📐 Class 6 Maths — Chapter 2: Lines and Angles Textbook Answers
Complete Answer Key for all "Figure it Out" Exercises & Chapter Mastery Questions
📘 Figure it Out - 2.1 (Page 44-46)
Q1. Ramu marked one point. He can draw countless (infinite) lines passing through that single point, because a single point does not fix a direction.
Venkat marked two points. Only one single line can be drawn passing through both of those points, because two distinct points fix exactly one direction.
Q2. Line segments in Fig. 2.4: LM, MP, PQ, QR.
Points on exactly one line segment: L and R (they are the outer end points).
Points on two line segments: M, P and Q (each is shared by two segments meeting there).
Q3. Rays in Fig. 2.5: ray TA and ray TB (point N lies on ray TB itself).
Yes, T is the starting point of both rays, since both rays begin at T and go outward in their own directions.
Q4. Rough figures (labelled as described):
Q5. In Fig. 2.6 (taking O as centre with rays going out to B, C, D, E and E-O-C forming one straight line):
- a. Five points: B, C, D, E, O
- b. A line: line ECe (points E, O, C are collinear, forming line EC)
- c. Four rays: OB, OC, OD, OE
- d. Five line segments: OB, OC, OD, OE, EC
Q6. a. Yes, we can also name it ray OB, because point B lies on the ray OA itself (between O and A), so ray OB and ray OA describe the exact same ray.
b. No, we cannot write OA as AO. Writing AO would mean a ray that starts at point A and passes through O — this points in the opposite direction and is a different ray altogether. The starting point must always be named first.
📗 Figure it Out - 2.2 (Page 52-54)
Q1. Bicycle: The frame bars form an angle where they meet near the seat/pedal area — vertex is the joint where the bars meet, arms are the two bars. Wooden panel: the diagonal wooden strips form angles where they cross, vertex at the crossing point, arms are the strips.
Q2. Angle with arms ST and SR:
This angle is named ∠TSR or ∠RST (vertex S written in the middle).
Q3. ∠APC cannot be labelled simply as ∠P because at vertex P there are more than one angle formed (∠APB, ∠BPC, ∠APC). Just writing "∠P" would not tell us which of these angles is meant — it is ambiguous, so we must name it using a point from each arm (∠APC).
Q4. The angles marked in the figure (rays TP, TQ, TR from vertex T) are: ∠PTQ, ∠QTR, ∠PTR.
Q5. Three points A, B, C not on one line:
Lines possible: 3 — line AB, line BC, line CA.
Angles possible: 3 — ∠ABC, ∠BCA, ∠CAB (marked with a small curve at each vertex).
Q6. Four points A, B, C, D with no three collinear:
Lines possible: 6 — AB, AC, AD, BC, BD, CD.
Angles possible: 12 in total — at each of the 4 points, 3 angles are formed by the lines meeting there (e.g. at A: ∠BAC, ∠CAD, ∠BAD), giving 4 × 3 = 12 angles.
📙 Figure it Out - 2.3 (Page 60)
Q1. When you fold a rectangular sheet and draw a line along the crease, the two angles formed between the fold and the sides of the paper can be compared by superimposing (folding along the crease again) — they will match exactly if the fold passes through a corner symmetrically. The largest angle is the widest opening you make, and the smallest is the narrowest fold near one edge.
Q2. Comparing (based on how far each ray has rotated from the base OB/OC):
a. ∠XOY is greater than ∠AOB (ray OX has turned further than ray OA).
b. ∠AOB is greater than ∠XOB is compared by seeing which arm (OA or OX) is farther from OB — the one with the wider rotation is bigger.
c. ∠XOC is greater than ∠XOB, since C is farther along the base ray than B, but the important thing to notice is that OX is the common arm and OC opens wider than OB.
(Tip for students: always compare using the common arm and see which second arm has rotated further.)
Q3. ∠XOY is greater than ∠AOB, because ray OX needs a larger rotation from the base ray to reach OY than ray OA needs to reach OB.
📒 Figure it Out - 2.4 (Page 72-74)
Q1. A rectangular classroom window usually has 4 right angles at its four corners. Yes — right angles can also be seen at door corners, book corners, table corners, and where the wall meets the floor.
Q2. Joining A to different grid points gives a straight angle whenever the two chosen points and A all lie on the same straight grid line (horizontal, vertical, or diagonal) with A exactly in between — there are several such ways depending on which line through A you pick.
Q3. To get a right angle at A, join A to two grid points such that the two lines drawn are perpendicular to each other (for example, one horizontal step-line and one vertical step-line, or two equal diagonal steps at 90° to each other, as shown by the hint figure with the dotted perpendicular line through A).
Q4. a. Right angles do exist in many classroom objects: window corners, door frames, book corners, floor tiles.
b. Fold description: Make any random slanting crease first, then fold the paper again so that one part of the first crease lies exactly on the other part — the new fold line will be perpendicular to the first crease, giving 4 right angles at the point where they cross.
Justification: When a straight angle (180°) is folded exactly in half, both halves become equal, and since 180° ÷ 2 = 90°, each of the 4 angles formed is a right angle.
Classifying Angles (in-text question): Common feature of the first group — all angles are less than a right angle (acute angles). Common feature of the third group — all angles are greater than a right angle but less than a straight angle (obtuse angles).
📕 Figure it Out - 2.5 (Page 76-78)
Q1. In the earlier figures: angles smaller than 90° are acute, angles exactly 90° are right angles (L-shaped), angles between 90° and 180° are obtuse, and angles that form a straight line (180°) are straight angles.
Q2. Sample acute and obtuse angles drawn in different orientations:
Q3. Acute means "sharp" — an acute angle looks thin and sharp, like the tip of a sharp needle. Obtuse means "blunt" — an obtuse angle looks wide and blunt, like a dull edge. The names were chosen because they describe how "sharp" or "wide-open" the angle looks.
Q4. Number of acute angles:
(i) 3 (ii) 6 (iii) 9
Pattern: The count increases by 3 each time (3, 6, 9, 12, ...). The next figure would have 12 acute angles, following the pattern of "multiples of 3."
📓 Figure it Out - 2.6 (Page 90-92)
Think! (in-text): ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI = 22.5°. This is because the straight angle (180°) was folded in half three times in a row (180° → 90° → 45° → 22.5°), dividing it into 8 exactly equal parts.
Q1. Using a protractor on the handmade angle sheets, students should find measures such as 30°, 60°, 90°, 120°, 150° depending on which rays are chosen — the exact reading should be verified with your own protractor since it depends on the printed figure.
Q2. This is a hands-on activity — measure a few real angles in your classroom (like a door's opening, a book's corner) using your own handmade protractor and record the readings.
📔 Figure it Out - 2.7 (Page 104-106)
Q1. a. At 1 o'clock, the angle between the hands is 30° because the clock face is divided into 12 equal parts of 360° ÷ 12 = 30° each, and the hour hand has moved just one part away from 12.
b. At 2 o'clock: 60°. At 4 o'clock: 120°. At 6 o'clock: 180° (a straight angle).
c. Similarly, at 3 o'clock the angle is 90°, at 5 o'clock it is 150°, and so on — each hour adds another 30°.
Q2. Yes, the amount by which a door is opened can be expressed as an angle. The vertex is the hinge of the door, and the arms are the closed position of the door (frame) and its current open position.
Q3. The angle is formed at the point where the swing's ropes are tied to the branch (the vertex), with the arms being the rope's resting (hanging straight down) position and the rope's position when pulled back before swinging.
Q4. Yes, angles can describe the slope of each slab. The vertex is the point where the slab touches the base of the toy. One arm is the slanted slab itself (this is visible), and the other arm is the flat horizontal base line, which is often not physically drawn (invisible/imaginary reference line).
Q5. Yes — the amount of rotation of the insect can be described using the angle between the horizontal reference line (touching both the original and rotated insect) and the tilted body line of the rotated insect. The vertex is the point where the body touches the horizontal line, and the arms are the original horizontal line and the rotated body line.
📖 Figure it Out - 2.8 (Page 112-113)
Q1. In Fig. 2.23 (two lines AB and CD crossing at P, with two more rays PL and PS below), the angles you can list include: ∠APC, ∠CPB, ∠BPD, ∠DPA (formed by the crossing lines) as well as ∠APL, ∠LPS, ∠SPB and other combinations with the extra rays. After guessing, always confirm the exact degree by measuring with a protractor on your own printed copy.
Q2. Draw angles of the given measures using a protractor:
(a. 110° b. 40° c. 75° d. 112° e. 134° — each drawn by placing the protractor's centre on the vertex, aligning one arm to 0°, and marking the given degree.)
Q3. To draw an angle equal to ∠HIJ: place the protractor centre on a new point, draw one base ray, then measure ∠HIJ with the protractor first to find its degree value, and mark the same degree value from your new base ray, then join to form the equal angle.
Steps followed: (1) Measure the given angle ∠HIJ using the protractor. (2) Draw a base ray from a new vertex. (3) Place the protractor's centre on the new vertex, aligned to 0°. (4) Mark the same degree reading. (5) Join the vertex to this mark to complete the equal angle.
📘 Figure it Out - 2.9 (Page 116-118)
Q1. Joining point A to different grid points to form:
a. An acute angle — join A to two nearby grid points so the angle between the two lines is less than 90° (a "narrow" turn).
b. An obtuse angle — join A to two grid points so that the turn between the two lines is wider than 90° but less than 180°.
c. A reflex angle — join A to two grid points, then mark the angle measured on the "outside" (larger) side, going more than 180° around — mark this larger curve with the curved arrow to show it is the reflex angle.
Q2. Using a protractor on the rays from T (to P, R, W and Q):
| Angle | Measure | Type |
|---|---|---|
| a. ∠PTR | ≈ 40° | Acute |
| b. ∠PTQ | ≈ 140° | Obtuse |
| c. ∠PTW | ≈ 250° | Reflex |
| d. ∠WTQ | ≈ 50° | Acute |
(Exact readings should be verified with a protractor on the actual printed figure — the values above show the expected type of each angle.)
Q3. On the grid: PTR, PTQ, PTW rays as points — classify using the same protractor method as Q2 above (each corresponding to acute, right, obtuse, or reflex depending on the exact position marked).
∠BET = 180° − 80° = 100°.
Since ∠SER = 90° (right angle), ∠SET = 90° − 80° = 10°.
📗 Figure it Out - 2.10 (Page 120-121)
Q1. Draw angles with measures: 140°, 82°, 195°, 70°, 35° using a protractor (195° is a reflex angle — draw it by measuring 360° − 195° = 165° on the other side, or extend the protractor scale appropriately).
Q2. Estimate first, then measure with a protractor and classify:
| Angle | Type |
|---|---|
| a | Obtuse |
| b | Obtuse |
| c | Acute |
| d | Acute |
| e | Obtuse |
| f | Reflex |
(Verify exact degree readings with your own protractor on the printed figure.)
Q3. Draw any closed figure (like a kite or arrow shape) that has 3 acute angles, 1 right angle, and 2 obtuse angles — for example, a 6-sided shape where three corners are sharp, one corner is a perfect "L" shape, and two corners are wide.
Q4. Letter 'M' with side angles 40° each and middle angle 60°:
Q5. Letter 'Y' with angles 150°, 60°, and 150°:
Q6. Ashoka Chakra has 24 spokes, so the angle between two adjacent spokes = 360° ÷ 24 = 15°.
The largest acute angle possible between two spokes must be a multiple of 15° that is still less than 90° — that is 75° (15° × 5).
Q7. Puzzle: Let the acute angle measure be x.
For 2x, 3x, 4x to remain acute: 4x < 90° → x < 22.5°
For 5x to be obtuse: 90° < 5x < 180° → 18° < x < 36°
Combining both conditions: 18° < x < 22.5° — so my measure is any angle strictly between 18° and 22.5° (for example, 20°).
🏆 Chapter Mastery (Page 122-126)
1) How many lines can be drawn through given two points? [A]
(A) Only one (B) 2 (C) 4 (D) Countless
2) Find the "False" statement. [D]
(A) Two lines intersect in a point
(B) The line segment has two end points
(C) The ray has one initial point
(D) A Ray has two end points — (this is false; a ray has only ONE end point/starting point and extends endlessly in one direction)
3) Match the following. [B]
| (i) Indefinite length in both directions | → | (b) Line |
| (ii) Has no size but shows position | → | (c) Point |
| (iii) Countless points, part of a line, two end points | → | (d) Line Segment |
| (iv) Indefinite length in one direction | → | (a) Ray |
Answer: (i)→(b), (ii)→(c), (iii)→(d), (iv)→(a) — Option (B)
4) Assertion (A): If angles a and b form a straight angle and a = 40°, then b = 150°.
Reason (R): Sum of a straight-line pair of angles is always 180°. [D]
Working: a + b = 180°, so b = 180° − 40° = 140° (not 150°). So Assertion (A) is False, but Reason (R) is True (this general rule is correct).
Answer: (D) A is False, R is true.
5) Match the following. [A]
| (i) Straight Angle | → | (c) Half of a revolution |
| (ii) Right Angle | → | (d) One-fourth of a revolution |
| (iii) Obtuse Angle | → | (e) Between 1/4 and 1/2 of a revolution |
| (iv) Reflex Angle | → | (b) More than half a revolution |
Answer: (i)→(c), (ii)→(d), (iii)→(e), (iv)→(b) — Option (A)
6) Two differences between line and line segment:
1. A line has no end points and extends endlessly in both directions, while a line segment has two fixed end points.
2. A line has infinite (indefinite) length and cannot be measured, while a line segment has a definite, measurable length.
7) In the given figure (rays R, Q, Y, X, P from centre O, with a right angle marked at ∠ROX):
- (a) ∠QOY — Acute angle
- (b) ∠YOP — Obtuse angle
- (c) ∠ROX — Right angle (90°, as marked)
- (d) ∠QOX — Obtuse angle
- (e) ∠POQ — Straight angle
(Please verify each classification against your printed figure, since the exact position of each ray may vary slightly.)
8) Fraction of clockwise revolution the hour hand turns through:
(A) 3 to 9 → 6 hours → 1/2 revolution
(B) 4 to 7 → 3 hours → 1/4 revolution
(C) 7 to 10 → 3 hours → 1/4 revolution
(D) 12 to 9 (clockwise) → 9 hours → 3/4 revolution
9) Number of right angles turned through by the hour hand going from 3 to 6 (clockwise) = 1 right angle (3 hours = 90°).
10) Rough sketches:
