Geometry / Unit 3: Parallel and Perpendicular Lines
3.4 Proving Lines Parallel
Notes
The converses of the parallel line theorems let you prove that lines are parallel.
- If corresponding angles are congruent, the lines are parallel.
- If alternate interior (or alternate exterior) angles are congruent, the lines are parallel.
- If same-side interior angles are supplementary, the lines are parallel.
- If two lines are parallel to the same line, they are parallel to each other.
- If two coplanar lines are perpendicular to the same line, they are parallel.
Practice
A transversal crosses lines $p$ and $q$. Alternate interior angles measure $64^\circ$ and $64^\circ$. What can you conclude?
Congruent alternate interior angles mean the lines are parallel.
Same-side interior angles measure $(4x)^\circ$ and $(2x + 30)^\circ$. For what value of $x$ are the lines parallel?
They must be supplementary: $6x + 30 = 180$, so $x = 25$.
Same-side interior angles measure $75^\circ$ and $95^\circ$. Are the lines parallel?
For parallel lines, same-side interior angles must be supplementary.
Ready for more? Take the Unit 3 test.