Criteria for Parallel Lines
Grade 7 Geometry โ Understanding when two lines are parallel
๐ Part 1: Introduction โ Transversals and Angle Names
What is a Transversal?
When a transversal $c$ intersects two lines $a$ and $b$, it creates eight angles โ four angles at each intersection point. These angles have special names based on their positions.
Numbering the Angles
We label the angles at the first intersection point (with line $a$) as $\angle 1, \angle 2, \angle 3, \angle 4$, and the angles at the second intersection point (with line $b$) as $\angle 5, \angle 6, \angle 7, \angle 8$.
๐ Part 2: Special Angle Pairs
1. Alternate Interior Angles
Angles on opposite sides of the transversal, between the two lines.
Pairs:
- $\angle 3$ and $\angle 5$
- $\angle 4$ and $\angle 6$
They form a "Z" shape (or reversed Z).
2. Co-Interior Angles (Same-Side Interior)
Angles on the same side of the transversal, between the two lines.
Pairs:
- $\angle 4$ and $\angle 5$
- $\angle 3$ and $\angle 6$
They form a "C" or "U" shape.
3. Corresponding Angles
Angles in the same position at each intersection โ one at each intersection point.
Pairs:
- $\angle 1$ and $\angle 5$
- $\angle 2$ and $\angle 6$
- $\angle 3$ and $\angle 7$
- $\angle 4$ and $\angle 8$
They form an "F" shape (or reversed/rotated F).
| Angle Pair Type | Position | Pairs | Shape |
|---|---|---|---|
| Alternate Interior | Opposite sides, between lines | $\angle 3$ & $\angle 5$; $\angle 4$ & $\angle 6$ | Z-shape |
| Co-Interior (Same-Side) | Same side, between lines | $\angle 4$ & $\angle 5$; $\angle 3$ & $\angle 6$ | C/U-shape |
| Corresponding | Same position at each point | $\angle 1$ & $\angle 5$; $\angle 2$ & $\angle 6$; $\angle 3$ & $\angle 7$; $\angle 4$ & $\angle 8$ | F-shape |