Transversal and Parallel Lines
When a line crosses two or more lines at different points, it is called a transversal. The transversal creates several angles at each point where it crosses. When the two lines being crossed are parallel, these angles follow special patterns.
Understanding transversals and parallel lines is important in geometry. It helps us find unknown angles, prove lines are parallel, and solve problems about shapes and constructions.
In Class 7 Mathematics (NCERT), this topic is studied in the chapter Lines and Angles. You will learn about corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles formed when a transversal cuts parallel lines.
What is Transversal and Parallel Lines?
Definition: A transversal is a line that intersects two or more lines at distinct (different) points.
Key terms:
- Transversal: The line that crosses two or more other lines.
- Parallel lines: Two lines in the same plane that never meet, no matter how far they are extended. They are always the same distance apart.
- Interior angles: Angles formed between the two lines (inside the two parallel lines).
- Exterior angles: Angles formed outside the two lines.
Angles formed:
- When a transversal crosses two lines, it creates 8 angles in total (4 at each point of intersection).
- These 8 angles are grouped into special pairs: corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles.
Notation:
- The two parallel lines are usually called l and m (or line 1 and line 2).
- The transversal is usually called t or n.
- We write l ∥ m to mean "l is parallel to m."
Transversal and Parallel Lines Formula
Angle relationships when a transversal cuts parallel lines:
1. Corresponding Angles (F-angles):
Corresponding angles are EQUAL when lines are parallel.
These are angles in the same position at each intersection — both on the same side of the transversal and both above (or both below) their respective lines.
2. Alternate Interior Angles (Z-angles):
Alternate interior angles are EQUAL when lines are parallel.
These are angles between the two lines, on opposite sides of the transversal.
3. Alternate Exterior Angles:
Alternate exterior angles are EQUAL when lines are parallel.
These are angles outside the two lines, on opposite sides of the transversal.
4. Co-interior Angles (Allied Angles or Same-side Interior Angles):
Co-interior angles are SUPPLEMENTARY (add up to 180°) when lines are parallel.
These are angles between the two lines, on the same side of the transversal.
Derivation and Proof
Understanding why these angle relationships hold:
Why corresponding angles are equal:
- Draw two parallel lines l and m, cut by a transversal t.
- At the first intersection, the transversal makes some angle (say 60°) with line l.
- Since l and m point in the same direction (they are parallel), the transversal makes the same angle with line m.
- So the angles in the same position at both intersections must be equal.
Why alternate interior angles are equal:
- Let angle 3 and angle 6 be alternate interior angles.
- Angle 3 and angle 2 are supplementary (they form a linear pair): angle 3 + angle 2 = 180°.
- Angle 2 and angle 6 are corresponding angles (so angle 2 = angle 6).
- Therefore angle 3 + angle 6 = 180°? No — since angle 2 = angle 6, and angle 3 + angle 2 = 180°, we get that angle 3 and angle 6 are equal only if we use the vertically opposite angle instead.
- More directly: angle 3 = angle 7 (corresponding), and angle 7 = angle 6 (vertically opposite). So angle 3 = angle 6.
Why co-interior angles add up to 180°:
- Let angle 3 and angle 5 be co-interior angles.
- Angle 3 and angle 4 form a linear pair: angle 3 + angle 4 = 180°.
- Angle 4 and angle 5 are alternate interior angles: angle 4 = angle 5.
- Substituting: angle 3 + angle 5 = 180°.
Types and Properties
Types of angle pairs formed by a transversal:
- Same position at each intersection.
- One is interior, one is exterior, both on the same side of the transversal.
- Think of the letter F — corresponding angles make an F-shape.
- If lines are parallel: equal.
- Between the two lines (interior).
- On opposite sides of the transversal.
- Think of the letter Z — alternate interior angles make a Z-shape.
- If lines are parallel: equal.
- Outside the two lines (exterior).
- On opposite sides of the transversal.
- If lines are parallel: equal.
4. Co-interior Angles (Allied Angles):
- Between the two lines (interior).
- On the same side of the transversal.
- Think of the letter U or C — they are on the same side.
- If lines are parallel: supplementary (add up to 180°).
5. Vertically Opposite Angles:
- At the same intersection point, opposite each other.
- Always equal (whether or not lines are parallel).
6. Linear Pair:
- Adjacent angles on a straight line at the same intersection.
- Always supplementary (add up to 180°).
Solved Examples
Example 1: Example 1: Finding corresponding angles
Problem: Two parallel lines are cut by a transversal. One of the angles formed is 65°. Find its corresponding angle.
Solution:
Given:
- Lines are parallel.
- One angle = 65°.
Rule:
- Corresponding angles are equal when lines are parallel.
Answer: The corresponding angle = 65°.
Example 2: Example 2: Finding alternate interior angles
Problem: Lines l ∥ m are cut by transversal t. If one of the alternate interior angles is 110°, find the other.
Solution:
Given:
- l ∥ m (parallel lines)
- One alternate interior angle = 110°
Rule:
- Alternate interior angles are equal when lines are parallel.
Answer: The other alternate interior angle = 110°.
Example 3: Example 3: Finding co-interior angles
Problem: Two parallel lines are cut by a transversal. One of the co-interior angles is 75°. Find the other co-interior angle.
Solution:
Given:
- Lines are parallel.
- One co-interior angle = 75°
Rule:
- Co-interior angles are supplementary (add up to 180°).
Calculate:
- Other angle = 180° − 75° = 105°
Answer: The other co-interior angle is 105°.
Example 4: Example 4: Finding all angles
Problem: Lines AB ∥ CD are cut by transversal PQ. The angle at the intersection with AB is 70° (above the transversal, on the left). Find all 8 angles.
Solution:
At the intersection with AB:
- Angle 1 = 70° (given)
- Angle 2 = 180° − 70° = 110° (linear pair with angle 1)
- Angle 3 = 110° (vertically opposite to angle 2)
- Angle 4 = 70° (vertically opposite to angle 1)
At the intersection with CD (using parallel line properties):
- Angle 5 = 70° (corresponding to angle 1)
- Angle 6 = 110° (corresponding to angle 2)
- Angle 7 = 110° (corresponding to angle 3)
- Angle 8 = 70° (corresponding to angle 4)
Answer: The angles are 70°, 110°, 110°, 70° at one intersection and 70°, 110°, 110°, 70° at the other.
Example 5: Example 5: Using algebra to find angles
Problem: Lines l ∥ m are cut by a transversal. Two co-interior angles are (2x + 10)° and (3x + 20)°. Find x and the angles.
Solution:
Given:
- Co-interior angles: (2x + 10)° and (3x + 20)°
Rule: Co-interior angles are supplementary:
- (2x + 10) + (3x + 20) = 180
- 5x + 30 = 180
- 5x = 150
- x = 30
Find the angles:
- First angle = 2(30) + 10 = 70°
- Second angle = 3(30) + 20 = 110°
Verify: 70° + 110° = 180° ✓
Answer: x = 30. The angles are 70° and 110°.
Example 6: Example 6: Checking if lines are parallel
Problem: A transversal cuts two lines. The corresponding angles formed are 125° and 125°. Are the lines parallel?
Solution:
Given:
- Corresponding angles = 125° and 125°
Test:
- If corresponding angles are equal, the lines are parallel.
- 125° = 125° ✓
Answer: Yes, the lines are parallel.
Example 7: Example 7: Using alternate angles to find unknown angle
Problem: In the figure, PQ ∥ RS. Transversal t makes an angle of 55° with PQ. Find the alternate interior angle at RS.
Solution:
Given:
- PQ ∥ RS
- Angle at PQ = 55° (interior angle on one side of transversal)
Rule:
- Alternate interior angles are equal when lines are parallel.
Answer: The alternate interior angle at RS = 55°.
Example 8: Example 8: Algebraic problem with corresponding angles
Problem: Two parallel lines are cut by a transversal. Corresponding angles are (5y − 20)° and (3y + 40)°. Find y and the angles.
Solution:
Given:
- Corresponding angles: (5y − 20)° and (3y + 40)°
Rule: Corresponding angles are equal:
- 5y − 20 = 3y + 40
- 5y − 3y = 40 + 20
- 2y = 60
- y = 30
Find the angles:
- 5(30) − 20 = 150 − 20 = 130°
- 3(30) + 40 = 90 + 40 = 130°
Verify: Both angles are 130°, which are equal ✓
Answer: y = 30. Each corresponding angle is 130°.
Example 9: Example 9: Real-life application — railway tracks
Problem: Two railway tracks are parallel. A road crosses both tracks. The road makes an angle of 60° with one track. What angle does it make with the other track?
Solution:
Given:
- Railway tracks are parallel lines.
- The road is the transversal.
- Angle with first track = 60°
The angle at the second track is a corresponding angle:
- Corresponding angles are equal for parallel lines.
Answer: The road makes an angle of 60° with the other track.
Example 10: Example 10: Checking parallel lines using co-interior angles
Problem: A transversal cuts two lines. The co-interior angles are 95° and 80°. Are the lines parallel?
Solution:
Given:
- Co-interior angles = 95° and 80°
Test: Co-interior angles must add to 180° for parallel lines:
- 95° + 80° = 175°
- 175° ≠ 180°
Answer: No, the lines are not parallel because the co-interior angles do not add up to 180°.
Real-World Applications
Real-world uses of transversal and parallel lines:
- Railway tracks: Railway tracks are parallel lines. When a road crosses them, it acts as a transversal. Engineers use angle relationships to design safe crossings.
- Architecture: Floors of a building are parallel. Staircases and ramps act as transversals. Angle calculations help in structural design.
- Road design: When a road intersects two parallel roads, traffic engineers use angle measurements for turns and intersections.
- Carpentry: Carpenters use parallel cuts and transversal measurements to ensure shelves, frames, and joints are properly aligned.
- Map reading: Latitude lines are parallel. Lines of longitude act as transversals. Understanding angles helps in navigation.
- Sports: Lines on a football field, basketball court, or tennis court are parallel. Understanding transversal angles helps in field design and marking.
Key Points to Remember
- A transversal is a line that cuts two or more lines at different points.
- When a transversal cuts two lines, it forms 8 angles (4 at each intersection).
- For parallel lines cut by a transversal:
- Corresponding angles are equal (F-shape).
- Alternate interior angles are equal (Z-shape).
- Alternate exterior angles are equal.
- Co-interior angles are supplementary (add up to 180°).
- If corresponding angles (or alternate angles) are equal, the lines are parallel.
- If co-interior angles add up to 180°, the lines are parallel.
- These properties can be used to prove lines are parallel or to find unknown angles.
Practice Problems
- Lines AB ∥ CD are cut by a transversal. If one angle is 48°, find all 8 angles.
- Two parallel lines are cut by a transversal. Co-interior angles are (4x + 5)° and (6x − 15)°. Find x.
- A transversal cuts two lines making corresponding angles of 72° and 72°. Are the lines parallel?
- Lines l ∥ m are cut by transversal t. Alternate interior angles are (3a + 15)° and (5a − 25)°. Find a.
- Two parallel lines are cut by a transversal. One exterior angle is 135°. Find the co-interior angle on the same side.
- A road crosses two parallel railway tracks. It makes an angle of 75° with one track. Find the angle with the other track and both co-interior angles.
- Can two co-interior angles both be acute (less than 90°) if the lines are parallel? Explain.
- If alternate interior angles formed by a transversal are 90° each, what can you say about the transversal?
Frequently Asked Questions
Q1. What is a transversal?
A transversal is a line that intersects (crosses) two or more lines at different points. It creates angles at each point of intersection.
Q2. How many angles are formed when a transversal cuts two lines?
A transversal cutting two lines creates 8 angles — 4 angles at each intersection point. These 8 angles form special pairs: corresponding, alternate interior, alternate exterior, and co-interior.
Q3. What are corresponding angles?
Corresponding angles are in the same position at each intersection — both above or both below their respective lines, on the same side of the transversal. They form an F-shape. When lines are parallel, corresponding angles are equal.
Q4. What are alternate interior angles?
Alternate interior angles are between the two lines (interior) and on opposite sides of the transversal. They form a Z-shape. When lines are parallel, alternate interior angles are equal.
Q5. What are co-interior angles?
Co-interior angles (also called allied angles or same-side interior angles) are between the two lines and on the same side of the transversal. When lines are parallel, they are supplementary — they add up to 180°.
Q6. How can you use these angle properties to prove lines are parallel?
If a transversal cuts two lines and any one of these conditions is true — corresponding angles are equal, alternate interior angles are equal, or co-interior angles add up to 180° — then the two lines are parallel.
Q7. Do these angle properties work if the lines are not parallel?
No. If the lines are not parallel, corresponding angles are NOT equal, alternate interior angles are NOT equal, and co-interior angles do NOT add up to 180°. The special relationships only hold for parallel lines.
Q8. What is the difference between interior and exterior angles?
Interior angles are between the two lines (in the region enclosed by the two parallel lines). Exterior angles are outside the two lines (above the top line or below the bottom line). Both types form special pairs with the transversal.
