# 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)

## 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.