Class 9 Mathematics Chapter 2 Lines and Angles: 18 Important Questions with Solutions

Lines and Angles (Chapter 2) is compulsory in CBSE Class 9 and typically carries 4–6 marks in the board exam. The 2024-25 rationalized syllabus emphasizes understanding geometric relationships over rote memorization. Questions now test: (1) ability to measure angles using protractors and construct them with compass and straightedge; (2) recognition of angle pairs and their properties in real-world contexts; (3) reasoning about vertically opposite angles, linear pairs, and their supplementary/complementary relationships. Board examiners increasingly ask mixed-topic questions that combine definitions with problem-solving. For example, you may see a diagram showing intersecting lines with algebraic expressions for angles, requiring you to find unknown angles using the vertically opposite angle theorem. Short-answer questions (2 marks) test whether you can explain why angles behave as they do, not just state rules. Long-answer questions (5 marks) often involve construction steps followed by proof or justification. Understanding these 18 important questions prepares you for all such variations and boosts confidence in geometry.

**Question 1:** Two lines intersect at a point. If one angle measures 35°, what is the measure of the vertically opposite angle? (A) 35° (B) 90° (C) 145° (D) 180° **Answer: (A) 35°** **Explanation:** Vertically opposite angles are always equal. When two lines intersect, the angle opposite to 35° is also 35°. **Question 2:** A ray is best defined as: (A) a line segment with two endpoints (B) a portion of a line with one endpoint, extending infinitely in one direction (C) a portion of a line between two points (D) a straight path without direction **Answer: (B)** **Explanation:** A ray has exactly one starting point (endpoint) and extends infinitely in one direction. The ray AB starts at A and passes through B, continuing beyond. **Question 3:** If two adjacent angles form a linear pair, their sum is: (A) 90° (B) 180° (C) 270° (D) 360° **Answer: (B) 180°** **Explanation:** A linear pair consists of two adjacent angles whose non-common arms form a straight line. By the linear pair postulate, their measures sum to 180° (supplementary). **Question 4:** Which of the following is NOT a type of angle? (A) Acute angle (B) Right angle (C) Perpendicular angle (D) Reflex angle **Answer: (C) Perpendicular angle** **Explanation:** An acute angle measures between 0° and 90°, a right angle is exactly 90°, and a reflex angle is between 180° and 360°. "Perpendicular" describes a relationship between lines, not a type of angle. **Question 5:** An angle measures 127°. What is its supplement? (A) 63° (B) 53° (C) 90° (D) 180° **Answer: (B) 53°** **Explanation:** Two angles are supplementary if their sum is 180°. Supplement of 127° = 180° − 127° = 53°.

**Question 1:** Define a line segment and a ray. How do they differ? **Answer:** A **line segment** is a part of a line bounded by two distinct endpoints. It has a definite length and does not extend beyond the endpoints. Example: segment AB with endpoints A and B. A **ray** is a part of a line that starts at one point (endpoint) and extends infinitely in one direction. It has no endpoint in the forward direction. Example: ray AB starts at A and passes through B, extending forever. **Difference:** A line segment has two endpoints and finite length, while a ray has one endpoint and infinite length. **Question 2:** Two adjacent angles measure (3x + 5)° and (2x − 10)°, and they form a linear pair. Find the value of x and the measures of both angles. **Answer:** Since the angles form a linear pair, their sum is 180°. (3x + 5) + (2x − 10) = 180 5x − 5 = 180 5x = 185 x = 37 First angle = 3(37) + 5 = 111 + 5 = 116° Second angle = 2(37) − 10 = 74 − 10 = 64° Verification: 116° + 64° = 180° ✓ **Question 3:** Three lines intersect at a point, forming six angles. If one angle measures 60°, name the angle that measures 60° and explain your reasoning. **Answer:** When three lines intersect at a point, they form six angles around that point. The angle vertically opposite to the 60° angle also measures 60°. Additionally, if the three lines are arranged such that one pair forms the 60° angle, the vertically opposite angle to it will always be 60° by the Vertically Opposite Angles Theorem. The remaining four angles will be supplementary to these, measuring 180° − 60° = 120° each. **Question 4:** An angle is three times its complement. Find the angle and its complement. **Answer:** Let the angle be x°. Its complement is (90 − x)°. Given: x = 3(90 − x) x = 270 − 3x 4x = 270 x = 67.5° Complement = 90 − 67.5 = 22.5° Verification: 67.5 = 3 × 22.5 ✓ **Question 5:** Classify the following angles: 45°, 90°, 135°, 210°, 359°. **Answer:** 45°: Acute angle (0° < 45° < 90°) 90°: Right angle (exactly 90°) 135°: Obtuse angle (90° < 135° < 180°) 210°: Reflex angle (180° < 210° < 360°) 359°: Reflex angle (180° < 359° < 360°)

**Question 1:** Two parallel lines are cut by a transversal. If one interior angle on the same side of the transversal measures (4x + 20)°, and the other measures (3x + 10)°, find the value of x. (Note: Co-interior angles are supplementary.) **Answer:** Co-interior angles (same-side interior angles) formed when a transversal cuts two parallel lines are supplementary. (4x + 20) + (3x + 10) = 180 7x + 30 = 180 7x = 150 x = 150/7 ≈ 21.43° First angle = 4(150/7) + 20 = 600/7 + 140/7 = 740/7 ≈ 105.71° Second angle = 3(150/7) + 10 = 450/7 + 70/7 = 520/7 ≈ 74.29° Verification: 105.71° + 74.29° = 180° ✓ **Question 2:** Using a protractor, explain how you would measure an angle of 120° and then construct the same angle using a compass and straightedge. **Answer:** **Measuring with a protractor:** Place the center of the protractor at the vertex of the angle. Align the baseline of the protractor with one arm of the angle. Read the scale where the other arm intersects. For a 120° angle, the reading should be 120° (use the appropriate scale). **Constructing with compass and straightedge:** 1. Draw a ray AB (base ray). 2. Place the compass point at A and draw an arc of any radius, cutting AB at point P. 3. Using the same radius, place the compass point at P and draw an arc cutting the first arc at Q. 4. Without changing radius, place the compass point at Q and draw another arc cutting the first arc at R. 5. Place the compass point at R and adjust it to meet the arc at Q (now the radius is longer). 6. From P, draw an arc with this new radius, intersecting the initial arc at S. 7. Draw the ray AS. Angle BAS = 120°. **Question 3:** Two lines AB and CD intersect at point O, forming four angles. If angle AOC measures (2a − 5)°, express the measures of the other three angles in terms of a, and verify that their sum is 360°. **Answer:** Let ∠AOC = (2a − 5)°. Vertically opposite angle: ∠BOD = (2a − 5)° (vertically opposite angles are equal). Adjacent angles: ∠AOD and ∠BOC are supplementary to ∠AOC. ∠AOD = 180 − (2a − 5) = 185 − 2a ∠BOC = 180 − (2a − 5) = 185 − 2a **Sum of all four angles:** (2a − 5) + (2a − 5) + (185 − 2a) + (185 − 2a) = 4a − 10 + 370 − 4a = 360° ✓ **Question 4:** Three rays OA, OB, and OC emanate from point O. If ∠AOB = 35°, ∠BOC = 50°, and ∠AOC = 85°, show that the three rays are coplanar. (Note: Coplanar rays lie in the same plane if one ray lies between the other two.) **Answer:** For three rays to be coplanar, the angle between the outermost two must equal the sum of the intermediate angles. Given: ∠AOB = 35°, ∠BOC = 50°, ∠AOC = 85°. Check: ∠AOB + ∠BOC = 35° + 50° = 85° = ∠AOC ✓ This means ray OB lies between rays OA and OC in the plane. Therefore, all three rays lie in the same plane, confirming they are coplanar.

**Question 1:** Four rays emanate from a point O, dividing the plane into four angles. The angles are in the ratio 1 : 2 : 3 : 4. Find the measure of each angle and construct the largest angle using a compass. **Solution:** Let the four angles be x, 2x, 3x, and 4x. Since these angles form a complete angle around point O: x + 2x + 3x + 4x = 360° 10x = 360° x = 36° The four angles are: 36°, 72°, 108°, and 144°. **Construction of 144° angle:** 1. Draw a ray OA (baseline). 2. Place compass at O, draw an arc cutting OA at P. 3. From P, with same radius, draw arcs at points Q₁, Q₂, Q₃, Q₄ along the arc (each representing 36° increments). 4. Mark the 4th division as point Q (representing 4 × 36° = 144°). 5. Join O to Q. Ray OQ makes an angle of 144° with OA. 6. Verify using protractor: angle AOQ = 144°. **Question 2:** Two straight lines PQ and RS intersect at point O. Another line TU passes through O, creating six angles around O. If the angle between PQ and TU is 28°, find all six angles and verify their sum is 360°. **Solution:** When three lines intersect at a point, they form six angles. Given: one angle = 28°. Vertically opposite angle = 28°. Remaining two angles on a straight line with 28°: each = 180° − 28° = 152°. Since three lines intersect, we have three pairs of vertically opposite angles. Angles: 28°, 152°, 28°, 152°, and two more angles. Actually, with three lines, the six angles are: 28°, a°, 152°, b°, c°, d°. For simplicity, if the three lines divide the plane such that adjacent angles alternate: 28°, 152°, 28°, 152° (for two lines), and the third line further divides one of these. Let's assume the third line divides one 152° angle into two equal parts: 76° and 76°. Angles: 28°, 76°, 76°, 28°, 76°, 76°. **Verification:** 28 + 76 + 76 + 28 + 76 + 76 = 360° ✓ **Question 3:** Using compass and straightedge, construct an angle of 75° and then construct its bisector. Verify that each half measures 37.5°. **Solution:** **Step 1: Construct 75°** 75° = 45° + 30°. Construct 60° (equilateral triangle method), then bisect to get 30°. Construct 90° (perpendicular), then bisect to get 45°. Combine: 45° + 30° = 75°. - Draw ray OA. - Construct 90° using perpendicular bisector, mark point B. - Bisect the 90° angle to get 45°; mark point C₁. - Construct 60° using equilateral triangle; mark point D. - Bisect 60° to get 30°; mark point E. - The angle between OC₁ and OE is 45° − 15° = 30°... (alternative: use combination method). **Step 2: Bisect the 75° angle** 1. With center O and any radius, draw an arc cutting both arms of the 75° angle at points P and Q. 2. With centers P and Q, draw arcs of equal radius (greater than half of PQ) intersecting at point R. 3. Draw ray OR. This is the angle bisector. 4. Angle AOR = Angle ROB = 75° ÷ 2 = 37.5°. **Verification:** Measure angles AOR and ROB with protractor; each should be 37.5°.

**Case-Study Question:** A surveyor is measuring angles at an intersection of two roads. Road 1 runs north-south, and Road 2 runs east-west, intersecting at point O. A third road (Bypass) makes an angle of 35° with Road 1 (measured clockwise from north). The surveyor needs to find: (a) The angle the Bypass makes with Road 2. (b) The reflex angle between Road 1 and the Bypass. (c) If a fourth road (Detour) is constructed parallel to the Bypass and intersects Road 2, what angle does the Detour make with Road 2? **Solution with Reasoning:** **Part (a): Angle between Bypass and Road 2** Road 1 and Road 2 are perpendicular (north-south and east-west). The Bypass makes 35° with Road 1. Angle between Bypass and Road 2 = 90° − 35° = 55° (since Road 1 ⊥ Road 2). Alternatively, if measured as the acute angle: The Bypass, Road 1, and Road 2 form a triangle with one 90° angle. The angle between Bypass and Road 2 is 90° − 35° = 55°. **Part (b): Reflex angle between Road 1 and Bypass** The non-reflex angle is 35°. The reflex angle is 360° − 35° = 325°. Or, considering the straight angle: 180° + 35° = 215° (if measured on the opposite side). Most accurately: Reflex angle = 360° − 35° = 325°. **Part (c): Angle between Detour and Road 2** The Detour is parallel to the Bypass. When two parallel lines (Bypass and Detour) are cut by a transversal (Road 2), corresponding angles are equal. Since the Bypass makes 55° with Road 2, the Detour also makes 55° with Road 2 (corresponding angles property). **Key Concepts Tested:** Complement/supplement of angles, properties of parallel lines, angle relationships at intersections, real-world application of geometry.

CBSETUTOR.ai is built specifically for CBSE Class 9 students preparing for board exams. Our AI tutor uses adaptive learning algorithms to track your strengths and weaknesses in Lines and Angles, then generates personalized drill sets daily. Here's how it works: **Daily Practice Drills:** Each session begins with 2–3 MCQs on recently taught topics. The AI tracks your response time and accuracy, flagging conceptual gaps instantly. If you struggle with vertically opposite angles, the tutor prioritizes more questions on that concept the next day. **Video Walkthroughs:** Every question type has an accompanying video explanation narrated in clear, conversational Hindi and English. For example, our tutor shows step-by-step compass construction of 120° angles, matching NCERT standards exactly. **Timed Mock Exams:** Weekly full-chapter tests simulate the board exam format, with a mix of 1-mark, 2-mark, 3-mark, and 5-mark questions. Your score is benchmarked against national averages, and weak areas are identified for targeted review. **Doubt Resolution:** Ask the AI tutor any question—"Why are vertically opposite angles equal?" or "How do I bisect a 65° angle?"—and receive instant, step-by-step replies with diagrams. No waiting for teacher email responses. **Spaced Repetition:** The platform uses spaced repetition schedules. Questions you answered incorrectly resurface after 3 days, then 1 week, then 2 weeks, ensuring long-term retention. **Board-Pattern Alignment:** All 18 questions on this page have been solved by thousands of students on cbsetutor.ai. Feedback data shows students who drill these patterns score 7–9 marks out of 10 in the Lines and Angles section of their board exams. **Start a 3-day free trial at cbsetutor.ai.** No credit card needed. Get instant access to all Chapter 2 questions, video solutions, and personalized drill schedules today.