Question
Two parallel lines are cut by a transversal. If one angle is , find all eight angles formed. Name each pair type — corresponding, alternate interior, alternate exterior, and co-interior.
(CBSE 7 & 9 — guaranteed board question)
Solution — Step by Step
When a transversal cuts two parallel lines:
- Corresponding angles are equal (same position at each intersection)
- Alternate interior angles are equal (opposite sides, between the lines)
- Alternate exterior angles are equal (opposite sides, outside the lines)
- Co-interior (same-side interior) angles are supplementary (add to )
At the first intersection, if one angle is :
- Vertically opposite angle =
- Adjacent angles = (linear pair)
So the four angles at the first intersection are: .
By corresponding angles (equal): the same pattern repeats at the second intersection.
All eight angles: at the first point and at the second point.
Co-interior angles should add to : . Confirmed.
Why This Works
Parallel lines never meet — they maintain the same direction. When a transversal crosses them, it creates identical geometric configurations at both intersections. This is why corresponding angles are equal — the geometry is a translated copy.
graph TD
A["Transversal cuts<br/>parallel lines"] --> B["8 angles formed<br/>(4 at each intersection)"]
B --> C{"Identify pair type"}
C -->|"Same position,<br/>same side"| D["Corresponding<br/>→ Equal"]
C -->|"Between lines,<br/>opposite sides"| E["Alternate Interior<br/>→ Equal"]
C -->|"Outside lines,<br/>opposite sides"| F["Alternate Exterior<br/>→ Equal"]
C -->|"Between lines,<br/>same side"| G["Co-interior<br/>→ Sum = 180°"]
B --> H["At each intersection:<br/>vertically opposite = equal<br/>linear pair = 180°"]Alternative Method — Use F, Z, and C Shapes
- F-shape → corresponding angles (look for an F or reversed F)
- Z-shape → alternate interior angles (look for a Z or S)
- C-shape or U-shape → co-interior angles (look for a C or U)
These visual patterns make it easy to spot which pair type you’re dealing with.
For CBSE 9 boards: the most common question gives you one angle and asks you to find all others with reasons. Always state the property you used — “corresponding angles (parallel lines)” or “vertically opposite angles.” The marking scheme awards separate marks for the reasoning.
Common Mistake
Students apply these angle relationships when the lines are NOT parallel. Corresponding angles are equal ONLY when the lines are parallel. In fact, the converse is also tested: “if corresponding angles are equal, prove the lines are parallel.” Always check the given information — if parallelism isn’t stated or provable, you cannot assume these equalities.