Parallel Lines
🎯 What will you learn on this page?
👉 Identify parallel lines and a transversal.
👉 Understand corresponding, alternate interior and same-side interior angles.
👉 Use angle relationships formed by a transversal across parallel lines.
👉 Find unknown angles using simple equations and short proofs.
👉 Use angle relationships to check whether two lines are parallel.
👉 Parallel Lines
Two straight lines in the same plane are called parallel lines if they never meet, no matter how far they are extended.
The perpendicular distance between two parallel lines remains constant. If AB ∥ CD, it means that line AB is parallel to line CD.
Parallel Lines
AB ∥ CD
Transversal of Parallel Lines
MN intersects AB and CD at distinct points
MN is a transversal
👉 Transversal
A line that intersects two or more lines at distinct points is called a transversal.
A transversal does not have to look slanted. Its important property is that it crosses the given lines at different points.
When a transversal intersects two parallel lines, several useful pairs of angles are formed.
👉 Corresponding Angles
When a transversal intersects two parallel lines, angles in the same relative position at the two intersections are called corresponding angles.
For parallel lines, corresponding angles are equal.
Corresponding Angles
∠MOB = ∠OPD
Angles in the same relative position are corresponding angles.
👉 Alternate Interior Angles
When a transversal intersects two parallel lines, the angles that lie between the two lines and on opposite sides of the transversal are called alternate interior angles.
For parallel lines, alternate interior angles are equal.
Alternate Interior Angles
∠AOP = ∠DPO
∠BOP = ∠CPO
| ∵ | ∠MOB = ∠OPD because they are corresponding angles. |
| Also, | ∠MOB = ∠AOP because they are vertically opposite angles. |
| ∴ | ∠AOP = ∠OPD |
| Similarly, ∠BOP = ∠CPO. |
👉 Interior Angles on the Same Side
When a transversal intersects two parallel lines, the interior angles that lie between the two lines and on the same side of the transversal add up to 180°.
These are also called same-side interior angles or co-interior angles.
Same-Side Interior Angles
∠BOP + ∠OPD = 180°
∠CPO + ∠AOP = 180°
| ∵ | ∠MOB = ∠OPD because they are corresponding angles. |
| Also, | ∠MOB + ∠BOP = 180° because they form a linear pair. |
| ∴ | ∠BOP + ∠OPD = 180° |
🔹 How to Check Whether Two Lines Are Parallel
The converse of the angle rules can be used to test whether two lines are parallel.
If a transversal cuts two lines and any one of the following is true, the two lines are parallel:
🔹 A pair of corresponding angles is equal.
🔹 A pair of alternate interior angles is equal.
🔹 A pair of same-side interior angles has a sum of 180°.
✏️ Solved Examples
✍️ Example 1: Find a Corresponding Angle
Suppose AB ∥ CD and a transversal crosses both lines. If one corresponding angle is 68°, find the other corresponding angle.
Corresponding angles are equal when the lines are parallel.
Required angle = 68°Therefore, the other corresponding angle is 68°.
✍️ Example 2: Find a Same-Side Interior Angle
One same-side interior angle formed by a transversal across two parallel lines is 126°. Find the other angle.
Same-side interior angles are supplementary.
Required angle = 180° - 126° = 54°Therefore, the other interior angle is 54°.
✍️ Example 3: Find an Angle Using an Equation
Two corresponding angles are (3x + 5)° and (5x - 35)°. The lines are parallel. Find x and the angle.
Since corresponding angles are equal,
3x + 5 = 5x - 35 40 = 2x x = 20Now,
3x + 5 = 3 × 20 + 5 = 65°Therefore, x = 20 and each corresponding angle is 65°.
✍️ Example 4: Interior Angles in a Ratio
Two same-side interior angles are in the ratio 4 : 5. Find the angles.
Let the angles be 4x and 5x.
4x + 5x = 180° 9x = 180° x = 20°Therefore, the angles are 80° and 100°.
✍️ Example 5: Are the Lines Parallel?
A transversal cuts two lines, and a pair of corresponding angles measures 74° and 74°. Are the lines parallel?
Yes. Since the corresponding angles are equal, the converse of the corresponding-angle rule tells us that the two lines are parallel.
🧩 Practice Questions
✍️ 1. One corresponding angle formed by a transversal across two parallel lines is 57°. Find its corresponding angle.
✍️ 2. One alternate interior angle is 72°. Find the other alternate interior angle.
✍️ 3. One same-side interior angle is 118°. Find the other interior angle.
✍️ 4. Two corresponding angles are (4x + 8)° and (6x - 22)°. Find x and the angle.
✍️ 5. Two same-side interior angles are (3x + 12)° and (5x - 8)°. Find x and both angles.
✍️ 6. Two same-side interior angles are in the ratio 7 : 5. Find the angles.
✍️ 7. A transversal cuts two lines. A pair of corresponding angles is 82° and 82°. Are the two lines parallel? Give a reason.
✍️ 8. Two interior angles on the same side of a transversal are 117° and 63°. Can you conclude that the two lines are parallel? Explain.
1. 57°
2. 72°
3. 62°
4. x = 15; each angle = 68°
5. x = 22; angles = 78° and 102°
6. 105° and 75°
7. Yes. Equal corresponding angles imply that the lines are parallel.
8. Yes. 117° + 63° = 180°, so the converse of the same-side interior angle rule implies that the lines are parallel.
⚠️ Common Misconceptions
🔹 Corresponding angles are always equal.
Not always. They are guaranteed to be equal when the transversal intersects parallel lines.
🔹 A transversal must always be slanted.
No. A transversal simply needs to intersect two or more lines at distinct points.
🔹 All interior angles are equal.
No. Alternate interior angles are equal, while same-side interior angles add up to 180°.
🔹 Two lines can be called parallel just because they look parallel in a diagram.
No. Use a parallel mark, a given fact, or a valid angle relationship to justify that the lines are parallel.
✅ Quick Revision
🔹 Parallel lines lie in the same plane and never meet.
🔹 A transversal intersects two or more lines at distinct points.
🔹 Corresponding angles are equal when the lines are parallel.
🔹 Alternate interior angles are equal when the lines are parallel.
🔹 Same-side interior angles have a sum of 180°.
🔹 The converse of these angle relationships can be used to prove that two lines are parallel.
🔹 In unknown-angle problems, first identify the correct angle relationship, then form and solve the equation.
