Finding Angle Measures with Parallel Lines

The final set of angle rules that commonly come up in tests are parallel line rules. 

It is important to remember that whenever two lines are crossed by a transversal, specific angles (corresponding, alternate interior, alternate exterior) are created.  However, those angles are EQUAL when the lines cut by the transversal are parallel.  First, some vocab:

• Parallel Lines: Lines that have the same slope and will never intersect.  Line n||m means that lines are parallel. In figures, lines marked with arrows (like the two parallel lines below) are parallel.
• Transversal: A line that cuts through two parallel lines. Note the diagonal line cutting through the parallel lines below.

Parallel lines cut by transversals create several sets of equal angles (remember, if the lines are not parallel, these angles are not equal!). In the figure below:

• Angles marked with 😁 are alternate exterior angles.  They are outside the pathway created by the parallel lines and are on opposite sites of the transversal. Alternate exterior angles are equal to each other. 
• Angles marked with a 🧡 are alternate interior angles.  They are inside the pathway created by the parallel lines and they are opposite sites of the transversal.  All alternate interior angles are equal to each other. 
• Angles marked with a 🌞 are corresponding angles.  They are on the same side of the transversal, in the same position, relative to each of the parallel lines.  All corresponding angles are equal to each other.

Knowing which sets of angles are equal to each other helps you to solve a variety of questions about angles in parallel lines. 

Example: In the figure below, what is the value of x?

If you know your angle rules, you'll know that the angle marked $60^{\circ}$ and the angle marked $x^{\circ}$ are alternate exterior angles (outside the pathway, on opposite sides of the transversal), so they are equal.

$x=60$

However, it's important to remember that all of the vertical angle rules work together and in conjunction with the vertical angles rule (angles opposite each other in an X are equal) and the linear/supplementary angles rule (angles that form a line add up to $180^{\circ}$.  So, if you can't remember all of the rules, use the ones you know!  Eventually, you'll get to your answer.

Let's look at the figure above again, and pretend we aren't sure about the exterior angles rule.

We know that, due to the vertical angles rule, the angle opposite the $60^{\circ}$ is also $60^{\circ}$.

We also know that the angle marked with the star is a vertical angle to the $x^{\circ}$, so it is also $x^{\circ}$.  The second $60^{\circ}$ and the star angle are alternate interior angles, so they are equal.  Once again: 

$x=60$

The fewer rules you know, the more steps you might have to take to solve these problems, but you'll get there eventually!

Note: sometimes all of the angle rules (and types of angle rules) come into play in the same problem.  A triangle may be embedded within parallel lines.  The parallel lines of a quadrilateral may be a critical part of a problem (remember, lines go on forever, so never hesitate to draw lines extending past the boundaries set in the figure, sometimes that helps you to see more!).  Always keep your eyes open for parallel lines and the equal angles that come with them.