You know vertical and supplementary angles. When a line, called a transversal, crosses two parallel lines, it makes eight angles, and they come in just two sizes.
Why it works
Slide the top crossing down along the transversal (a translation). Because the two lines are parallel, it lands exactly on the bottom crossing, so each angle at the top matches the angle in the same position at the bottom: corresponding angles are equal. Add what you already know (vertical angles are equal, angles on a straight line add to $180^\circ$), and all the other pairs follow.
Key idea
When two parallel lines are cut by a transversal, corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to $180^\circ$. Every angle is one of two sizes, and the two sizes add to $180^\circ$.
Worked examples
Worked example: Find the others
In the picture, $\angle 1 = 65^\circ$. Find $\angle 5$, $\angle 3$ and $\angle 6$.
- $\angle 5$ is in the same position at the other crossing: corresponding, so $65^\circ$.
- $\angle 3$ is vertical to $\angle 1$: $65^\circ$.
- $\angle 6$ sits next to $\angle 5$ on a straight line: $180^\circ - 65^\circ = 115^\circ$.
Answer: $65^\circ$, $65^\circ$, $115^\circ$
Worked example: With algebra
Two alternate interior angles measure $(3x + 10)^\circ$ and $(5x - 30)^\circ$. Find $x$ and the angles.
- Alternate interior angles are equal: $3x + 10 = 5x - 30$.
- Subtract $3x$ and add 30: $40 = 2x$, so $x = 20$.
- Each angle is $3 \cdot 20 + 10 = 70^\circ$.
Answer: $x = 20$; both angles are $70^\circ$
Common mistake
Using the angle-pair rules when the lines are not parallel. The rules come from sliding one crossing onto the other, which only works for parallel lines. Look for the arrow marks (or the word “parallel”) before you use them.