Two lines that lie in the same plane but do not intersect are parallel. Think of these lines as railroad tracks; they appear side-by-side and have the same slant (slope), so they never cross.
Note that two lines that don’t intersect but are in different planes are skew lines. Also, two planes that don’t intersect are parallel planes.
Angle Relationships
Note the following diagram that contains lines $ l$ and $ m$ and the transversal $ t$ that intersects them. Here are the names for the angle pairs that are formed by the lines:
| Name | Angle Pairs |
| Interior Angles | $ \displaystyle \angle \,3$, $ \displaystyle \angle \,4$, $ \displaystyle \angle \,5$, and $ \displaystyle \angle \,6$ |
| Alternate Interior Angles (AIA) | $ \displaystyle \angle \,4$ and $ \displaystyle \angle \,6$; $ \displaystyle \angle \,3$ and $ \displaystyle \angle \,5$ |
| Corresponding Angles | $ \displaystyle \angle \,2$ and $ \displaystyle \angle \,6$; $ \displaystyle \angle \,1$ and $ \displaystyle \angle \,5$; $ \displaystyle \angle \,3$ and $ \displaystyle \angle \,7$; $ \displaystyle \angle \,4$ and $ \displaystyle \angle \,8$ |
| Exterior Angles | $ \displaystyle \angle \,1$, $ \displaystyle \angle \,2$, $ \displaystyle \angle \,7$, and $ \displaystyle \angle \,8$ |
| Alternate Exterior Angles (AEA) | $ \displaystyle \angle \,1$ and $ \displaystyle \angle \,7$; $ \displaystyle \angle \,2$ and $ \displaystyle \angle \,8$ |
Parallel Lines and Transversals
The reason we care about all these angles is if the two lines are parallel, certain angles cut by their transversal are congruent or supplementary, as shown below. Note when doing problems or preparing proofs, the problem must specify the angles are parallel (with text or arrows); don’t rely on the lines just looking parallel.
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- Corresponding angles are congruent. Example: $ \displaystyle \angle \,2\cong \angle \,6$. See how this sort of forms an “$ F$”?
- Alternate interior angles (AIA) are congruent. Example: $ \displaystyle \angle \,3\cong \angle \,5$. See how this sort of forms a “$ Z$”?
- Alternate exterior angles are congruent. Example: $ \displaystyle \angle \,1\cong \angle \,7$. In my geometry classes, this isn’t used as much.
- Consecutive interior, or same side interior (SSI) angles are supplementary. Note that consecutive angles are those that are formed when a transversal intersects two parallel lines. Example: $ \displaystyle \angle 4$ is supplementary to $ \displaystyle \angle 5$, which means that $ \displaystyle \text{m}\,\angle \,4+ \text{m}\,\angle \,5=180$. See how this sort of forms a “$ C$”?
- Corresponding angles are congruent. Example: $ \displaystyle \angle \,2\cong \angle \,6$. See how this sort of forms an “$ F$”?
Note that the converse of the above statements are true also, such as if two corresponding angles are congruent, then the lines in which the transversal intersect are parallel.
Problems and Solutions
Problem: $ \displaystyle m\angle 1={{x}^{2}}$ and $ \displaystyle m\angle 2=4x+148$. Find $ \displaystyle m\angle 1$, if $ \displaystyle l\parallel m$. 
Note that the drawing may not be at all to scale!
Solution: Since $ \angle \text{ }\!\!’\!\!\text{ s }1\,\,\text{and }3$ are vertical, they are congruent. Since $ \angle \text{ }\!\!’\!\!\text{ s }2\,\,\text{and }3$ are corresponding, they are supplementary. Thus, $ \displaystyle m\angle 1+m\angle 2=180$ (substitution). We have a quadratic:
$ \displaystyle \begin{array}{c}m\angle 1+m\angle 2=180\\{{x}^{2}}+4x+148=180;\,\,\,{{x}^{2}}+4x-32=0\\\left( {x+8} \right)\left( {x-4} \right)=0;\,\,\,x=-8,4\\m\angle 1={{x}^{2}}={{4}^{2}}=16\,\,\,\,\,\,(m\angle 2=\text{164)}\\\text{or }m\angle 1={{x}^{2}}={{\left( {-8} \right)}^{2}}=64\,\,\,\,\,\,(m\angle 2=\text{116)}\end{array}$
Note that we have two answers, since it’s a quadratic. But if any of the angles had turned out to have negative values, we’d have to throw out those $ x$-values. Always check answers, especially with quadratics!!
Problem: What pair of line(s) have to be parallel if $ \displaystyle \angle \,4\cong \angle \,10$? 
Solution: $ \displaystyle \angle \,4\cong \angle \,2$, since they are vertical angles, and since $ \displaystyle \angle \,4\cong \angle \,10$, then $ \displaystyle \angle \,2\cong \angle \,10$ (transitive property of congruence). $ \displaystyle \angle \,2$ and $\angle \,10$ are corresponding angles, and since they are congruent, lines $ n$ and $ o$ are parallel. We can’t tell if lines $ l$ and $ m$ are parallel or not.
Problem: Find $ y$: 
Solution: This is tricky, but much easier to see if we draw a line parallel to the two existing parallel lines that goes through the vertex of the $ 48{}^\circ $ angle:
With the new parallel line drawn, we can see that the original $ 48{}^\circ $ can be turned into two angles of $ 20{}^\circ $ and $ 28{}^\circ $, using vertical and alternate interior angles (AIA). Again, using AIA and vertical angles, we see that $ y=28{}^\circ $. Note that there are other ways to do this problem, but we should arrive at the same answer!
Problem:
Given: $ \displaystyle n\parallel o$, $ \displaystyle \angle \,6\cong \angle \,2$ 
Prove: $ \displaystyle \angle \,16$ is supplementary to $ \displaystyle \angle \,3$
Solution:
Proof:
| Statements | Reasons |
| 1. $ \displaystyle \angle \,6\cong \angle \,2$ | 1. Given |
| 2. $ \displaystyle l\parallel m$ | 2. Corr $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong \displaystyle \,\,\,\Rightarrow $ lines are $ \parallel$ |
| 3. $ \displaystyle \angle \,16\cong \angle \,12$ | 3. Lines are $ \parallel \,\,\Rightarrow $ corr $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong$ |
| 4. $ \displaystyle \text{m}\,\angle \,16= \text{m}\,\angle \,12$ | 4. Def of $\cong$ |
| 5. $ \displaystyle n\parallel o$ | 5. Given |
| 6. $ \displaystyle \angle \,12$ and $ \displaystyle \angle \,3$ are supplementary | 6. Lines are $ \parallel \,\,\Rightarrow $ SSI (same side interior) $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}$ supp |
| 7. $ \displaystyle \text{m}\,\angle \,12+ \text{m}\,\angle \,3=180$ | 7. Def of Supp |
| 8. $ \displaystyle \text{m}\,\angle \,16+ \text{m}\,\angle \,3=180$ | 8. Substitution |
| 9. $ \displaystyle \angle \,16$ is supplementary to $ \displaystyle \angle \,3$ | 9. Def of Supp |
Problem:
Given: $ \displaystyle n\parallel o$, $ \displaystyle \angle \,16\cong \angle \,4$
Prove: $ \displaystyle l\parallel m$
Solution:
Proof:
| Statements | Reasons |
| 1. $ \displaystyle n\parallel o$ | 1. Given |
| 2. $ \displaystyle \angle \,16\cong \angle \,6$ | 2. Lines are $ \parallel \,\,\Rightarrow $ AIA (Alternate Interior Angle) $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong$ |
| 3. $ \displaystyle \angle \,16\cong \angle \,4$ | 3. Given |
| 4. $ \displaystyle \angle \,6\cong \angle \,4$ | 4. Transitive |
| 5. $ \displaystyle l\parallel m$ | 5. AIA (Alternate Interior Angle) $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong \displaystyle \,\,\,\Rightarrow $ lines are $ \parallel$ |
On to Congruent Triangles!