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Parallelism

Author:
Jorge Cássio
Topic:
Angles, Straight Lines

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)

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Angles determined by parallel and transversal lines

Question 2

In the previous structure, which pairs of angles are congruent?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 3

In the previous structure, which of the following pairs of angles are supplementary?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

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.

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