Parallelism
Parallel Lines - Definition
Two lines are parallel if, and only if, they coincide (that is, equal) or are coplanar and have no common point.
Question 1
Which pairs of lines are parallel? (use the "Show / Hide angle marks" box to help you)
Angles determined by parallel and transversal lines
Question 2
In the previous structure, which pairs of angles are congruent?
Question 3
In the previous structure, which of the following pairs of angles are supplementary?
Parallelism theorem
Alternate Interior Angle Theorem (Alternate for the previous theorem)
Constructing a parallel line
In the following GeoGebra applet, follow the steps below:
- Select the POINT (Window 2) and draw a point B on line r.
- Select the COMPASS tool (Window 6). Then click on point A and point B (it will open the compass) and again on point A (it will close the compass and form a circle). After that click on point B and point A (it will
open the compass) and again on B (it will close the compass and form a second circle).
- Select the INTERSECT (Window 3) and mark the intersection C of the last circle with the line r.
- Select the COMPASS (Window 6). Then click on point C and point A (it will open the compass) and again on point B (it will close the compass and form a circle).
- Select the option INTERSECT (Window 3) and mark point D, which is the upper intersection of the first circunference with the third circunference.
-Select the option LINE (Window 3) and click on point A and point D. Label this line s.
- Select the option SHOW / HIDE OBJECT (Window 7) and hide the circles, points B, C and D, leaving only the lines and point A.
-Select the option RELATION (Window 8) and click on the two lines. What happens?
- Select the option MOVE (Window 1) move point A or line r. What can you see?
Analysis
Write an argument to justify the construction.
Exterior Angles of a Triangle
Triangle Exterior Angle Theorem
Interior angles of the triangle (source: https://www.geogebra.org/luisclaudio)
Question 4
Move the selector "t". Also move the vertices of the triangle. What can you see?
Question 5
Explain the previous property.