Two lines are [b]perpendicular[/b] if, and only if, they are concurrent and form complementary adjacent congruent angles. These lines form right angles (90º). [br][br][b][center][math]\huge{a⊥b⇔a∩b=D \ and\ A\^{D}C=C\^{D}B=90º}[/math][/center][/b][br]
Which pairs of straight lines are perpendicular?
Two straight lines are oblique when they are concurrent, but not perpendicular.[br][br][b][center][math]\huge{a∩b=D \ e\ a\ ∡ \ b}[/math][/center][/b][br]
In the following GeoGebra applet, follow the steps below: [br]- Select the [b]COMPASS [/b][b](Window 5)[/b]. Then click on the segment [b]AB [/b](opening of the compass) and on [b]E[/b] (compass point). [br]- Select the option [b]INTERSECT (Window 3) [/b]and mark the intersections [b]F [/b]and [b]G [/b]of the circumference with the line g. [br]- Select the [b]COMPASS [/b][b](Window 6)[/b]. Then click on point [b]F [/b]and point[b] G[/b] (it will open the compass) and again[br]on point [b]F[/b] (it will close the compass and form a circle). After that, click on point [b]G [/b]and point[b] F[/b] (it will[br]open the compass) and again on [b]G[/b] (it will close the compass and form a second circle). [br][b]- [/b]Select the option [b]INTERSECT (Window 3)[/b] and mark a point [b]H[/b], point of intersection of the last two circunferences. [br]-Select the option [b]LINE (Window 4)[/b] and click on point [b]E [/b]and point [b]H.[/b] It will create the intended perpendicular line. Let us analyse it. [br][b]- [/b]Select the option [b]INTERSECT (Window 3)[/b] and mark point [b]I[/b], point of intersection of points [b]h [/b]and [b]g[/b]. [br][b]-[/b] Select the option[b] ANGLE (Window 6)[/b]. Click on points [b]E[/b],[b] I [/b]and[b] C[/b] to mark the angle [b]EIC[/b] (the vertex of the angle will always be the second point clicked). What is the measurement of this angle? [br]- Select the option [b]SHOW / HIDE OBJECT (Window 7)[/b] and hide the circles, points [b]H[/b], [b]F [/b]and [b]G[/b], leaving only the lines and point [b]E.[/b] [br]-Select the option [b]RELATION tool (Window 8) [/b]and click on the two lines. What happens? - Select the option [b]MOVE (Window 1)[/b] move point [b]E[/b] or line [b]g[/b]. What can you see?
Write an argument to justify the construction. Use the perpendicular bisector property: [list][*]line that passes perpendicularly through the midpoint; [/*][*]geometric location of points equidistant (same length) from the endpoints of a segment.[/*][/list][*][br][/*]
- Select the [b]COMPASS (Window 5)[/b]. Then click on the segment [b]AB [/b](opening of the compass) and on [b]E[/b] (compass point). [br][b]- [/b]Select the option [b]INTERSECT (Window 3)[/b] and mark the intersections [b]F [/b]and [b]G[/b] of the circumference with the line [b]g[/b]. [br]- Select the [b]COMPASS (Window 6)[/b]. Then click on point [b]F [/b]and point[b] G[/b] (it will open the compass) and again on point [b]F[/b] (it will close the compass and form a circle). After that, click on point [b]G [/b]and point[b] F[/b] (it will[br]open the compass) and again on [b]G[/b] (it will close the compass). [br][b]- [/b]Select the option [b]INTERSECT (Window 3)[/b] and mark a point [b]H[/b], point of intersection of the last two circunferences. [br]-Select the option [b]LINE (Window 4)[/b] and click on point [b]E [/b]and point [b]H.[/b] It will create the intended[br]perpendicular line. [br][b]-[/b] Select the option[b] ANGLE (Window 9)[/b]. Click on points [b]H[/b],[b] E [/b]and[b] C[/b] to mark the angle [b]HEC[/b] (the vertex of the angle will always be the second point clicked). What is the measurement of this angle?-[br]Select the option [b]SHOW / HIDE OBJECT (Window 7)[/b] and hide the circles, points [b]H[/b], [b]F [/b]and [b]G[/b], leaving only the lines and point [b]E.[/b] [br]-Select the option [b]RELATION tool (Window 8) [/b]and click on the two lines. What happens? -[br]Select the option [b]MOVE tool (Window 1)[/b] and move point [b]E[/b] or line [b]g[/b]. What can you see? [br][br][br]
Write an argument to justify the construction. Repeat the use of Perpendicular bisector properties. [br]