Geometry for Elementary School/Parallel lines

Definition
Let us recall what we learnt from the lines chapter. Parallel lines are straight lines that never intersect, which means that they never cross. Notice that when we look at parallel parts of shapes there is no place where they intersect even if we extend the lines.

As we have learnt from the plane shapes chapter, parallelograms, including squares, rhombi and rectangles, have two pairs of parallel sides. Also, there is one pair in trapeziums. Can you name any other polygons that contain parallel lines?

Transversals
A transversal is a line segment that cut through two line segments. (They can, of course, be lines and rays as well.) When the two line segments are parallel, the eight angles produced will have some special properties.

In such a case, imagine that one of the angles on one of the segments is called x. It is the top left angle. The top left angle of the other segment would be its corresponding angle, or corr. ∠. Corresponding angles are always the same size. Then imagine the vertically opposite angle of angle x. Let's call it y. The corresponding angle of y would be the alternate angle, or alt. ∠, of x. Bear in mind that alternate angles are also the same in size as they are vertically opposite angles with the corresponding angles. What about the adjacent angles of x? As you can see, these angles are adjacent angles on a straight line with x, so they must be supplementary. These are called interior angles on the same side of the transversal (int. ∠s). They are different from the kind of interior angle in a polygon, so do not mix them up.

When you are doing your sums, you will find out that these three kinds of angles pop up very commonly and are very useful. The reference for these angles are corr./alt./int. ∠s, AB//CD. When you want to prove that two lines are congruent, then use corr./alt. ∠s equal or int. ∠s supp..

Look at the figure. Given that m and n are parallel, angles 1 and 5, 2 and 6, 3 and 7, 4 and 8 are corresponding angles, and are therefore equal. 5 and 3, 6 and 4 are alternate angles, so they must be equal as well. 5 and 4, 3 and 6 are interior angles, so they must be supplementary. Note that there are also two sets of angles at a point and eight pairs of adjacent angles on a straight line.

Let us look at an example. Given that DE and GB are parallel, how can we find x, y and z?

$$\begin{align} \angle EFC &= \angle BGC \text{ (corr. } \angle \text{s, DE} / / \text{GB)} \\ z &= 125^\circ \end{align} $$ $$\begin{align} \angle AFD &= \angle BGC \text{ (alt. } \angle \text{s, DE} / / \text{GB)} \\ y &= 125^\circ \end{align} $$ $$\begin{align} \angle AGB + \angle BGC &= 180^\circ \text{ (adj. } \angle \text{s on st. line)}\\ \angle AGB + 125^\circ &= 180^\circ \\ \angle AGB &= 180^\circ - 125^\circ \\ &= 55^\circ \end{align}$$ $$\begin{align} \angle AGB + \angle EFC &= 180^\circ \text{ (int. } \angle \text{s, DE} / / \text{GB)} \\ 55^\circ + x &= 180^\circ \\ x &= 180^\circ - 55^\circ \\ &= 125^\circ \end{align} $$

Geometria per scuola elementare/Linee parallele