Geometric Constructions Review 
                                                                                              
                In Geometry, constructions utilize only two tools, a straightedge (unmarked ruler) and the compass.  You 
                do not measure or draw freehand when doing constructions! 
                 
                Basic Geometrical Constructions: 
                         I.     Copy a Line Segment 
                        II.     Copy an Angle 
                      III.      Bisect a Line Segment (Construct a Perpendicular Bisector) 
                      IV.       Bisect an Angle (Construct an Angle Bisector) 
                       V.       Construct Parallel Lines 
                      VI.       Construct Perpendicular Lines 
                 
                 
                I. Copy a Line Segment: 
                                                                                                              A                                                    B 
                Given:  AB   
                Task:   To construct a line segment congruent to  AB  
                 
                Directions: 
                1)  Draw a reference line slightly longer than  AB . 
                2)  Draw point C on one end of the reference line. 
                3)  Place the point of the compass on point A. 
                4)  Stretch the compass so the pencil is exactly on point B. 
                5)  Using this same compass setting, place the compass point on C and draw an arc that crosses the 
                      reference line.  Label the intersection D.  AB CD. 
                 
                 
                II. Copy an Angle: 
                 
                Given: A  
                Task:  To construct an angle congruent to A 
                 
                Directions:                                                                                          A 
                1)  Draw a ray. Label the endpoint D.  
                2)  Place the compass tip at the vertex of A.  
                      Draw an arc across both sides of A.  
                3)  Label the points of intersection with the rays B and C.  
                4)  Using this same compass setting, place the compass tip at  
                      point D (on the new ray) and draw a long arc across the ray.  
                5)  Label the intersection point E.  
                6)  Set the compass so that it is the width of BC.  
                7)  Using this same compass setting, place the compass tip at  
                      point E and draw an arc, intersecting the arc from step 4.  
                8)  Label the intersection F.  
                9)  Draw DF.  EDF BAC  
                 
                 
            III. Bisect a Line Segment (Construct a Perpendicular Bisector): 
                     
            Given:  AB  
            Task:  Bisect  AB 
             
            Directions: 
            1)  Place your compass point on A and stretch the  
                compass slightly more than half way to point B,  
                but not beyond B.   
            2)  With this length, swing a large arc that will go                     A                                      B 
                BOTH above and below AB.  
            3)  Using this same compass setting, place the compass  
                point on B and swing the arc again BOTH above  
                and below AB.  The two pairs of arcs you created should  
                intersect. 
            4)  Label the points of intersection as C and D. 
            5)  With your straightedge, connect C and D.  
                 CD bisects AB and CD AB.   
            6)  Label the intersection of  AB  and CD as point E.   AE  EB and CE  ED. 
             
             
             
                 
            IV. Bisect an Angle (Construct an Angle Bisector): 
                     
            Given: A 
            Task:  Bisect A 
             
            Directions: 
            1)  Place the compass tip at point A. Draw an arc that  
                intersects both rays of the angle. Label the points  
                of intersection B and C.  
            2)  Place the compass tip at point B and draw an arc in                  A 
                the interior of A.  
            3)  Using this same compass setting, place the compass  
                tip at point C and draw an arc that intersects the arc  
                you drew in step 2.  
            4)  Label the point of intersection Q.  
            5)  Use a straightedge to draw  AQ .  
                This is the angle bisector of A.  BAQ QAC   
             
             
             
             
             
             
             
             
             
            V.  Parallel Lines: 
             
            Given: Point P and line m                                                     P 
            Task:  Construct a line through P parallel to line m. 
             
            Directions: 
            1)  With your straightedge, draw a transversal through                                                       m 
                line m and point P.  This is simply a straight line  
                which runs through P and intersects line m. 
            2)  Construct a copy of the angle formed by the transversal  
                and the given line such that the copy is located UP at point P.   
                The vertex of your copied angle will be point P. 
            3)  When the copy of the angle is complete, you will have two parallel lines. 
             
             
            VI.  (a) Perpendicular Lines – from a point ON a line: 
                    
            Given: Point A on line m 
            Task:  Construct a line through A perpendicular to line m. 
             
            Directions: 
            1)  Place compass tip at point A. Using any compass opening                                                   m 
                less than the length of m, draw two arcs intersecting                       A 
                line m on both sides of A. Label these points C and D.  
            2)  Place compass tip at point C. Adjust the compass so that it is  
                slightly longer than ½ CD and draw an arc above m.  
            3)  Using the same compass opening, place compass point  
                at point D. Draw an arc above m intersecting your first arc.  
            4)  Label the intersection point X.  
            5)  Use a straightedge to draw line  AX .  AX is perpendicular to m through point A.  
             
             
            VI.  (b) Perpendicular Lines – from a point OFF a line: 
                    
            Given: line m and point P NOT on the line                                              P 
            Task:  Construct a perpendicular from point P to line m. 
             
            Directions: 
            1)  Place compass tip at point P. Draw arcs intersecting line m  
                at two points. Label these points A and B.                                                                 m 
            2)  Using a compass opening greater than ½  AB, place compass 
                point at point A. Draw an arc below line m. 
            3)  Using the same compass opening, place compass tip at point B 
                and draw an arc from point B below line m, intersecting  the arc  
                from step 2. 
            4)  Label the intersection point X. 
            5)  Draw line PX . PX is perpendicular to m through point P.r