Question
Two parallel lines are cut by a transversal. One of the angles formed is 65°. Find all eight angles. Identify each pair as corresponding, alternate interior, alternate exterior, or co-interior angles.
Solution — Step by Step
When a transversal cuts two parallel lines, it creates 8 angles (4 at each intersection point). If one angle is 65°, its vertically opposite angle is also 65°. Its supplementary angles are .
graph TD
A["Given angle = 65°"] --> B["Vertically opposite = 65°"]
A --> C["Linear pair = 115°"]
A --> D["Corresponding angle at other intersection = 65°"]
D --> E["Its vertically opposite = 65°"]
D --> F["Its linear pair = 115°"]
C --> G["Corresponding angle at other intersection = 115°"]
G --> H["Its vertically opposite = 115°"]All 8 angles: 65°, 115°, 65°, 115° (at first intersection) and 65°, 115°, 65°, 115° (at second intersection).
| Angle Pair Type | Relationship | Example |
|---|---|---|
| Corresponding | Equal (same position at each intersection) | Both 65° in matching corners |
| Alternate interior | Equal (opposite sides of transversal, between lines) | 65° = 65° |
| Alternate exterior | Equal (opposite sides, outside the lines) | 65° = 65° |
| Co-interior (same-side interior) | Supplementary (add to 180°) | 65° + 115° = 180° |
Why This Works
When lines are parallel, corresponding angles are equal because the geometry at both intersection points is identical — the transversal meets each parallel line at the same angle. Alternate angles are equal because they are formed by the same transversal “zigzagging” between the parallel lines. Co-interior angles are supplementary because together they form a “C” or “U” shape that spans 180°.
The converse is equally powerful: if corresponding angles (or alternate angles) are equal, the lines must be parallel. This is how we prove lines are parallel in geometry proofs — a very common CBSE board question.
Alternative Method
The “F, Z, C” shortcut for remembering angle pairs:
- F-shape: Corresponding angles (look like the letter F when you trace the transversal and one parallel line)
- Z-shape: Alternate interior angles (trace a Z between the parallel lines)
- C-shape (or U-shape): Co-interior angles (trace a C between the parallel lines)
Draw these letters on your figure during exams — it helps identify angle pairs instantly and avoids confusion.
Common Mistake
Applying these properties when lines are NOT parallel. Corresponding angles are equal ONLY when the lines are parallel. If the lines are not parallel, these relationships do not hold. Students sometimes use alternate angle properties without first confirming that the lines are parallel. In proofs, always state “since AB is parallel to CD” before using any parallel line angle property. Missing this statement loses marks in CBSE boards.