7.01 Parallel lines and transversals
A line that intersects two (or more) other lines is called a transversal. Angles formed on the same side of a transversal are called consecutive angles.
In the applet below, $f$f is a transversal of $g$g and $h$h. Moving the slider changes the orientation of $g$g. Move the slider in the applet below and notice how the measures of the highlighted angles change:
When $g$g and $h$h are parallel, the measure of the interior angles add to $180^\circ$180°. When they aren't parallel, $g$g intersects $h$h forming a third angle, and adding the measures of the two interior angles and this third angle always makes $180^\circ$180°.
Consecutive angles
Let's use the applet below to explore the different pairs of consecutive angles formed by parallel lines. Notice that angles between the parallel lines are called interior angles, and angles outside the parallel lines are exterior angles.
Formally, we can say that if two lines are parallel, then consecutive interior angles are supplementary. We refer to this as the consecutive interior angles postulate. Using properties of linear pairs allows us to prove that consecutive exterior angles are also supplementary, and this is referred to as the consecutive exterior angles theorem.
The consecutive interior angles postulate is assumed to be true, so we call it a postulate. Other results, such as the consecutive exterior angle theorem, are derived from the postulates and other results, so we call it a theorem.
Alternate angles
If we look at a pair of angles on opposite sides of the transversal, we can refer to them as alternate angles. Alternate angles between the parallel lines are alternate interior angles, while alternate angles outside the parallel lines are alternate exterior angles.
Use the applet to explore the different pairs of alternate angles. What do you notice about their values?
We can use properties of linear pairs and vertical angles to prove that if two lines are parallel, then pairs of alternate interior angles are congruent - this is called the alternate interior angles theorem. It's also true that pairs of alternate exterior angles are congruent - this is called the alternate exterior angles theorem.
Corresponding angles
Corresponding angles are the angles that occupy the same relative position at each intersection. We can slide the slider in the applet below to see all four pairs of corresponding angles. What relationship do these angles have?
Using similar results to those we discussed above, we can prove that if two lines are parallel, then the corresponding angles are congruent - this is called the corresponding angles theorem.
If two lines are parallel, then:
- Consecutive interior angles are supplementary (the consecutive interior angles postulate)
- Consecutive exterior angles are supplementary (the consecutive exterior angles theorem)
- Alternate interior angles are congruent (the alternate interior angles theorem)
- Alternate exterior angles are congruent (the alternate exterior angles theorem)
- Corresponding angles are congruent (the corresponding angles theorem)
All angle relationships summarized in a diagram
Practice questions
Question 1
Consider the diagram below.
Two parallel vertical lines are intersected by a single diagonal line, slanting from the lower left to the upper right, creating two intersections. Small arrow symbols on the vertical lines indicate that they are parallel.
Two angles are marked with blue arcs, one on each intersection. The angle on the left intersection is marked with a double-arc and the angle on the right intersection is marked with a single-arc. Both angles are on the same side, above the diagonal line. Both angles are inside the parallel lines.
Which relationship describes the marked angles?
Alternate exterior angles
AAlternate interior angles
BConsecutive interior angles
CConsecutive exterior angles
DCorresponding angles
E
Question 2
Consider the diagram below.
Solve for $x$x.
Question 3
Consider the diagram below.
Two parallel diagonal lines are intersected by a horizontal transversal such that the parallel lines are right on top of the transversal. The parallel line on the left intersects the left endpoint of the transversal line, forming an angle marked as $p$p degrees. The parallel line on the right intersects one of the points along the transversal line, forming an angle marked as $q$q degrees. To the right is the other parallel line and it forms an angle marked as $q$q degrees with the transversal line. Both the $p$p-degree and $q$q-degree angles are above the transversal and on the left side of their respective parallel lines, indicating that they are corresponding angles. At the left endpoint of the transversal line, a vertical line extends, forming a right angle as indicated by a blue square. At the same time, the vertical line forms an angle with the left parallel line, and it measures $26^\circ$26° as labeled.
Solve for $p$p.
Solve for $q$q.