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Transcript

Constructing Parallelograms There are five ways to construct a quadrilateral that is a parallelogram. The following are methods using a compass and straightedge to construct a parallelogram using one of these five methods, based on having an original angle to start from. 1. 2. 3. 4. 5. Corresponding Angles (CA’s)are congruent Alternate Exterior Angles (AEA’s) are congruent Alternate Interior Angles (AIA’s) are congruent Two lines are perpendicular to the same line (perpendicular transversals) Same Side Interior Angles (SSIA’s) are congruent (This construction in not shown, since it is drawn from CA’s) Construction steps for #1 (Both pairs of opposite sides are parallel using Corresponding Angles - CA) (1) Draw an original angle B C (2) Pick a random point along one side of angle B and label it “C” and another random point A on the other side. B 1 A (3) Copy the angle inside B (1) to the same direction to point C (corresponding angles along transversal BC) and to point A (corresponding angles along transversal BA) (a) Make congruent measuring arcs in the same direct 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 D 2 1 3 A 1 2, CA, AB ǁ CD 1 3, CA, BC ǁ AD ABCD Construction steps for #2 (Both pairs of opposite sides are parallel using Alternate Exterior Angles - AEA) (1) Draw an original angle B C (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 must extend the lines past point B for each line to copy the External Angles (3) Copy the angle outside B (1) to the opposite direction to point C (corresponding angles along transversal BC) and to point A (corresponding angles along transversal BA) (a) Make congruent measuring arcs in the same direct 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. B 1 A D C 2 B 1 1 2, AEA, AB ǁ CD 1 3, AEA, BC ǁ AD 3 A ABCD Construction steps for #3 (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 Construction steps for #4 (Both pairs of opposite sides are parallel using Perpendicular Transversals) (1) Draw an original angle B (2) Pick a random point along one side of angle B and label it “W”. B W Note: You should choose the points farther away to avoid crossing the semicircles (3) Make a perpendicular transversal from point W (a) Make semicircle about point W (b) From each intersections for the semicircle to the line WB make congruent arcs that intersect (c) Extend a line from point W thru the intersection of the arc in (b) (4) Pick a random point on this transversal “X” and make another perpendicular line from that point by repeating steps (a) thru (c) above (a) Make semicircle about point X (b) From each intersections for the semicircle to the line XW make congruent arcs that intersect (c) Extend a line from point X thru the intersection of the arc in (b) Make perpendicular box X Make perpendicular box B W (5) Extend the line from B to cross this parallel line (thru X), Label this point C and Repeat steps (3) and (4) for this line BC (6) Pick a random point Y on BC Make a perpendicular transversal from point Y (a) Make semicircle about point Y (b) From each intersections for the semicircle to the line BC make congruent arcs that intersect (c) Extend a line from point Y thru the intersection of the arc in (b) (7) Pick a random point on this transversal “Z” and make another perpendicular line from that point by repeating steps (a) thru (c) above X C Y B Make perpendicular box W Z Make perpendicular box (8) Label the two points where this final line (perpendicular to ZY, thru point Z) intersects the two parallel lines thru points X and W as points A and D X C D Y A B W Z (9) Make a construction statement in a box showing that opposite sides are parallel AB WX, CD WX so AB ǁ CD BC YZ, AD YZ so BC ǁ AD ABCD