass at point Q. Now construct of the compass, center it at the point where the second arc an angle with vertex P, point P and draw another intersects transversal line PQ. congruent to one of the arc 180⁰ in the opposite Mark the arc intersection point two interior angles (3 direction (make sure they R. or 4) at Q. both pass thru the transversal). P • P • 3 4 3 • k Q P • (or cursive l) and the two 3 • Q Q 3 k 4 • Q k 6. Box a statement to show the l 5 P • 4 • k 5. Label the new line PR angles (example: 3 and 5). P • two angles are congruent, Alternate Interior Angles (AIA), 4 3 5, AIA, l ǁ k so line l is parallel to line k. k *Note: these pair of angles can be either the obtuse or acute angles (in pairs) from the two vertices Q and P Constructing a perpendicular to a line from a point OFF the line (“Dropping a Perpendicular”) After doing this 1 Your work should look like this After doing this Start with a line and point R which is not on that line. 2 Set the compasses' width to a approximately 50% more than the distance to the line. The exact width does not matter. Place the compasses on the given external point R. 3 Draw an arc across the line on each side of R, making sure not to adjust the compasses' width in between. Label these points P and Q Your work should look like this 4 At this point, you can adjust the compasses' width. Recommended: leave it as is. From each point P,Q, draw an arc below the line so that the arcs cross. 5 Place a straightedge between R and the point where the arcs intersect. Draw the perpendicular line from R to the line, or beyond if you wish. 6 Done. This line is perpendicular to the first line and passes through the point R. It also bisects the segment PQ (divides it into two equal parts) Constructing perpendicular from point ON a line (“Pulling” a perpendicular) After doing this Start with a line and point K on that line. 1 Set the compasses' width to a medium setting. The actual width does not matter. 2 Without changing the compasses' width, mark a short arc on the line at each side of the point K, forming the points P,Q. These two points are thus the same distance from K. 3 Increase the compasses to almost double the width (again the exact setting is not important). 4 From P, mark off a short arc above K Your work should look like this After doing this 5 Without changing the compasses' width repeat from the point Q so that the the two arcs cross each other, creating the point R 6 Using the straight edge, draw a line from K to where the arcs cross. 7 Done. The line just drawn is a perpendicular to the line at K Your work should look like this Advanced Construction (Both pairs of opposite sides are parallel using Alternate Interior Angles - AIA) (1) Draw an original angle B (2) Pick a random point along one side of angle B and label it “C” and another random point A on the other side. Note: You should choose the points farther away to avoid crossing the measuring arcs. (3) Copy the angle inside B (1) to the opposite direction to point C (NE and SW along transversal BC) and to point A (NE and SW along transversal BA) (a) Make congruent measuring arcs in the same opposite directions from each point (b) Span the arc between the two sides of 1 and copy this span onto the arc about point C and point A (c) Extend a line from point C thru the intersection of the measuring arc in (a) and the span in (b) and likewise with the arcs about point A (d) Label the new angle by points C and A as 2 and 3 (4) Where the two lines from points C and A intersect is your final vertex. Label it “D” (5) Make a construction statement in a box showing that opposite sides are parallel. C B 1 A C 2 B D 1 A 3 1 2, AIA, AB ǁ CD 1 3, AIA, BC ǁ AD ABCD hw24 Do each Geometric Construction twice (using a compass and straight edge). Mark all new ANGLES, LINES, and congruent parts and box a statement describing the construction. Construct eight pairs of parallel line pairs, given a line and a point off the line. Using Converse of CAP (twice: one acute /. one obtuse) (1) Draw a line from the NW to SE and a point OFF the line above it to the NE (2) Draw a transversal thru the line and the point approximately horizontal (3) Pick one acute angle and copy it in the same direction onto the point off the line. (use the transversal as your “image ray”) Repeat, this time choosing the obtuse angle in step 3 Using Converse of AEAT (twice: one acute /. one obtuse) (1) Draw a line from the NE to SW and a point OFF the line above it to the NW (2) Draw a transversal thru the line and the point approximately horizontal (3) Pick the EXTERNAL acute angle and copy it in the opposite direction onto the point off the line. (use the transversal as your “image ray”) Repeat, this time choosing the obtuse angle in step 3 Using Converse of AEAT (twice: one acute /. one obtuse) (1) Draw a vertical line towards the left side of your paper and a point OFF the line far to the right towards the SE (2) Draw a transversal thru the line and the point going NW to SE (3) Pick the INTERNAL acute angle and copy it in the opposite direction onto the point off the line. (use the transversal as your “image ray”) Repeat, this time choosing the obtuse angle in step 3 Using perpendicular transversals (twice: diagonal lines /. Horizontal and vertical lines) (1) Draw a vertical line towards the left side of your paper and a point “P” OFF the line to the right near its center. (2) Drop a perpendicular from this point thru the line (3) Pull another perpendicular from point P to the line you just drew in step 2 Repeat, this time starting with a diagonal line going NW to SE and the point OFF toward the NE in step 1.