Lines and Angles — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Lines and Angles" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Lines and Angles" — 8 important questions with detailed answers for CBSE board exam…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 9 Mathematics — Lines and Angles. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Lines and Angles — All 8 Questions Solved
Q1. If a transversal intersects two lines and makes angles of 65° and 115° on the same side of the transversal, determine if the lines are parallel.
Given: Transversal intersects two lines
Angles on the same side of transversal = 65° and 115°
Determine: Are the lines parallel?
Step 1: Recall the co-interior angles property
Co-interior angles (also called consecutive interior angles or allied angles):
- Located on the same side of the transversal
- Between the two lines (interior)
- Property: If two lines are parallel, co-interior angles are supplementary (sum to 180°)
Step 2: Check if the angles are supplementary
Sum of angles = 65° + 115° = 180°
Step 3: Apply the converse of the co-interior angle theorem
If co-interior angles are supplementary, then the lines are parallel.
Since 65° + 115° = 180°, the lines ARE parallel.
Step 4: Conclusion
The two lines are parallel to each other.
Answer: Yes, the lines are parallel because the co-interior angles sum to 180°.
Q2. Two lines intersect. If one angle formed is 48°, find all four angles.
Given: Two lines intersect
One angle = 48°
Find: All four angles
Step 1: Understand vertical angles
When two lines intersect, they form four angles.
Vertical angles (opposite angles) are equal.
Step 2: Identify the angles
Let's label the angles as ∠1, ∠2, ∠3, ∠4 going around.
Given: ∠1 = 48°
Step 3: Find the vertically opposite angle
Vertical angle to ∠1 is ∠3.
∠3 = ∠1 = 48° (vertically opposite angles are equal)
Step 4: Find adjacent angles using linear pair property
Adjacent angles are supplementary (sum to 180°).
∠2 = 180° - ∠1 = 180° - 48° = 132°
Step 5: Find the fourth angle
∠4 = 180° - ∠3 = 180° - 48° = 132°
Or, ∠4 = ∠2 = 132° (vertically opposite angles)
Step 6: Verify
∠1 + ∠2 + ∠3 + ∠4 = 48° + 132° + 48° + 132° = 360° ✓
Answer: The four angles are 48°, 132°, 48°, and 132°
Q3. In triangle ABC, if ∠A = 60°, ∠B = 70°, find ∠C and verify that the sum equals 180°.
Given: Triangle ABC with ∠A = 60°, ∠B = 70°
Find: ∠C
Step 1: Use the angle sum property of a triangle
The sum of all interior angles in a triangle = 180°
∠A + ∠B + ∠C = 180°
Step 2: Substitute known values
60° + 70° + ∠C = 180°
130° + ∠C = 180°
∠C = 180° - 130°
∠C = 50°
Step 3: Verify the sum
∠A + ∠B + ∠C = 60° + 70° + 50° = 180° ✓
Step 4: Geometric interpretation
The three angles are 60°, 70°, and 50°.
Since all angles are less than 90°, this is an acute-angled triangle.
Answer: ∠C = 50°, and the sum of angles = 180°
Q4. An exterior angle of a triangle is 125°. One of the non-adjacent interior angles is 65°. Find the other non-adjacent interior angle.
Given: Exterior angle = 125°
One non-adjacent interior angle = 65°
Find: The other non-adjacent interior angle
Step 1: Recall the exterior angle property
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Exterior angle = ∠1 + ∠2 (where ∠1 and ∠2 are non-adjacent interior angles)
Step 2: Apply the property
125° = 65° + (other non-adjacent angle)
Other non-adjacent angle = 125° - 65° = 60°
Step 3: Verify using triangle angle sum
Let the interior angle adjacent to the exterior angle = x
Then: 65° + 60° + x = 180° (angle sum property)
x = 180° - 65° - 60° = 55°
Also verify exterior angle = interior adjacent angle
Exterior angle = 180° - x = 180° - 55° = 125° ✓
Answer: The other non-adjacent interior angle is 60°
Q5. A line makes an angle of 120° with the positive x-axis. Find the angle it makes with the negative x-axis.
Given: Line makes 120° with positive x-axis
Find: Angle with negative x-axis
Step 1: Visualize the problem
The positive x-axis and negative x-axis form a straight line.
They are collinear and opposite in direction.
Step 2: Understand the angle relationship
If a line makes angle θ with the positive x-axis (measured counterclockwise),
then it makes angle (180° - θ) with the negative x-axis.
Step 3: Apply the formula
Angle with positive x-axis = 120°
Angle with negative x-axis = 180° - 120° = 60°
Step 4: Verify using geometry
The positive x-axis and negative x-axis are supplementary (form a straight line).
If the line makes 120° on one side, it makes 180° - 120° = 60° on the other side.
Alternative verification:
Angle from positive x-axis (counterclockwise) = 120°
Angle from negative x-axis (counterclockwise) = 120° - 180° = -60° (or 60° clockwise)
Magnitude = 60°
Answer: The line makes an angle of 60° with the negative x-axis
Q6. Two parallel lines are cut by a transversal. If one of the alternate interior angles is (2x + 15)° and the other is (3x - 5)°, find x and both angles.
Given: Two parallel lines cut by transversal
Alternate interior angles: (2x + 15)° and (3x - 5)°
Find: Value of x and both angles
Step 1: Apply the alternate interior angles property
When two parallel lines are cut by a transversal, alternate interior angles are equal.
2x + 15 = 3x - 5
Step 2: Solve for x
2x + 15 = 3x - 5
15 + 5 = 3x - 2x
20 = x
x = 20
Step 3: Find the first angle
First angle = 2x + 15 = 2(20) + 15 = 40 + 15 = 55°
Step 4: Find the second angle
Second angle = 3x - 5 = 3(20) - 5 = 60 - 5 = 55°
Step 5: Verify
Both angles = 55° ✓ (they are equal as expected for alternate interior angles)
Answer: x = 20, and both angles = 55°
Q7. Angles on a straight line: If three angles form a linear pair with measures (x + 10)°, (2x - 30)°, and (3x - 40)°, find all three angles.
Given: Three angles on a straight line
Measures: (x + 10)°, (2x - 30)°, (3x - 40)°
Find: All three angles
Step 1: Apply the property of angles on a straight line
Angles on a straight line sum to 180°.
(x + 10) + (2x - 30) + (3x - 40) = 180
Step 2: Combine like terms
x + 2x + 3x + 10 - 30 - 40 = 180
6x - 60 = 180
Step 3: Solve for x
6x = 180 + 60
6x = 240
x = 40
Step 4: Find all three angles
First angle = x + 10 = 40 + 10 = 50°
Second angle = 2x - 30 = 2(40) - 30 = 80 - 30 = 50°
Third angle = 3x - 40 = 3(40) - 40 = 120 - 40 = 80°
Step 5: Verify
Sum = 50° + 50° + 80° = 180° ✓
Answer: The three angles are 50°, 50°, and 80°
Q8. Prove that the sum of angles in a quadrilateral is 360°.
Statement: The sum of interior angles of a quadrilateral = 360°
Proof:
Step 1: Draw a quadrilateral ABCD
Label the interior angles as ∠A, ∠B, ∠C, ∠D.
Step 2: Divide the quadrilateral into two triangles
Draw diagonal AC from vertex A to vertex C.
This divides quadrilateral ABCD into two triangles: △ABC and △ACD.
Step 3: Apply the angle sum property to triangle ABC
In △ABC: ∠CAB + ∠ABC + ∠BCA = 180° ... (1)
Step 4: Apply the angle sum property to triangle ACD
In △ACD: ∠CAD + ∠ACD + ∠ADC = 180° ... (2)
Step 5: Add equations (1) and (2)
(∠CAB + ∠ABC + ∠BCA) + (∠CAD + ∠ACD + ∠ADC) = 180° + 180°
Step 6: Rearrange terms
(∠CAB + ∠CAD) + ∠ABC + (∠BCA + ∠ACD) + ∠ADC = 360°
Step 7: Recognize that combined angles form quadrilateral angles
∠A = ∠CAB + ∠CAD
∠B = ∠ABC
∠C = ∠BCA + ∠ACD
∠D = ∠ADC
Step 8: Conclusion
∠A + ∠B + ∠C + ∠D = 360°
Therefore, the sum of interior angles of a quadrilateral is 360°.
Lines and Angles — 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. What is the measure of an angle that is twice its complement?
- A. 30°
- B. 45°
- C. 60°
- D. 90°
Answer: C. 60° — Let the angle be x. Its complement is 90-x. According to the question, x = 2(90-x). Solving this gives x = 180 - 2x, so 3x = 180, and x = 60°.
MCQ 2. If two angles are supplementary and one angle is 50° greater than the other, what is the measure of the smaller angle?
- A. 65°
- B. 55°
- C. 75°
- D. 85°
Answer: A. 65° — Let the angles be x and x+50. Since they are supplementary, x + (x+50) = 180°. So, 2x + 50 = 180°, 2x = 130°, and x = 65°. The smaller angle is 65°.
MCQ 3. Two adjacent angles form a linear pair. If one angle is 120°, what is the measure of the other angle?
- A. 60°
- B. 30°
- C. 90°
- D. 180°
Answer: A. 60° — Angles forming a linear pair are supplementary, meaning their sum is 180°. If one angle is 120°, the other angle is 180° - 120° = 60°.
MCQ 4. If a transversal intersects two parallel lines, and one of the alternate interior angles is 75°, what is the measure of the other alternate interior angle?
- A. 75°
- B. 105°
- C. 90°
- D. 180°
Answer: A. 75° — When a transversal intersects two parallel lines, the alternate interior angles are equal. Thus, the other alternate interior angle is also 75°.
MCQ 5. If l || m and a transversal t intersects them, and ∠1 = 80°, then what is the measure of ∠5 (corresponding to ∠1)?
- A. 80°
- B. 100°
- C. 10°
- D. 180°
Answer: A. 80° — When a transversal intersects two parallel lines, corresponding angles are equal. Since ∠1 and ∠5 are corresponding angles, ∠5 = ∠1 = 80°.
MCQ 6. If a transversal intersects two parallel lines, and the ratio of two interior angles on the same side of the transversal is 2:7, what are the measures of these angles?
- A. 40° and 140°
- B. 20° and 70°
- C. 80° and 100°
- D. 50° and 130°
Answer: A. 40° and 140° — Interior angles on the same side of a transversal are supplementary. Let the angles be 2x and 7x. So, 2x + 7x = 180°, which means 9x = 180°, and x = 20°. The angles are 40° and 140°.
MCQ 7. In a triangle ABC, if ∠A = 60° and ∠B = 70°, what is the measure of ∠C?
- A. 50°
- B. 60°
- C. 70°
- D. 80°
Answer: A. 50° — The sum of angles in a triangle is 180°. So, ∠C = 180° - (∠A + ∠B) = 180° - (60° + 70°) = 180° - 130° = 50°.
MCQ 8. In triangle PQR, side QR is produced to S. If ∠PRS = 110° and ∠QPR = 60°, what is the measure of ∠PQR?
- A. 50°
- B. 60°
- C. 70°
- D. 80°
Answer: A. 50° — By the exterior angle property of a triangle, the exterior angle ∠PRS is equal to the sum of the interior opposite angles ∠PQR and ∠QPR. So, 110° = ∠PQR + 60°. Therefore, ∠PQR = 110° - 60° = 50°.
MCQ 9. If OM bisects ∠AOB and ∠AOM = 35°, what is the measure of ∠AOB?
- A. 35°
- B. 70°
- C. 105°
- D. 140°
Answer: B. 70° — An angle bisector divides an angle into two equal parts. Since OM bisects ∠AOB, ∠AOM = ∠MOB. Therefore, ∠AOB = ∠AOM + ∠MOB = 35° + 35° = 70°.
MCQ 10. If AB || CD, and a point P lies between AB and CD such that ∠ABP = 30° and ∠CDP = 50°, what is the measure of ∠BPD?
- A. 70°
- B. 80°
- C. 90°
- D. 100°
Answer: B. 80° — Draw a line XY through P parallel to AB and CD. Since AB || XP, ∠BPX = ∠ABP = 30° (alternate interior angles). Since CD || YP, ∠DPY = ∠CDP = 50° (alternate interior angles). Therefore, ∠BPD = ∠BPX + ∠DPY = 30° + 50° = 80°.
MCQ 11. The measure of an angle is 20° less than its complement. Find the angle.
- A. 35°
- B. 45°
- C. 55°
- D. 65°
Answer: A. 35° — Let the angle be x. Its complement is 90-x. According to the question, x = (90-x) - 20. So, x = 70 - x, which gives 2x = 70, and x = 35°.
MCQ 12. If an angle is 70°, what is the measure of its reflex angle?
- A. 110°
- B. 290°
- C. 20°
- D. 180°
Answer: B. 290° — A reflex angle is the angle greater than 180° but less than 360°. To find the reflex angle, subtract the given angle from 360°. So, 360° - 70° = 290°.
MCQ 13. Three angles on a straight line are in the ratio 2:3:5. Find the measure of the largest angle.
- A. 36°
- B. 54°
- C. 90°
- D. 180°
Answer: C. 90° — Angles on a straight line sum up to 180°. Let the angles be 2x, 3x, and 5x. So, 2x + 3x + 5x = 180°. This means 10x = 180°, so x = 18°. The largest angle is 5x = 90°.
MCQ 14. Lines AB and CD intersect at O. If ∠AOC + ∠BOD = 120°, find ∠AOD.
- A. 60°
- B. 120°
- C. 150°
- D. 90°
Answer: B. 120° — Vertically opposite angles are equal, so ∠AOC = ∠BOD. Given ∠AOC + ∠BOD = 120°, it implies 2∠AOC = 120°, so ∠AOC = 60°. Angles ∠AOC and ∠AOD form a linear pair, so ∠AOC + ∠AOD = 180°. Therefore, ∠AOD = 180° - 60° = 120°.
MCQ 15. If l || m, and a transversal intersects them such that one of the corresponding angles is 2x - 10° and the other is x + 30°, find the value of x.
- A. 40°
- B. 50°
- C. 60°
- D. 70°
Answer: A. 40° — When a transversal intersects two parallel lines, corresponding angles are equal. So, 2x - 10° = x + 30°. Solving for x, we get 2x - x = 30° + 10°, which gives x = 40°.
MCQ 16. If two parallel lines are intersected by a transversal, and one of the alternate exterior angles is 115°, what is the measure of its vertically opposite angle?
- A. 65°
- B. 115°
- C. 135°
- D. 90°
Answer: B. 115° — Vertically opposite angles are always equal. If an alternate exterior angle is 115°, its vertically opposite angle will also be 115°.
MCQ 17. In ΔABC, ∠A + ∠B = 100° and ∠B + ∠C = 120°. Find ∠B.
- A. 40°
- B. 60°
- C. 70°
- D. 80°
Answer: A. 40° — We know that ∠A + ∠B + ∠C = 180°. Substituting ∠A + ∠B = 100° gives 100° + ∠C = 180°, so ∠C = 80°. Now substitute ∠C = 80° into ∠B + ∠C = 120° to get ∠B + 80° = 120°, which means ∠B = 40°.
MCQ 18. If (2x+10)° and (3x-20)° are two angles forming a linear pair, find the value of x.
- A. 36°
- B. 38°
- C. 40°
- D. 42°
Answer: B. 38° — Angles forming a linear pair sum to 180°. So, (2x+10) + (3x-20) = 180. This simplifies to 5x - 10 = 180, so 5x = 190, and x = 38°.
MCQ 19. If two parallel lines are intersected by a transversal such that the interior angles on the same side of the transversal are in the ratio 4:5, what is the measure of the smaller angle?
- A. 60°
- B. 80°
- C. 100°
- D. 120°
Answer: B. 80° — Interior angles on the same side of a transversal are supplementary, meaning their sum is 180°. Let the angles be 4x and 5x. So, 4x + 5x = 180°, which means 9x = 180°, and x = 20°. The angles are 4*20 = 80° and 5*20 = 100°. The smaller angle is 80°.
MCQ 20. An exterior angle of a triangle is 105°. If its interior opposite angles are in the ratio 2:5, find the larger of the two interior opposite angles.
- A. 30°
- B. 45°
- C. 75°
- D. 90°
Answer: C. 75° — By the exterior angle property, the exterior angle is equal to the sum of the two interior opposite angles. Let the interior opposite angles be 2x and 5x. So, 2x + 5x = 105°. This means 7x = 105°, and x = 15°. The larger angle is 5x = 75°.
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.