Class 6 Mathematics: Chapter 2 — Lines and Angles (Complete Solutions)
Comprehensive step-by-step solutions for all exercises, in-text questions, "Figure it Out" sections, and Chapter Mastery from Chapter 2.
Figure it Out - 2.1
1. (a) Ramu marked a point on a piece of paper. How many lines can he draw that pass through the point? (b) Venkat marked two points on a piece of paper. How many different lines can he draw that pass through both of the points?
Answer:
- Ramu's case (1 point): Infinitely many (countless) lines can pass through a single point.
- Venkat's case (2 points): Exactly one unique line can pass through two distinct points.
2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one line segment? Which are on two?
Answer:
- Line Segments: LM, MP, PQ, QR (Total = 4 line segments).
- Points on exactly one segment: Points L and R (the two end points of the zigzag chain).
- Points on two segments: Points M, P, and Q (junction vertices connecting consecutive segments).
3. Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?
Answer:
- Rays: Ray TA and Ray TB (or Ray TN, since N lies along ray TB).
- Is T the starting point? Yes, point T is the common starting (initial) point for both rays.
4. Draw a rough figure and write labels appropriately:
- (a) Lines OP and OQ meet at O: Two straight lines intersecting at common point O.
- (b) Line XY and line PQ intersect at point M: An 'X'-shaped intersection labeled at center M.
- (c) Line l contains points E and F but not D: A straight line with dots E and F on it, and dot D placed completely off the line.
- (d) Point P lies on segment AB: A line segment with endpoints A and B, containing dot P anywhere between them.
5. In Fig. 2.6, name: (a) Five points, (b) A line, (c) Four rays, (d) Five line segments.
Answer:
- (a) Five points: O, B, C, D, E.
- (b) A line: Line ED (or Line OE).
- (c) Four rays: Ray OB, Ray OC, Ray OD (or Ray OE), and the upward unnamed ray.
- (d) Five line segments: OE, ED, OD, OB, OC.
6. Here is a ray OA passing through B (Fig. 2.7):
(a) Can you also name it as ray OB? Why?
(b) Can we write ray OA as ray AO? Why or why not?
(a) Can you also name it as ray OB? Why?
(b) Can we write ray OA as ray AO? Why or why not?
Answer:
- (a) Yes: Because it starts at initial point O and extends infinitely in the exact same direction passing through B.
- (b) No: In ray notation, the first letter must always denote the starting (initial) point. Ray AO would mean a ray starting at point A and going through O, which is completely different.
Figure it Out - 2.2
1. Angles in the given pictures (Bicycle & Wooden crate):
- Bicycle: Vertex A with arms along chainstay and seatstay; Vertex B with arms along top tube and seat tube; Vertex C with arms along top tube and down tube; Vertex D at bottom bracket.
- Wooden Crate: The intersecting wooden planks form right angles (90°) at cross joints and corners.
2. Draw and label an angle with arms ST and SR:
Vertex is S; arms are rays ST and SR. Angle is denoted as ∠TSR or ∠RST.
3. Explain why ∠APC cannot be labelled simply as ∠P?
Answer: Because multiple angles share vertex P (such as ∠APQ, ∠QPC, etc.). Using only "∠P" creates ambiguity as to which angle is being referred to. Three letters uniquely specify the arms.
5. Mark 3 non-collinear points A, B, C. (a) How many lines do you get? Name them. (b) How many angles?
Answer:
- Number of lines: 3 lines (Line AB, Line BC, Line CA).
- Number of angles: 3 interior angles (∠ABC, ∠BCA, ∠CAB).
6. Mark 4 points A, B, C, D (no three collinear). (a) How many lines? (b) How many angles?
Answer:
- Number of lines: (4 × 3) / 2 = 6 lines (Line AB, Line BC, Line CD, Line DA, Line AC, Line BD).
- Number of angles: 4 corner points with 3 angles each + central intersection angles ≈ 12 or more angles formed by the intersecting line pairs.
Figure it Out - 2.3
2. In each case, determine which angle is greater and why:
- (a) ∠AOB or ∠XOY: ∠AOB is greater because ray OA requires a larger amount of rotation from the base ray than ray OX.
- (b) ∠AOB or ∠XOB: ∠AOB is greater because ray OA lies further away from OB than ray OX.
- (c) ∠XOB or ∠XOC: ∠XOC is greater since ray OC is opened wider from ray OX than ray OB.
3. Crane mouths: Which crane is making the bigger angle?
Answer: The first crane makes the bigger angle because its bill opens with a greater amount of circular rotation between its jaws.
Special Angles & Figure it Out - 2.4 / 2.5
Figure it Out - 2.4 (Classroom & Grid Questions):
- Q1: Standard rectangular classroom windows contain 4 right angles at the four corners. Other examples: blackboard corners, book edges, doors, tiled floor intersections.
- Q4 (Perpendicular crease):
- (a) We get 4 right angles around the intersection point.
- (b) Folding Method: Fold the paper along the first crease onto itself so that the crease lies exactly on top of itself; the resulting new fold line is perpendicular (90°) to the original crease.
Figure it Out - 2.5 (Angle Meanings & Triangle Patterns):
- Q3: Why "acute" and "obtuse"? "Acute" comes from Latin meaning sharp/pointed (angles < 90° look sharp), while "obtuse" means blunt/dull (angles > 90° look broad and blunt).
- Q4: Acute angles pattern in nested triangles:
- Figure (i) [1 triangle]: 3 acute angles.
- Figure (ii) [1 inverted triangle inside, total 5 triangles]: 5 × 3 = 15 acute angles.
- Figure (iii) [further nested]: Follows the linear formula for nested equilateral subdivisions.
Degree Measures of Circle Fractions & Handmade Protractor
Degree measures for circle divisions (Page 82):
| Parts | Fraction of Turn | Calculation | Degree Measure |
|---|---|---|---|
| 1 part (Full turn) | 1 | 1 × 360° | 360° |
| 2 parts (Half turn) | 1/2 | 360° / 2 | 180° (Straight angle) |
| 3 parts | 1/3 | 360° / 3 | 120° |
| 4 parts (Quarter turn) | 1/4 | 360° / 4 | 90° (Right angle) |
| 5 parts | 1/5 | 360° / 5 | 72° |
| 6 parts | 1/6 | 360° / 6 | 60° |
| 8 parts | 1/8 | 360° / 8 | 45° |
| 9 parts | 1/9 | 360° / 9 | 40° |
| 10 parts | 1/10 | 360° / 10 | 36° |
| 12 parts | 1/12 | 360° / 12 | 30° |
Protractor Readings (Pages 84 & 86):
- Why two sets of numbers? To easily measure angles opening from the right (counter-clockwise using inner scale) as well as from the left (clockwise using outer scale).
- Page 84 Measures: ∠POQ = 30°, ∠POR = 90°, ∠POS = 135°, ∠POT = 160°, ∠POU = 180°.
- ∠TOS: 55° - 20° = 35° (or using inner scale: 160° - 125° = 35°).
- Page 86 Measure (∠AOB): Line aligns at 0° on right, ray passes through 80° ⇒ 80°.
Figure it Out - 2.6
Questions 4, 6, 7 & 9:
- Q4 (Reflex angle measurement): Measure the inner non-reflex angle θ using the protractor, then subtract it from a full turn: Reflex Angle = 360° - θ.
- Q6 (Page 96 Protractor):
- ∠BXE = 180° - 65° = 115°
- ∠CXE = 180° - 95° = 85°
- ∠AXB = 65°
- ∠BXC = 95° - 65° = 30°
- Q7 (Page 96 Rays): ∠PQR ≈ 45°, ∠PQS ≈ 70°, ∠PQT ≈ 140°.
- Q9 (Triangle Angle Sum):
In all triangles (a), (b), and (c), the sum of the three interior angles is always equal to 180°.
Mind the Mistake, Mend the Mistake! (Page 100)
| Angle | Student's Value | Error Identified | Correct Measure |
|---|---|---|---|
| ∠U | 35° | Base ray is on the right (0°), but student read outer scale instead of inner scale. | 145° (Obtuse) |
| ∠V | 80° | Vertex of angle is not placed at the center point of the protractor. | Re-align vertex to center mark |
| ∠W | 70° | Base ray is not aligned with the 0° baseline. | Re-align base arm to 0° line |
| ∠X | 150° | Read obtuse number for an acute angle (used wrong scale). | 30° (Acute) |
| ∠Y | 120° | Base ray is on the left (0°), but student read inner scale instead of outer scale. | 60° (Acute) |
| ∠Z | 85° | Angle does not start at 0°; spans between 20° and 105°. Direct subtraction needed: 105° - 20°. | 85° (Calculated correctly by subtraction) |
Figure it Out - 2.7 (Real-World Angles)
1. Angles in a Clock:
- (a) Why 30° at 1 o'clock? A full circle is 360° divided into 12 hours. Angle per hour = 360° / 12 = 30°. At 1:00, hands are 1 hour apart ⇒ 1 × 30° = 30°.
- (b) Other times:
- At 2 o'clock: 2 × 30° = 60°
- At 4 o'clock: 4 × 30° = 120°
- At 6 o'clock: 6 × 30° = 180° (Straight angle)
2 to 5: Doors, Swings, and Slopes:
- Q2 (Door): Yes, the angle of opening is measured at the door hinge (Vertex), between the closed door frame (Fixed Arm) and the moving door face (Rotating Arm).
- Q3 (Swing): The angle is formed between the vertical resting position (imaginary vertical line) and the inclined rope at the highest point.
- Q4 (Toy Slopes): Yes; the angle of slope is formed between the tilted slab surface (visible arm) and the horizontal ground line (invisible reference arm).
- Q5 (Insect Rotation): The angle describes rotation between the insect's starting orientation line and its new rotated axis.
Figure it Out - 2.9 & 2.10
Figure it Out - 2.9 (Q2: Measure & Classify):
- ∠PTR ≈ 30° ⇒ Acute Angle
- ∠PTQ ≈ 50° ⇒ Acute Angle
- ∠PTW ≈ 105° ⇒ Obtuse Angle
- ∠WTQ ≈ 45° ⇒ Acute Angle
- Outer full revolution turn indicated by circular arrow ⇒ Reflex Angle
Let's Explore (Page 120): ∠TER = 80° on straight line BER:
- ∠BET = 180° - 80° = 100°
- Since ∠SER = 90°, ∠SET = 90° - 80° = 10°
Ashoka Chakra & Acute Angle Puzzle (Page 122):
- Q6 (Ashoka Chakra):
- Angle between two adjacent spokes = 360° / 24 = 15°.
- Largest acute angle formed by spokes must be < 90°. Multiples of 15°: 15°, 30°, 45°, 60°, 75°. Thus, the largest acute angle is 75° (spanning 5 intervals).
- Q7 (Puzzle):
Let the angle be x.
- 4x < 90° ⇒ x < 22.5°
- 5x > 90° ⇒ x > 18°
Answer: The angle measure is any value strictly between 18° and 22.5° (e.g., 19°, 20°, 21°, 22°).
Chapter Mastery Solutions (Pages 122–126)
1. How many lines can be drawn through given two points?
Answer: (A) Only one
2. Find "False" statement:
Answer: (D) A Ray has two end points (A ray has only 1 initial end point and goes infinitely in one direction).
3. Match properties with figures:
(i) Indefinite length in both directions → (b) Line
(ii) Has no size but shows position → (c) Point
(iii) Part of line with two end points → (d) Line Segment
(iv) Indefinite length in one direction → (a) Ray
Answer: (B) (i) → (b), (ii) → (c), (iii) → (d), (iv) → (a)
4. Assertion & Reason:
Assertion: a + b = 180° ⇒ 40° + b = 180° ⇒ b = 140° ≠ 150°. So A is False, R is True.
Answer: (D) A is False, R is true
5. Match the following angles:
(i) Straight Angle → (c) Half of a revolution (180°)
(ii) Right Angle → (d) One-fourth of a revolution (90°)
(iii) Obtuse Angle → (e) Between 1/4 and 1/2 of a revolution (90° to 180°)
(iv) Reflex Angle → (b) More than half a revolution (>180°)
Answer: (A) (i) → (c), (ii) → (d), (iii) → (e), (iv) → (b)
6. Two differences between line and line segment:
| Line | Line Segment |
|---|---|
| Extends indefinitely in both directions (infinite length). | Has a fixed, measurable length. |
| Has no endpoints. | Has exactly two endpoints. |
7. Classify the angles in Fig on page 126:
- (a) ∠QOY: Acute angle (< 90°)
- (b) ∠YOP: Obtuse angle (between 90° and 180°)
- (c) ∠ROX: Right angle (= 90°)
- (d) ∠QOX: Obtuse angle (> 90°)
- (e) ∠POQ: Straight angle (= 180°, straight line through origin)
8. Fraction of a clockwise revolution turned by hour hand:
- (A) 3 to 9: 6 hours ⇒ 6/12 = 1/2 revolution
- (B) 4 to 7: 3 hours ⇒ 3/12 = 1/4 revolution
- (C) 7 to 10: 3 hours ⇒ 3/12 = 1/4 revolution
- (D) 12 to 9: 9 hours ⇒ 9/12 = 3/4 revolution
9. Right angles turned from 3 to 6:
Clock moves 3 hours = 90° ⇒ 1 Right Angle.
10. Rough sketches:
