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. [br][b][center][math]\huge{\left(a\subset\alpha, b\subset\alpha, a\cap b=\varnothing\right)\Rightarrow a\slash\slash b}[/math][/center][/b]

Question 1

Which pairs of lines are parallel? (use the "Show / Hide angle marks" box to help you)

None of them only [b]a[/b] and [b]b[/b] [b]a[/b] and [b]b[/b], [b]b[/b] and [b]c[/b], [b]a[/b] and [b]c[/b]. It is not possible to answer this question

Angles determined by parallel and transversal lines

Question 2

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

only [math]\Large{\lambda\ and\ \delta;\ \alpha \ and\ \theta; \lambda\ and\ \theta;\ \alpha\ and\ \delta}[/math]. Any pairs of angles none of them [math]\Large{\lambda\ and\ \delta;\ \alpha\ and\ \theta; \lambda\ and\ \theta;\ \alpha\ and\ \delta; \beta\ and\ \zeta; \epsilon\ and\ \eta; \beta$\ and\ \eta; \epsilon\ and\ \zeta} [/math][br]

Question 3

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

Any pairs of angles somente [math]\Large{$\lambda$\ and\ \beta;\ \lambda\ and\ \zeta; \lambda\ and\ \epsilon;\ \eta\ and\ \lambda; \beta$\ and\ \alpha; \epsilon\ and\ \alpha; \alpha\ and\ \zeta; \alpha\ and\ \eta} [/math] All angles are complementary [math]\Large{\lambda\ and\ \beta;\ \lambda\ and\ \zeta; \lambda\ and\ \epsilon;\ \eta\ and\ \lambda; \beta\ and\ \alpha; \epsilon\ and\ \alpha; \alpha\ and\ \zeta$; \alpha\ and\ \eta} [/math]

Parallelism theorem

Alternate Interior Angle Theorem (Alternate for the previous theorem)

Constructing a parallel line

In the following GeoGebra applet, follow the steps below: [br]- Select the [b]POINT [/b](Window 2) and draw a point B on line r. [b] [/b] [b] [/b] [br]- Select the [b]COMPASS tool (Window 6)[/b]. Then click on point [b]A [/b]and point[b] B [/b](it will open the compass) and again on point [b]A [/b](it will close the compass and form a circle). After that click on point [b]B [/b]and point[b] A [/b](it will[br]open the compass) and again on [b]B [/b](it will close the compass and form a second circle). [br][b]- [/b]Select the [b]INTERSECT (Window 3)[/b] and mark the intersection [b]C [/b]of the last circle with the line [b]r[/b]. [br]- Select the [b]COMPASS (Window 6)[/b]. Then click on point [b]C[/b] and point [b]A[/b] (it will open the compass) and again on point [b]B[/b] (it will close the compass and form a circle). [br][b]- [/b]Select the option [b]INTERSECT (Window 3)[/b] and mark point [b]D[/b], which is the upper intersection of the first circunference with the third circunference. [br]-Select the option [b]LINE (Window 3)[/b] and click on point [b]A [/b]and point [b]D.[/b] Label this line [b]s[/b]. [br]- Select the option [b]SHOW / HIDE OBJECT (Window 7)[/b] and hide the circles, points [b]B[/b], [b]C [/b]and [b]D[/b], leaving only the lines and point [b]A.[/b] [br]-Select the option [b]RELATION (Window 8) [/b]and click on the two lines. What happens? [br]- Select the option [b]MOVE (Window 1) [/b]move point [b]A[/b] or line [b]r[/b]. 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.[br]