# Parallel, Perpendicular, Line Bisector and Angle Bisector

- Author:
- Jorge Cássio

- 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?## Parallel Line

## 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 not belonging to the line)

## 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?## Perpendicular line (point belonging to line)

## 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?## Drawing the Line segment Bisector

## 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 with visible vertex

## 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?