Lesson 3 — The Parallel Postulate
Big idea: Through a point not on a line, there is exactly one line parallel to the given line.
Words to know
| Word | Meaning |
|---|---|
| Parallel Postulate | Through a point not on a line, there is exactly one line parallel to the line. |
| Construction 7 | Construct a line parallel to a given line through a given point. |
Concepts to know
- The Parallel Postulate is Postulate 7.
- “Exactly one” means there is one parallel line, not zero and not two.
- This postulate is special: for centuries mathematicians tried to prove it from Euclid’s other postulates.
- Theorem 18: In a plane, two lines parallel to a third line are parallel to each other.
- A transversal can be used to construct a parallel line by copying an angle.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Construct a parallel line | Copy a corresponding angle at the given point. |
| Use the Parallel Postulate | State that there is exactly one parallel through the point. |
| Prove two lines are parallel | Show both are parallel to the same third line. |
| Explain why a construction works | Equal corresponding angles make the lines parallel. |
| Use “exactly one” | Do not claim more than one parallel line. |
Worked example
Given line ℓ and point P not on ℓ.
- Draw a transversal through P that crosses ℓ.
- Copy the corresponding angle at P.
- Draw the line through P with that angle.
The new line is parallel to ℓ by the corresponding-angles test. The Parallel Postulate says this is the only line through P parallel to ℓ.
Watch out
- The point must not be on the line.
- “Exactly one” is part of the postulate; do not ignore it.
- The construction uses equal corresponding angles.
- In non-Euclidean geometry, this postulate can fail — but in this course, it holds.