Point — a location (capital letter). Line segment AB — shortest path, two end points. Line — no end points. Ray — one starting point, goes on in one direction (OA ≠ AO).
Through one point: infinitely many lines; through two points: exactly one line.
Angle — two rays with a common starting point (vertex). Name it with the vertex in the middle: ∠ABC.
Size of an angle = amount of turn. Arm length does NOT change the angle.
Full turn 360° · straight angle 180° · right angle 90° · acute 0°–90° · obtuse 90°–180° · reflex 180°–360°.
Using a protractor: centre on the vertex, one arm on 0°, read the SAME scale.
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๐ Page 14Think · Line segment
Mark two points A and B on paper and join them by different routes. What is the shortest route from A to B?
1We can go from A to B along curved, bent or zig-zag paths.
2Every bent path is longer than the straight path.
3The straight path from A to B (including A and B) is the shortest. It is called the line segment AB (or BA).
The dashed curve is longer; the straight segment AB is the shortest path.
✅ Answer: The straight path — the line segment AB — is the shortest route from A to B.
๐ Page 14Think · Line
Can you draw a complete picture of a line? Why?
1A line goes on without end in BOTH directions.
2Paper (or a board) always has an end, so we can never draw all of it.
3So we draw only a part and put arrowheads at both ends to show that it continues.
✅ Answer: No. A line has no end points — it goes on forever in both directions — so only a part of it can be drawn.
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๐ Page 15Figure it Out · Q1
Rihan marked one point on paper. How many lines can he draw through it? Sheetal marked two points. How many different lines can she draw through both points?
1Through ONE point we can draw lines in every direction — as many as we like.
2So Rihan can draw countless (infinitely many) lines.
3Through TWO points only one straight line can pass. Any other line would bend or miss one point.
4So Sheetal can draw exactly one line.
Through one point: as many lines as we like. Through two points: only one.
✅ Answer: Rihan: infinitely many (countless) lines. Sheetal: exactly one line.
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๐ Page 16Figure it Out · Q2
Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?
Fig. 2.4 (redrawn): a broken path L–M–P–Q–R.
1The path goes L → M → P → Q → R.
2Line segments: LM, MP, PQ and QR.
3L is only on LM and R is only on QR → on exactly one segment.
4M is on LM and MP; P is on MP and PQ; Q is on PQ and QR → on two segments.
✅ Answer: Segments: LM, MP, PQ, QR. On exactly one: L and R. On two: M, P and Q.
๐ Page 16Figure it Out · Q3
Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?
Fig. 2.5 (redrawn).
1One ray starts at T and passes through A → ray TA.
2The other ray starts at T and passes through N and B → ray TN (it can also be called TB).
3Both rays start at T.
✅ Answer: Rays TA and TN (= TB). Yes, T is the starting point of both rays.
⚠️ Common mistake: Writing ray NT or AT. The starting point must be written FIRST.
๐ Page 16Figure it Out · Q4
Draw a rough figure and label it: (a) lines OP and OQ meet at O (b) lines XY and PQ intersect at M (c) line l contains E and F but not D (d) point P lies on AB.
1(a) Draw two lines crossing at O; mark P on one and Q on the other.
2(b) Draw two lines crossing at M; mark X, Y on one line and P, Q on the other.
3(c) Draw a line, name it l, mark E and F on it; mark D away from the line.
4(d) Draw AB and mark P on it between (or beyond) A and B.
(a) lines OP and OQ meet at O · (b) lines XY and PQ cross at M · (c) line l through E and F, D outside · (d) P lies on AB.
✅ Answer: See the four rough figures.
๐ Page 16Figure it Out · Q5
In Fig. 2.6, name: (a) five points (b) a line (c) four rays (d) five line segments.
Fig. 2.6 (redrawn): D, E, O, B lie on one line; one ray goes up from O and one goes through C.
1(a) Points: D, E, O, C, B.
2(b) D, E, O and B lie on one line → line DB (also called line EB, DO, …).
3(c) Rays: OB, OC, OD (same as OE) and ED (it starts at E and goes through D). Others: BD, DB, EB.
4(d) Line segments: DE, EO, OB, OC, DB (others: DO, EB, EO …).
✅ Answer: (a) D, E, O, C, B (b) line DB (c) OB, OC, OD, ED (d) DE, EO, OB, OC, DB — other correct names are possible.
๐ก Tip: The ray going straight up from O has no second point named, so we cannot name it with letters.
๐ Page 16Figure it Out · Q6
Ray OA starts at O and passes through A and B (Fig. 2.7). (a) Can you also name it as OB? Why? (b) Can we write OA as AO? Why or why not?
Fig. 2.7 (redrawn): ray OA also passes through B.
1(a) A ray is named by its starting point and ANY other point on it. B is on the same ray, so OB names the same ray.
2(b) In AO the starting point would be A, and the ray would go from A towards O and beyond — the opposite direction.
3So OA and AO are different rays.
✅ Answer: (a) Yes — OA and OB are the same ray (same starting point O, same direction). (b) No — AO starts at A and goes the other way.
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๐ Page 17Think · Vidya’s book
Vidya opens the cover of her book more and more (Cases 1 to 6). Do you see angles? Which angle is greater — Case 1 or Case 2?
1The spine of the book is the vertex.
2The cover and the pages are the two arms.
3In Case 2 she turns the cover more than in Case 1.
4More turning (rotation) means a bigger angle.
Size of an angle = amount of turn.
✅ Answer: Yes, each case makes an angle. The angle in Case 2 is greater because the cover is turned more.
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๐ Page 18Think · Angles around us
Identify the arms and vertex of the angle in (a) a compass or divider (b) a pair of scissors.
1(a) Compass: the two legs are the arms; the joint (hinge) at the top is the vertex.
2(b) Scissors: the two blades are the arms; the screw where the blades are joined is the vertex.
3Opening them wider (turning one arm more) makes a bigger angle.
Mark four points A, B, C, D so that no three are on one line. Draw all lines through pairs. How many lines? How many angles can you name using A, B, C, D?
1Pairs: AB, AC, AD, BC, BD, CD → 6 lines.
2At each point, three lines meet, giving 3 angles at that point.
3At A: ∠BAC, ∠CAD, ∠BAD. At B: ∠ABC, ∠ABD, ∠CBD.
4At C: ∠ACB, ∠ACD, ∠BCD. At D: ∠ADB, ∠ADC, ∠BDC.
54 points × 3 angles = 12 angles.
4 points (no three on one line) → 6 lines: AB, AC, AD, BC, BD, CD.
✅ Answer: 6 lines and 12 angles.
⚠️ Common mistake: Counting only the 4 corner angles of ABCD. The diagonals AC and BD make more angles at every corner.
๐ Page 21Math Talk · Comparing angles
Here are some angles. Label each one. How will you compare them?
Four angles in different positions. Trace one and place it on another (vertex on vertex, one arm on one arm) to compare.
1Label each angle with three letters, the vertex in the middle (for example ∠ABC).
2The angles point in different directions, so it is hard to compare them just by looking.
3Trace one angle on tracing paper and place it on another so that the vertices match and one arm lies on one arm.
4The angle whose second arm lies outside is bigger. (In this figure the order from smallest is Angle 3, Angle 1, Angle 2, Angle 4.)
✅ Answer: Label them and compare by superimposing (placing one on top of the other). Here: Angle 3 (50°) < Angle 1 (75°) < Angle 2 (105°) < Angle 4 (130°).
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๐ Page 22Think · Equal angles
∠AOB and ∠XOY are placed one on top of the other: OA lies on OX and OB lies on OY. Which angle is greater?
1When superimposed, the vertices match.
2Both arms also lie exactly on top of each other.
3So the same amount of turn is needed in both angles.
When the vertex and both arms match, the angles are equal.
✅ Answer: Neither — the angles are equal.
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๐ Page 23Math Talk
Where else do we use superimposition (placing one thing on another) to compare?
1Comparing the lengths of two pencils by keeping them side by side with ends together.
2Checking if two coins, stamps or tiles are the same size and shape.
3A tailor places one cloth piece on another to cut the same shape.
✅ Answer: To compare lengths, sizes and shapes — pencils, coins, tiles, cloth pieces, leaves, etc.
๐ Page 23Figure it Out · Q1
Fold a rectangular sheet of paper and draw a line on the fold. Name and compare the angles between the fold and the sides of the paper. Make different angles by folding. Which is the largest and smallest angle you made?
1If the fold is parallel to a side, it meets the other sides at right angles — all the angles are equal (90°).
2If the fold is slanting, at each side it makes one smaller (acute) angle and one larger (obtuse) angle.
3The two angles on the same side together make a straight angle (180°).
4The more slanting the fold, the smaller the acute angle and the larger the obtuse angle.
✅ Answer: A straight-across fold gives four right angles; a slanting fold gives acute and obtuse pairs that add up to 180°. (Your largest/smallest angles depend on your folds.)
๐ Page 23Figure it Out · Q2
In each case, which angle is greater and why? (a) ∠AOB or ∠XOY (b) ∠AOB or ∠XOB (c) ∠XOB or ∠XOC
Rays OA, OX, OY and OB (with C) start at O.
1(a) Rays OX and OY lie inside ∠AOB, so ∠XOY is a part of ∠AOB → ∠AOB is greater.
2(b) Ray OX lies inside ∠AOB, so ∠XOB is a part of ∠AOB → ∠AOB is greater.
3(c) B and C are on the same ray from O, so ∠XOB and ∠XOC are the same angle → they are equal.
B and C lie on the same ray from O, so ∠XOB and ∠XOC are the same angle.
✅ Answer: (a) ∠AOB (b) ∠AOB (c) equal.
⚠️ Common mistake: Thinking ∠XOC is bigger because OC is longer. The length of the arms does NOT change the angle.
๐ Page 23Figure it Out · Q3 (Math Talk)
Which angle is greater: ∠XOY or ∠AOB? Give reasons.
Which is greater: ∠XOY or ∠AOB?
1Trace ∠AOB and place it on ∠XOY so that O is on O and OB lies along OY.
∠XOY ≈ 110° (blue) and ∠AOB ≈ 54° (red). ∠XOY is greater.
✅ Answer: ∠XOY is greater.
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๐ Page 25Think · Cranes
Two cranes argue about who opens its mouth wider. Using a transparent circle (centre on the vertex), which crane makes the bigger angle?
1Put the circle on Crane 1’s angle with its centre on the vertex. Mark A and B where the arms cut the circle.
2Move the circle to Crane 2: centre on the vertex and one arm through A.
3Crane 2’s other arm crosses the circle at Y, beyond B.
4So Crane 2’s mouth is turned more.
Place the circle with its centre on the vertex and one arm through A. Crane 2’s upper arm passes OUTSIDE B (at Y), so Crane 2 opens wider.
✅ Answer: Crane 2 makes the bigger angle.
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๐ Page 26Think · Passing through a slit
Which rotating arm can pass through the angle-shaped slit? Does the length of the arms matter?
1If the slit’s angle is bigger than the arms’ angle, the arms do not fit.
2If the slit’s angle is smaller, the arms do not fit either.
3Only arms with exactly the same angle as the slit pass through.
4Longer or shorter arms do not matter (as long as they fit in the slit) — only the angle matters.
✅ Answer: Only the rotating arm whose angle is EQUAL to the slit’s angle passes through. Length does not matter.
๐ Textbook Page 27
๐ Page 27Challenge
A boy cuts the straws of a rotating arm shorter to ‘reduce the angle’. Has the angle become smaller?
1Cutting the straws makes the arms shorter.
2But the amount of turn between the arms is the same.
3The size of an angle does not depend on the length of its arms.
✅ Answer: No. The angle is still the same; only the arms are shorter.
⚠️ Common mistake: Thinking that longer arms make a bigger angle.
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๐ Page 28Math Talk · Straight angle
Is it possible to draw ray OC so that the two angles ∠AOC and ∠COB of the straight angle AOB are equal? How?
1Mark the straight angle AOB on the edge of a paper.
2Fold the paper so that OB lies exactly on OA (the fold passes through O).
3The crease OC divides the straight angle into two angles that lie exactly on top of each other when folded.
4So ∠AOC = ∠COB. Each of them is a right angle.
Fold so that OB lies on OA: the crease OC splits the straight angle into two equal right angles (90° each).
✅ Answer: Yes. Fold OB onto OA; the crease OC makes two equal angles — two right angles.
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๐ Page 29Think
A straight angle is half of a full turn. How much of a full turn is a right angle?
1A straight angle = ½ turn.
2A right angle is half of a straight angle.
3½ of ½ turn = ¼ turn.
✅ Answer: A right angle is a quarter (¼) of a full turn.
๐ Page 29Figure it Out · Q1
How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
1A rectangular window frame has 4 corners → 4 right angles.
2If it has bars, every place where a bar meets the frame or another bar makes more right angles (often 4 at each crossing).
3Other right angles: corners of the blackboard, door, desk top, books, tiles, walls and floor.
✅ Answer: A plain rectangular window has 4 right angles (more if it has bars). Doors, boards, books and tiles also have right angles.
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๐ Page 30Figure it Out · Q2
Join A to other grid points by a straight line to get a straight angle with AB. What are all the different ways?
1A straight angle means the new line continues the line BA straight on, beyond A.
2Left grid (AB is flat): the points to the left of A in the same row → 3 ways.
3Right grid (AB slants down): the points up-left of A on the same slant → 2 ways.
Straight angle: the new line must continue BA straight on, beyond A. Left grid: 3 ways. Right grid: 2 ways.
✅ Answer: Left grid: 3 ways. Right grid: 2 ways (all on the line BA extended beyond A).
๐ Page 30Figure it Out · Q3
Now join A to other grid points to get a right angle with AB. What are all the different ways?
1A right angle divides the straight angle CAB (C on BA extended) into two equal parts.
2Left grid: the line must go straight up or straight down from A: 2 points above + 3 points below = 5 ways.
3Right grid: AB goes 1 right, 1 down each step. The line at right angles goes 1 right, 1 up (or 1 left, 1 down): 2 points up-right + 2 points down-left = 4 ways.
Right angle: the new line must go straight up or down from A, square to AB. Left grid: 5 ways. Right grid: 4 ways.
✅ Answer: Left grid: 5 ways. Right grid: 4 ways.
๐ Textbook Page 31
๐ Page 31Figure it Out · Q4
Make a slanting crease on paper. Then make another crease perpendicular to it. (a) How many right angles do you have? Justify. (b) Describe how you folded.
1Fold any slanting crease and open it.
2Fold the paper again so that the slanting crease lies exactly on top of itself (one part of the crease falls on the other part).
3Press to make the new crease, then open.
4(a) The new crease divides each straight angle on the slanting crease into two equal parts, so all four angles are right angles.
5They are exact right angles because, when folded, the two parts lay exactly on top of each other.
✅ Answer: (a) 4 right angles. (b) Fold the slanting crease onto itself, press, and unfold.
๐ Page 31Figure it Out · Q1 (Classifying)
Identify acute, right, obtuse and straight angles in the previous figures.
1Acute (less than a right angle): Vidya’s book in Cases 1–2, the cranes’ beaks, the slanting window bars, Fig. 2.4 at Q.
2Right: the folds of paper, the window frames, the grid answers with a right angle.
3Obtuse (between right and straight): Vidya’s book in Case 4, ∠XOY on page 23, the open scissors when wide.
4Straight: Vidya’s book opened flat on the table (Case 5), ∠AOB in Fig. 2.11.
✅ Answer: See the examples in the steps.
๐ Page 31Figure it Out · Q2
Make a few acute angles and a few obtuse angles. Draw them in different orientations.
1Acute: draw two arms that open less than an ‘L’ — for example 30°, 50°, 80°.
2Obtuse: arms that open more than an ‘L’ but less than a straight line — for example 100°, 135°, 160°.
3Turn your notebook and draw some with the base arm slanting or pointing left — the angle stays the same type.
Different orientations — the type depends only on the opening.
✅ Answer: See the examples: acute angles (30°, 50°, 80°) and obtuse angles (100°, 135°, 160°).
๐ Textbook Page 32
๐ Page 32Figure it Out · Q3
Acute means ‘sharp’ and obtuse means ‘blunt’. Why do you think these words were chosen?
1An acute angle has a narrow opening, so its corner is pointed — like the tip of a needle or a knife (sharp).
2An obtuse angle opens wide, so its corner is flat and not pointed (blunt).
✅ Answer: Acute angles look sharp and pointed; obtuse angles look blunt and wide.
๐ Page 32Figure it Out · Q4
Find the number of acute angles in each figure (a triangle, then the triangle with its midpoints joined, and so on). What will be the next figure and how many acute angles will it have? Do you see a pattern?
How many acute angles are there in each figure?
1Figure 1: a triangle with 3 angles of 60° → 3 acute angles.
2Figure 2: 3 new points (midpoints). At each one, the lines make 3 angles of 60° on one side → 3 × 3 = 9 more. Total 3 + 9 = 12.
3Figure 3: the middle triangle is divided again → 3 new points × 3 = 9 more. Total 21.
4Next figure: divide the newest middle triangle again → 21 + 9 = 30.
Each new figure adds 3 new points, each with 3 angles of 60° → 9 more acute angles: 3, 12, 21, 30, …
✅ Answer: 3, 12, 21; the next figure has 30 acute angles. The number goes up by 9 each time.
๐ก Tip: The 120° and 180° angles at the midpoints are not acute, so do not count them.
๐ Textbook Page 33
๐ Page 33Think · Measures of special angles
What is the measure of a full turn, a straight angle and a right angle in degrees?
1A full turn is divided into 360 equal parts of 1° → 360°.
2A straight angle is half a turn → 360° ÷ 2 = 180°.
3A right angle is half a straight angle → 180° ÷ 2 = 90°.
✅ Answer: Full turn = 360°, straight angle = 180°, right angle = 90°.
๐ Textbook Page 34
๐ Page 34Think · Dividing a circle
A circle is divided into 1, 2, 3, 4, 5, 6, 8, 9, 10 and 12 equal parts. Write the degree measure of each angle.
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