Subject: Geometry

Parallels

The Breakdown

What are parallel lines?

Parallel lines are lines that are the same distance apart and in the same plane. They are the same distance apart perpendicularly all the way along the lines.

Parallel lines never meet, touch or intersect.

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Figure 1: PARALLEL LINES, SETS OF 2 AND MORE. ARROWS TO SHOW SAME DISTANCE

Line GT is parallel to line HS. Lines AZ, BY, CX and DW are all parallel. Lines EV and FU are not parallel with each other or with any others.

A line that intersects two or more parallel lines is called a transversal.

In the section on Pairs of Lines it was mentioned that if the two inner angles on one side of a transversal equalled 180° then those two lines are parallel.

Lets look a little deeper into why this is. In the following diagram we have two lines cut at exactly right angles. The two interior angles are right angles, their total is exactly 180° - they are supplementary.

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Figure 2: TWO LINES CUT BY ANOTHER AT RIGHT ANGLES

In a triangle, the sum of the three angles will always total 180°. Therefore if two lines are cut by a third and the interior angles on one side of the transversal are less than 180° we conclude that a triangle is formed, in other words the lines will eventually meet and are therefore not parallel. The following diagram illustrates this.

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Figure 3: TWO LINES CUT BY ANOTHER BUT THEY’RE GETTING CLOSER TOGETHER

If the sum of the two interior angles on one side of a transversal is greater than 180° the lines are not parallel either. Why do you think this is?

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Figure 4: REVERSED, ENLARGED, HIGHLIGHTED ANGLES ON THE OTHER SIDE

The key to appreciating this is to think about what is occurring on the other side of the highlighted angles. These two angles will total less than 180° and will meet on their side, a triangle is formed and the two lines can not be parallel.

Angles of Parallel Lines

Corresponding angles of parallel lines are congruent. In the following diagram, all \angle{a} \cong \angle{a} and all \angle{b} \cong \angle{b}.

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Figure 5: PARALLEL LINES, TRANSVERSAL, SHOW CONGRUENT ANGLES

This includes alternate interior angles, which are nonadjacent and formed on the opposite sides of a transversal that cuts two parallel lines. They are always congruent. If they are not congruent, they lines cut by the transversal are not parallel. In the following diagram, \angle{a} \cong \angle{b} and all \angle{c} \cong \angle{d}.

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Figure 6: ALTERNATE INTERIOR ANGLES

Alternate exterior angles of two parallel lines cut by a transversal are congruent. In the following diagram, \angle{a} \cong \angle{b} and all \angle{c} \cong \angle{d}.

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Figure 7: PARALLEL LINES, EXTERIOR ANGLES ALTERNATE

Lines are parallel if they are parallel or perpendicular to the same line. In the following diagram, lines AB and CD are parallel to line YZ, and lines EF, and GH are perpendicular to line YZ.

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Figure 8: LINES PARALLEL AND PERP TO A LINE

If two sets of lines that create a pair of angles are parallel, then the pair of angles are congruent. In the following diagram, \angle{a} \cong \angle{b}.

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Figure 9: TWO ANGLES, LINES ARE PARALLEL, ANGLES EQUAL