Parallel and Intersecting Lines — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Parallel and Intersecting Lines" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Parallel and Intersecting Lines" — 8 important questions with detailed answers for…
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NCERT Solutions for Class 7 Mathematics — Parallel and Intersecting Lines. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Parallel and Intersecting Lines — All 8 Questions Solved
Q1. Define parallel lines and give one example.
Step 1: Definition of parallel lines:
Two lines in a plane that never intersect and are always the same distance apart are called parallel lines.
Step 2: They remain equidistant (equal distance) at all points.
Step 3: Examples:
- The opposite edges of a rectangle
- Railway tracks
- The top and bottom lines on a notebook page
- Opposite sides of a parallelogram
Answer: Parallel lines are lines that never meet and are always equidistant. Example: Railway tracks or opposite sides of a rectangle.
Q2. What is a transversal? Name the angles formed when a transversal crosses two parallel lines.
Step 1: Definition:
A transversal is a line that intersects two or more other lines at different points.
Step 2: When a transversal crosses two parallel lines, it forms 8 angles:
At the first intersection: 4 angles
At the second intersection: 4 angles
Step 3: Types of angles formed:
- Corresponding angles (equal when lines are parallel)
- Alternate interior angles (equal when lines are parallel)
- Alternate exterior angles (equal when lines are parallel)
- Co-interior angles or consecutive interior angles (supplementary when lines are parallel)
Answer: A transversal is a line crossing two or more lines. It forms corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles.
Q3. If two parallel lines are cut by a transversal, and one angle is 60°, find all other angles.
Step 1: When a transversal crosses two parallel lines, angles are formed at two intersection points.
Step 2: If one angle is 60°:
- The angle adjacent to it = 180° - 60° = 120° (supplementary angles on a line)
Step 3: At the first intersection:
Angles formed: 60°, 120°, 60°, 120° (opposite angles are equal)
Step 4: At the second intersection (with parallel line):
Corresponding angles are equal, so angles are also: 60°, 120°, 60°, 120°
Answer: The angles are 60°, 120°, 60°, 120° at the first intersection and 60°, 120°, 60°, 120° at the second intersection.
Q4. Explain the difference between intersecting lines and perpendicular lines.
Step 1: Intersecting lines:
- Two lines that cross each other at a point
- They can meet at any angle (not necessarily 90°)
- The angle can be acute, right, or obtuse
Step 2: Perpendicular lines:
- Two lines that intersect at exactly 90° (right angle)
- Perpendicular lines are a special case of intersecting lines
- The symbol ⊥ is used to denote perpendicular lines
Example: AB ⊥ CD means AB is perpendicular to CD
Step 3: Relationship:
All perpendicular lines are intersecting lines, but not all intersecting lines are perpendicular.
Answer: Intersecting lines cross at any angle; perpendicular lines cross at exactly 90°. All perpendicular lines are intersecting, but not vice versa.
Q5. If two angles on a straight line are (3x + 10)° and (5x - 30)°, find the value of x.
Step 1: Angles on a straight line are supplementary (sum to 180°):
(3x + 10)° + (5x - 30)° = 180°
Step 2: Combine like terms:
3x + 5x + 10 - 30 = 180
8x - 20 = 180
Step 3: Add 20 to both sides:
8x = 200
Step 4: Divide by 8:
x = 25
Step 5: Verify:
First angle: 3(25) + 10 = 75 + 10 = 85°
Second angle: 5(25) - 30 = 125 - 30 = 95°
Sum: 85° + 95° = 180° ✓
Answer: x = 25
Q6. Two lines intersect forming four angles. If one angle is 45°, find the other three angles.
Step 1: When two lines intersect, they form 4 angles.
Let the four angles be A, B, C, D (in order around the intersection point).
Step 2: Vertically opposite angles are equal.
If angle A = 45°, then angle C = 45° (opposite to A)
Step 3: Adjacent angles are supplementary (sum to 180°):
Angle B = 180° - 45° = 135°
Angle D = 180° - 45° = 135° (also opposite to B)
Step 4: Verify: 45° + 135° + 45° + 135° = 360° ✓
Answer: The four angles are 45°, 135°, 45°, 135°.
Q7. State the properties of parallel lines cut by a transversal.
Step 1: When a transversal cuts two parallel lines, the following properties hold:
Property 1: Corresponding angles are equal
- Angles on the same side of the transversal and in the same position at each intersection are equal
Property 2: Alternate interior angles are equal
- Interior angles on opposite sides of the transversal are equal
Property 3: Alternate exterior angles are equal
- Exterior angles on opposite sides of the transversal are equal
Property 4: Co-interior angles (consecutive interior angles) are supplementary
- Interior angles on the same side of the transversal sum to 180°
Answer: (1) Corresponding angles are equal, (2) Alternate interior angles are equal, (3) Alternate exterior angles are equal, (4) Co-interior angles sum to 180°.
Q8. Two adjacent angles form a linear pair. If one angle measures 72°, what is the other angle?
Step 1: A linear pair consists of two adjacent angles on a straight line.
Step 2: Angles in a linear pair are supplementary (sum to 180°).
Step 3: If one angle = 72°, then:
Other angle = 180° - 72° = 108°
Step 4: Verify:
72° + 108° = 180° ✓
Answer: The other angle is 108°.
Parallel and Intersecting Lines — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which of the following describes lines that never intersect, no matter how far they are extended?
- A. Intersecting lines
- B. Parallel lines
- C. Perpendicular lines
- D. Transversal lines
Answer: B. Parallel lines — Parallel lines are defined as lines that lie in the same plane and never meet, even if extended indefinitely.
MCQ 2. A line that intersects two or more distinct lines at different points is called a:
- A. Parallel line
- B. Perpendicular line
- C. Transversal
- D. Ray
Answer: C. Transversal — A transversal is a line that cuts across two or more other lines at distinct points, forming various angles.
MCQ 3. When two lines intersect, the angles opposite to each other at the point of intersection are known as:
- A. Corresponding angles
- B. Alternate interior angles
- C. Vertically opposite angles
- D. Consecutive interior angles
Answer: C. Vertically opposite angles — Vertically opposite angles are formed when two lines intersect, and they are always equal.
MCQ 4. If two angles form a linear pair, their sum is always:
- A. 90 degrees
- B. 180 degrees
- C. 270 degrees
- D. 360 degrees
Answer: B. 180 degrees — A linear pair consists of two adjacent angles that form a straight line, meaning their sum is 180 degrees.
MCQ 5. If two parallel lines are cut by a transversal, which pair of angles are equal?
- A. Corresponding angles
- B. Alternate interior angles
- C. Alternate exterior angles
- D. All of the above
Answer: D. All of the above — When two parallel lines are intersected by a transversal, corresponding angles, alternate interior angles, and alternate exterior angles are all equal.
MCQ 6. Two lines are said to be intersecting if they have how many common points?
- A. Zero
- B. Exactly one
- C. Two
- D. Infinitely many
Answer: B. Exactly one — Intersecting lines cross each other at precisely one common point.
MCQ 7. What is the relationship between consecutive interior angles when two parallel lines are intersected by a transversal?
- A. They are always equal
- B. They are always supplementary
- C. They are always complementary
- D. They are vertically opposite
Answer: B. They are always supplementary — Consecutive interior angles (also known as co-interior or allied angles) add up to 180 degrees when the lines are parallel.
MCQ 8. If an angle is 65 degrees, what is the measure of its vertically opposite angle?
- A. 25 degrees
- B. 115 degrees
- C. 65 degrees
- D. 130 degrees
Answer: C. 65 degrees — Vertically opposite angles are always equal, so if one angle is 65 degrees, its vertically opposite angle is also 65 degrees.
MCQ 9. If two parallel lines are intersected by a transversal and one corresponding angle is 70 degrees, what is the measure of its pair?
- A. 20 degrees
- B. 70 degrees
- C. 110 degrees
- D. 180 degrees
Answer: B. 70 degrees — When lines are parallel, corresponding angles are equal. So, if one is 70 degrees, the other is also 70 degrees.
MCQ 10. Two angles form a linear pair. If one angle measures 100 degrees, what is the measure of the other angle?
- A. 10 degrees
- B. 80 degrees
- C. 100 degrees
- D. 180 degrees
Answer: B. 80 degrees — The sum of angles in a linear pair is 180 degrees. So, 180 - 100 = 80 degrees.
MCQ 11. In a diagram, angles on opposite sides of the transversal and between the two lines are called:
- A. Corresponding angles
- B. Alternate interior angles
- C. Alternate exterior angles
- D. Consecutive interior angles
Answer: B. Alternate interior angles — Alternate interior angles are located between the two lines and on opposite sides of the transversal.
MCQ 12. When a transversal intersects two lines, how many angles are formed in total?
- A. 2
- B. 4
- C. 6
- D. 8
Answer: D. 8 — At each of the two intersection points, four angles are formed, making a total of 4 + 4 = 8 angles.
MCQ 13. If lines l and m are parallel and a transversal t intersects them such that an alternate interior angle is 40 degrees, what is the measure of its paired alternate interior angle?
- A. 140 degrees
- B. 50 degrees
- C. 40 degrees
- D. 90 degrees
Answer: C. 40 degrees — For parallel lines intersected by a transversal, alternate interior angles are equal. Thus, the other angle is also 40 degrees.
MCQ 14. Lines p and q are parallel. A transversal intersects them. If a pair of consecutive interior angles are (x + 20)° and (2x - 50)°, what is the value of x?
- A. 50
- B. 70
- C. 60
- D. 40
Answer: C. 60 — Consecutive interior angles are supplementary, so (x + 20) + (2x - 50) = 180. This simplifies to 3x - 30 = 180, leading to 3x = 210, and x = 70.
MCQ 15. If two lines are intersected by a transversal such that the corresponding angles are not equal, then the lines are:
- A. Parallel
- B. Perpendicular
- C. Intersecting
- D. Congruent
Answer: C. Intersecting — If corresponding angles are not equal, then the lines are not parallel; therefore, they must be intersecting.
MCQ 16. Which of the following is true for lines that are perpendicular?
- A. They never intersect
- B. They intersect at an acute angle
- C. They intersect at a right angle
- D. They intersect at two points
Answer: C. They intersect at a right angle — Perpendicular lines are a special case of intersecting lines where they meet at a 90-degree angle (a right angle).
MCQ 17. In a figure, line AB is parallel to line CD. A transversal EF intersects AB at G and CD at H. If ∠EGB = 110°, find ∠GHC.
- A. 70°
- B. 110°
- C. 20°
- D. 180°
Answer: A. 70° — ∠EGB and ∠AGH are vertically opposite, so ∠AGH = 110°. ∠AGH and ∠GHC are consecutive interior angles (or allied angles). Their sum is 180°. So ∠GHC = 180° - 110° = 70°.
MCQ 18. If two lines are intersected by a transversal such that alternate exterior angles are equal, then the lines are:
- A. Perpendicular
- B. Parallel
- C. Skew
- D. Converging
Answer: B. Parallel — The converse of the alternate exterior angles theorem states that if alternate exterior angles are equal, the lines are parallel.
MCQ 19. Given parallel lines l and m, and a transversal t. If ∠1 is an angle and ∠2 is its linear pair. If ∠1 corresponds to ∠3, and ∠3 = 85°, then what is ∠2?
- A. 85°
- B. 95°
- C. 180°
- D. 105°
Answer: B. 95° — Since lines are parallel, corresponding angles are equal, so ∠1 = ∠3 = 85°. Since ∠1 and ∠2 form a linear pair, ∠1 + ∠2 = 180°. Therefore, ∠2 = 180° - 85° = 95°.
MCQ 20. Which of the following statements is always true?
- A. All intersecting lines are perpendicular.
- B. All parallel lines are straight lines.
- C. All straight lines are parallel.
- D. All perpendicular lines are parallel.
Answer: B. All parallel lines are straight lines. — Parallel lines are always straight lines that extend infinitely without meeting. Other options are incorrect definitions.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.