Lesson 2 — Proving Lines Parallel
Big idea: If a transversal makes certain angle pairs equal or supplementary, the two lines are parallel.
Words to know
| Word | Meaning |
|---|---|
| Parallel lines | Coplanar lines that do not intersect. |
| Transversal | A line that intersects two or more lines. |
| Corresponding angles | Angles in matching positions at the two intersections. |
| Alternate interior angles | Interior angles on opposite sides of the transversal. |
| Interior angles on the same side | Interior angles on the same side of the transversal. |
The parallel tests
| If… | Then… |
|---|---|
| Corresponding angles are equal | The lines are parallel. |
| Alternate interior angles are equal | The lines are parallel. |
| Interior angles on the same side are supplementary | The lines are parallel. |
| Two lines are perpendicular to the same line | The lines are parallel. |
| Two lines are parallel to the same line | The lines are parallel. |
Theorem 17: Equal corresponding angles mean that lines are parallel.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Prove lines are parallel | Find one of the angle relationships above. |
| Identify corresponding angles | Match positions at the two intersections. |
| Identify alternate interior angles | They are inside the two lines and on opposite sides of the transversal. |
| Identify same-side interior angles | They are inside and on the same side. |
| Use supplementary angles | Show they add to 180°. |
| Use a figure | Mark the transversal and the two lines first. |
Worked example
A transversal crosses lines m and n. The corresponding angles are both 65°.
Because the corresponding angles are equal, m ∥ n by Theorem 17.
If alternate interior angles are 70° and 70°, the lines are also parallel. If same-side interior angles are 110° and 70°, they are supplementary, so the lines are parallel.
Watch out
- Corresponding angles are in matching positions, not necessarily equal unless the lines are parallel.
- Alternate interior angles are inside the two lines.
- Same-side interior angles must be supplementary, not equal.
- Do not assume lines are parallel from the drawing.