# Perpendicularity

## Perpendicular Lines - Definition

Two lines are

**perpendicular**if, and only if, they are concurrent and form complementary adjacent congruent angles. These lines form right angles (90º).## Question 1

Which pairs of straight lines are perpendicular?

## Oblique lines - Definition

Two straight lines are oblique when they are concurrent, but not perpendicular.

## Move point A

## Constructing a Perpendicular line from a point outside the line.

In the following GeoGebra applet, follow the steps below:
- Select the

**COMPASS****(Window 5)**. Then click on the segment**AB**(opening of the compass) and on**E**(compass point). - Select the option**INTERSECT (Window 3)**and mark the intersections**F**and**G**of the circumference with the line g. - Select the**COMPASS****(Window 6)**. Then click on point**F**and point**G**(it will open the compass) and again on point**F**(it will close the compass and form a circle). After that, click on point**G**and point**F**(it will open the compass) and again on**G**(it will close the compass and form a second circle).**-**Select the option**INTERSECT (Window 3)**and mark a point**H**, point of intersection of the last two circunferences. -Select the option**LINE (Window 4)**and click on point**E**and point**H.**It will create the intended perpendicular line. Let us analyse it.**-**Select the option**INTERSECT (Window 3)**and mark point**I**, point of intersection of points**h**and**g**.**-**Select the option**ANGLE (Window 6)**. 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 option**SHOW / HIDE OBJECT (Window 7)**and hide the circles, points**H**,**F**and**G**, leaving only the lines and point**E.**-Select the option**RELATION tool (Window 8)**and click on the two lines. What happens? - Select the option**MOVE (Window 1)**move point**E**or line**g**. What can you see?## Analysis 1

Write an argument to justify the construction. Use the perpendicular bisector property:

- line that passes perpendicularly through the midpoint;
- geometric location of points equidistant (same length) from the endpoints of a segment.

## Construction of the Perpendicular from a point on a line

- Select the

**COMPASS (Window 5)**. Then click on the segment**AB**(opening of the compass) and on**E**(compass point).**-**Select the option**INTERSECT (Window 3)**and mark the intersections**F**and**G**of the circumference with the line**g**. - Select the**COMPASS (Window 6)**. Then click on point**F**and point**G**(it will open the compass) and again on point**F**(it will close the compass and form a circle). After that, click on point**G**and point**F**(it will open the compass) and again on**G**(it will close the compass).**-**Select the option**INTERSECT (Window 3)**and mark a point**H**, point of intersection of the last two circunferences. -Select the option**LINE (Window 4)**and click on point**E**and point**H.**It will create the intended perpendicular line.**-**Select the option**ANGLE (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 option**SHOW / HIDE OBJECT (Window 7)**and hide the circles, points**H**,**F**and**G**, leaving only the lines and point**E.**-Select the option**RELATION tool (Window 8)**and click on the two lines. What happens? - Select the option**MOVE tool (Window 1)**and move point**E**or line**g**. What can you see?## Analysis 2

Write an argument to justify the construction. Repeat the use of Perpendicular bisector properties.