Parallel Lines Cut by Transversal — Alternate Interior Angles
The Question
Two parallel lines AB and CD are cut by a transversal EF. One of the angles at the first intersection is 70°. Find the alternate interior angle.
Setting Up the Picture
When a transversal cuts two parallel lines, it creates 8 angles — 4 at each intersection.
E
|
A ------+------- B ← Line 1
| ∠1 ∠2
| ∠3 ∠4 (angles at first intersection, between the lines)
|
C ------+------- D ← Line 2
| ∠5 ∠6
| ∠7 ∠8 (angles at second intersection)
|
FThe angles between the two parallel lines (angles 3, 4, 5, 6 in the figure above) are called interior angles.
Alternate interior angles are the interior angles on opposite sides of the transversal:
- ∠3 and ∠6 are alternate interior angles.
- ∠4 and ∠5 are alternate interior angles.
The Alternate Interior Angles Property
When two lines are parallel, alternate interior angles are equal.
This is a fundamental theorem in geometry. The lines being parallel is the key condition.
If l ∥ m and t is a transversal: Alternate interior angles are equal.
∠3 = ∠6 and ∠4 = ∠5
The Z-Shape Pattern
Here’s a visual trick to identify alternate interior angles: they form a Z shape (or reverse Z) with the transversal and the two parallel lines.
A --------\
\ ← This slant is the transversal
/-----
C ----/The angles in the “kinks” of the Z are the alternate interior angles. They are always equal.
Solution
Given: One angle at the intersection = 70°.
Since the lines are parallel, the alternate interior angle = 70° (equal).
Let’s also find all related angles at that intersection:
The angle of 70° and its linear pair:
Linear pair = 180° - 70° = 110°
Vertically opposite to 70° = 70° Vertically opposite to 110° = 110°
At both intersections, the angles are: 70°, 110°, 70°, 110°.
Distinguishing Corresponding, Alternate Interior, and Co-Interior
These three often get confused. Let’s clarify with a memory trick:
| Pair type | Memory shape | Equal or Supplementary? |
|---|---|---|
| Corresponding | F-shape | Equal |
| Alternate interior | Z-shape | Equal |
| Co-interior (same-side) | C-shape or U-shape | Supplementary (180°) |
Draw the transversal cutting the two parallel lines on rough paper. Physically trace the Z, F, and C shapes with your finger or pencil. Once you can see these shapes in the figure, identifying the angle pairs becomes very natural.
Common mistake: Applying these rules to non-parallel lines.
If the lines are NOT parallel, corresponding angles are NOT equal and alternate interior angles are NOT equal. Always check that the lines are parallel before using these properties!
Try These Similar Problems
Problem 1: Two parallel lines are cut by a transversal. An alternate interior angle is 55°. Find the co-interior angle on the same side.
The co-interior angle is on the same side as the 55° angle but at the other intersection.
Co-interior angles are supplementary: Co-interior angle = 180° - 55° = 125°
Problem 2: Two parallel lines are cut by a transversal. A corresponding angle is 80°. Find the alternate interior angle at the second intersection.
At the first intersection, corresponding angle = 80°. At the second intersection, the corresponding angle is also 80° (corresponding angles are equal).
The alternate interior angle to the angle at the first intersection: The interior angle at the first intersection = 80° (it’s on the interior side). The alternate interior angle at the second intersection = 80°.
Answer: 80°
Problem 3: Two parallel lines are cut by a transversal. One angle is (4x + 5)° and its alternate interior angle is (3x + 20)°. Find x and the angle.
Alternate interior angles are equal: 4x + 5 = 3x + 20 4x - 3x = 20 - 5 x = 15
Angle = 4(15) + 5 = 60 + 5 = 65° Alternate interior angle = 3(15) + 20 = 45 + 20 = 65° ✓
Exam tip: Questions on parallel lines are very common in CBSE Class 7 geometry. When you see a transversal crossing parallel lines, identify ALL eight angles — mark the equal ones and the supplementary ones. Examiners often ask for multiple angles in one question, so label your diagram clearly.