Chapter focus
Parallel and intersecting lines create predictable angle relationships. Use these relationships to calculate unknown angles and decide whether lines are parallel.
These original notes support revision of the named Ganita Prakash Part I chapter. Use the textbook diagrams and constructions for the full treatment.
1. Intersecting lines
When two straight lines cross, they form four angles. Vertically opposite angles are equal. Adjacent angles on a straight line form a linear pair and add to 180°.
Worked example
If one angle is 68°, the opposite angle is 68°. Each adjacent angle is 180° − 68° = 112°. Check: 68° + 112° = 180°.
2. Parallel lines and a transversal
A transversal crosses two or more lines. If it crosses a pair of parallel lines:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Interior angles on the same side of the transversal add to 180°.
These facts can be used in reverse: equality of a corresponding pair or alternate interior pair can establish that the lines are parallel.
3. Solve an angle problem step by step
Two parallel lines are cut by a transversal. One interior angle is 3x + 10 degrees, and its alternate interior partner is 100 degrees.
- Alternate interior angles are equal for parallel lines: 3x + 10 = 100.
- Subtract 10 from both sides: 3x = 90.
- Divide by 3: x = 30.
- The angle is 3(30) + 10 = 100°, matching the given partner.
4. Recognise angle relationships in a diagram
Mark parallel arrows, right-angle squares and equal angles on the figure. Name the relationship used: vertical angles, a linear pair, corresponding angles or alternate interior angles. If no parallel marks or given condition exists, do not assume parallel lines from how the picture looks.
Common errors to avoid
- Vertically opposite angles are equal; adjacent angles usually are not.
- Angle equality from corresponding or alternate interior positions requires parallel lines.
- A linear pair sums to 180°, not 360°.
- Show the geometric reason before using an equation.
Practice: state the property
1. Two lines intersect. One angle is 73°. Find its vertically opposite angle.
Answer: 73°, because vertically opposite angles are equal.
2. Find the angle adjacent to 73° on a straight line.
Answer: 180° − 73° = 107°.
3. Parallel lines are crossed by a transversal. An alternate interior angle is 64°. Find its partner.
Answer: 64°, because alternate interior angles between parallel lines are equal.
4. Two corresponding angles are 4x and 120°. Find x if the lines are parallel.
Answer: 4x = 120, so x = 30.
5. What does an interior same-side angle pair between parallel lines add to?
Answer: 180°.
Chapter summary
Intersecting lines produce equal vertical angles and supplementary linear pairs. With parallel lines, a transversal creates equal corresponding and alternate interior angles, while same-side interior angles are supplementary. Label each relationship and justify it.
Open Ganita Prakash Part I on NCERT · NCERT CIET Grade 7 resources