# Parallel, Perpendicular, Line Bisector and Angle Bisector

Topic:
Angles

## Drawing a parallel line

- Select the POINT tool (Window 2) and draw a point O on line r. - Select the COMPASS tool (Window 6). Then click on O and A (it represents the length opening of the compass) and again on O (represent the sharp end of the compass). -Select the INTERSECT tool (Window 3) and mark points B and C, which are intersection points between the circle with the line r. - Select the COMPASS tool (Window 6). Then click on point B and point A (it represents the length opening of the compass) and again on point C (it represents the sharp end of the compass). -Select the INTERSECT tool (Window 3) and mark point D, which is the point of intersection of the two circles. -Select the LINE tool (Window 3) and click on A and D. Label this line s. - Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles, points B, C and D, leaving only the lines and point A. -Select the RELATION (Window 8) and click on the two lines. What happens when you follow all the steps above? - Select the MOVE tool (Window 1). Move either point A or line r. What can you see?

## Perpendicular line (point not belonging to line)

- Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on E (represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark F and G. They are points of intersection of the circle with the line. - Select the COMPASS tool (Window 6). Then click on point F and point G (it represents the length opening of the compass) and again on point F (it represents the sharp end of the compass). After that, click on point G and point F (it represents the length opening of the compass) and again on G (it represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.   -Select the LINE tool (Window 4) and click on point E and point H. The intended perpendicular line will appear. Let us analyse it. - Select the INTERSECT tool (Window 3) and mark point I, point of intersection of points h and g.   - Select the  ANGLE tool (Window 9). Click on points E, I and C to mark the angle EIC (the vertex of the angle will always be the second point clicked). What is the measurement of this angle? - Select the SHOW / HIDE OBJECT tool (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E. -Select the RELATION tool (Window 8) and click on the two lines. What happens when you follow all the steps above? - Select the MOVE tool (Window 1). Move either point E or line g. What can you see?

## Perpendicular line (point belonging to line)

-Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on E (it represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark the intersections F and G of the circle with the line g.   - Select the COMPASS tool (Window 6). Then click on point F and point G (it represents the length opening of the compass) and again on point F (it represents the sharp end of the compass). After that, click on point G and point F (it represents the length opening of the compass) and again on G (represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.   -Select the LINE tool (Window 4) and click on point E and point H. The intended perpendicular line will appear. -Select the  ANGLE tool (Window 9). Click on points H, E and C to mark the angle HEC (the vertex of the angle will always be the second point clicked). What is the measurement of this angle? - Select the SHOW / HIDE OBJECT tool (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E. -Select the RELATION tool (Window 8) and click on the two lines. What happens when you follow all the steps above? - Select the MOVE tool (Window 1). Move either point E or line g. What can you see?

## Line Segment Bisector

- Select the COMPASS tool (Window 6). Then click on point A and point B (it represents the length opening of the compass) and again on point A (it represents the sharp end of the compass). After that click on point B and point A (it represents the length opening of the compass) and again on B (it represents the sharp end of the compass). - Select the option INTERSECT (Window 3) and mark C and F, which are the intersections between the two circles.   -Select the LINE tool (Window 4) and click on C and D. This is the aimed line bisector. - Select the INTERSECT (Window 3) and mark E, which is the intersection of g with segment AB.   - Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles, points C and D, leaving only the lines and point E. - Select the MOVE tool (Window 1)  Move point A or B. What can you see?

## Analysis 1

Proof that any point of the line has the same distance to A and to B.

## Analysis 2

Proof that the angle AEC measures 90º.

## Angle Bisector with visible vertex

-Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on C (it represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark F and G. They are the points of intersection between the circle and the rays (Half-lines) that form the angle.   - Select the COMPASS tool (Window 6). Then click on the segment AB (it represents the length opening of the compass) and on G (it represents the sharp end of the compass). Then click on segment AB (it represents the length opening of the compass) and on F (It represents the sharp end of the compass). - Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.   -Select the RAY (HALF-LINE) tool (Window 6) and click on C and H. This is the intended angle bisector. - Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles and the points F and G. -Select the ANGLE tool (Window 7). Click on points D, C and  H to mark the angle HEC (the vertex of the angle will always be the second point clicked). Also measure the HCE angle.  What can you see? - Select the MOVE tool (Window 1). Move either point E or line d. What can you see?

## Angle Bisector (invisible vertex)

In this construction suppose that we want to find the bisector of an angle whose vertex we are not seeing. -Select the LINE tool (Window 4) and draw a line f so that it intersects the rays (half-lines) i and  j. - Select the INTERSECTION tool (Window 3) and mark H and I, which are the points of intersection of line f with rays (half-lines)i and j, respectively. -Select the POINT tool (Window 2) and draw a point J on ray i (which is positioned to the left of the line f). Also draw a point K on ray j (which is positioned to the left of line f). - Select the ANGLE BISECTOR tool (Window 5). Click on J, H and I to create the angle bisector JHI.  Also click on K, I and H to draw the angle bisector KIH.   - Select the INTERSECTION tool (Window 3) and mark L, which is the intersection  of the two angle bisectors.   - Select the ANGLE BISECTOR tool (Window 5). Click on D, H and E to draw the angle bisector of angle DHE.  Also click on E, I and D to draw the angle bisector of angle EID.   - Select the INTERSECT tool (Window 3) and mark L, which is the intersection of the two angle bisectors.   -Select the LINE (Window 4) and click on L and M. This is the desired angle bisector. In order to verify this, check the HIDE / SHOW VERTEX box.   -Select the MOVE tool (Window 1) Move either point E or line d. What can you see?