Chapter 2 exercise answers: Lines and Angles

Open NCERT p. 15Matches NCERT’s answer

1. Rihan has fixed only a position. Nothing stops a line through that point from tilting to any angle, so try turning a ruler about the point — it always keeps passing through it.
2. There is no limit to how many different tilts (directions) a line through one point can have.
3. Sheetal has fixed a position AND a direction, because two points force the line to head from one exactly to the other.
4. Try drawing a second, different line through both of Sheetal's points — it cannot be done without moving one of the points.

AnswerRihan can draw infinitely many lines through his one point; Sheetal can draw exactly 1 line through her two points.

Watch this explained “Rihan's one point and Sheetal's two”, 3:35 into Line and ray: what changes when you refuse to stop · हिंदी में देखें

Open NCERT p. 15Matches NCERT’s answer

1. Fig. 2.4 joins five points, L, M, P, Q, R, one after another in a zig-zag, so consecutive points give the segments.
2. That is 4 segments for 5 points, because the path does not close up into a loop: LM, MP, PQ, QR.
3. L and R sit only at the two open ends of the zig-zag, so each of them lies on just 1 segment.
4. M, P and Q each sit in the middle, where one segment ends and the next begins, so each of them lies on 2 segments.

AnswerSegments: LM, MP, PQ, QR. L and R are on exactly one segment each; M, P and Q are on two segments each.

Watch this explained “Reading a figure: the L-M-P-Q-R zig-zag”, 5:48 into A point fixes a location; a segment is the shortest route between two points · हिंदी में देखें

Open NCERT p. 15Matches NCERT’s answer

1. T has two paths leaving it: one going up through A, and one going sideways through N and then B.
2. The path through A gives ray TA, starting at T.
3. The sideways path gives ray TB (the same ray could equally be called TN, since N is also on it), also starting at T.
4. But N itself is a point on that sideways path, and the part of the path beyond N, through B, is its own ray — ray NB, which starts at N, not at T.

AnswerThe rays are TA, TB and NB. T is the starting point of TA and TB, but not of NB, so T is not the starting point of every ray shown.

Watch this explained “Naming a ray: the starting point goes first”, 6:16 into Line and ray: what changes when you refuse to stop · हिंदी में देखें

Open NCERT p. 16One way to think about it

a OP and OQ meet at O.

1. OP and OQ have a double-headed arrow over them, so both are lines: draw each with an arrowhead at both ends.
2. Make the two lines cross at one point and label that point O.
3. Mark a point P on one line and a point Q on the other line.

In shortTwo lines, OP and OQ, crossing each other at the point O.

b XY and PQ intersect at point M.

1. XY has a single arrowhead over it, so it is a ray: start at X, pass through Y, and put one arrowhead beyond Y.
2. PQ has a double-headed arrow, so it is a line: draw it through two points P and Q with arrowheads at both ends.
3. Draw the line so that it crosses the ray (not behind the starting point X), and label the crossing point M.

In shortRay XY and line PQ crossing at a point M that lies on the ray.

c Line l contains points E and F but not point D.

1. Draw a line with arrowheads at both ends and name it l.
2. Mark two points E and F on the line.
3. Mark a point D away from the line, so that it is not on l.

In shortLine l passing through E and F, with the point D off the line.

d Point P lies on AB.

1. AB is written with no mark over it, so a line segment AB with no arrowheads is enough (a line AB would also be correct).
2. Mark a point P on AB, between A and B.

In shortSegment AB with the point P marked on it.

Watch this explained “Writing it down: the bar, the arrows, the small letter”, 1:41 into Line and ray: what changes when you refuse to stop · हिंदी में देखें

Open NCERT p. 16Checked by computer

a Five points

1. The figure has five labelled points.

AnswerO, B, C, D, E

b A line

1. D, E, O and B all lie on one straight path with an arrowhead at each end, so that path is a line.
2. Any two of its labelled points can be used to name it.

AnswerLine BD (the same line can also be called DE, EO, OB, and so on).

c Four rays

1. From O, the line gives two rays going opposite ways: ray OB and ray OD. (Ray OD can also be called ray OE, because E lies on it. It is one ray with two names, not two rays.)
2. The arrow from O through C gives ray OC.
3. A ray can also start at another point of the line: ray ED starts at E and goes on through D.
4. These four rays are all different, because any two of them differ in their starting point or their direction. The arrow going up from O is a ray too, but it has no labelled point on it to name it by.

AnswerOB, OC, OD, ED (other correct choices include DB and EB)

d Five line segments

1. Any two labelled points on the same drawn path, together with the part between them, make a line segment.
2. On the line: OB, OD, OE and ED. On the ray through C: OC.
3. The answer key at the back of the book lists OC, OB, OE and OD as the four rays, but E lies on the ray from O through D, so OE and OD are one ray with two names; ED is a different fourth ray.

AnswerOB, OC, OD, OE, ED (others such as DB or EB are also correct)

Watch this explained “Reading a figure with all of it at once”, 9:14 into Line and ray: what changes when you refuse to stop · हिंदी में देखें

Open NCERT p. 17Matches NCERT’s answer

a Can you also name it as OB? Why?

1. A ray is named by its starting point first, then any other point on it.
2. Ray OA starts at O, and B also lies on it, on the same side of O as A.
3. So "start at O and pass through B" describes exactly the same ray.

AnswerYes. B is on the same ray, which still starts at O, so OB names the same ray as OA.

b Can we write OA as AO? Why or why not?

1. Writing AO would mean a ray that starts at A instead of at O.
2. That ray runs from A through O and on beyond O, the opposite direction to ray OA, so it is a different ray, not just another name for the same one.

AnswerNo. Ray AO starts at A and goes the other way, so it is a different ray from ray OA.

Watch this explained “One ray, many second letters”, 7:26 into Line and ray: what changes when you refuse to stop · हिंदी में देखें

Open NCERT p. 19Checked by computerAnswers can differ: one example

1. In the bicycle picture, the book has already drawn three rays from one point D, where the frame meets the pedals: ray DA along the lower frame towards the back wheel, ray DB up the seat tube, and ray DC up the frame towards the handlebar.
2. Any two of these rays make an angle with vertex D. Take rays DA and DB.
3. Name the angle with the vertex in the middle: ∠ADB (or ∠BDA).

Answer∠ADB, with vertex D and arms DA and DB. (∠BDC and ∠ADC, also with vertex D, are equally correct, and the lattice, ladder and bridge pictures contain many more angles.)

Watch this explained “The vertex and the arms”, 0:52 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 20One way to think about it

1. Mark a point S — this is the vertex, since it is the letter shared by both arms.
2. Draw one ray starting at S and passing through a point T; draw a second ray starting at S and passing through a point R.
3. Mark the small curve between the two rays at S, and label the angle ∠TSR (or ∠RST) — the vertex letter S stays in the middle.

In shortTwo rays, ST and SR, drawn from the same point S; the angle between them is ∠TSR.

Watch this explained “The vertex and the arms”, 0:52 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 20One way to think about it

1. In the figure, three rays leave the point P: one through A, one through B, and one through C.
2. Any two of those three rays make an angle at P, so there are three different angles with vertex P: ∠APB, ∠BPC and ∠APC.
3. Writing just ∠P does not say which of those three is meant, so it fails to name one particular angle.
4. ∠APB avoids that problem because its outer letters, A and B, pick out exactly the two arms intended.

In shortBecause more than one angle has its vertex at P (∠APB, ∠BPC and ∠APC all do), the single letter ∠P is ambiguous, so the full name ∠APB is needed.

Watch this explained “When one letter will not do”, 3:16 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 20Matches NCERT’s answer

1. The vertex is T, and three rays leave it: TP, TQ and TR.
2. Two curves are marked at T, and both start on ray TR.
3. The smaller curve goes from TR to TQ, so it marks ∠QTR.
4. The larger curve goes from TR all the way round to TP, so it marks ∠PTR.

AnswerThe marked angles are ∠QTR and ∠PTR (also written ∠RTQ and ∠RTP).

Watch this explained “The little curve: which angle you mean”, 2:27 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 20Matches NCERT’s answer

1. Each pair of points gives one line: A and B, B and C, C and A. The three points are not on one line, so these are 3 different lines: AB, BC and CA.
2. An angle named with these letters has one point as its vertex (the middle letter), and the other two points lie on its arms.
3. At A, rays AB and AC make ∠BAC. At B, rays BA and BC make ∠ABC. At C, rays CA and CB make ∠ACB.
4. There is no other way to choose a vertex and two different arms from these points, so there are 3 angles. Mark each one with a small curve at its vertex.

Answer3 lines: AB, BC, CA. 3 angles: ∠BAC, ∠ABC, ∠ACB.

Watch this explained “Counting angles: three points, then four”, 10:12 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 21Matches NCERT’s answer

1. Each pair of the four points gives one line, and because no three points are on one line, all of these are different: AB, AC, AD, BC, BD, CD. That is 6 lines.
2. At A there are three rays: AB, AC and AD. Any two of them make an angle, and there are 3 ways to choose two: ∠BAC, ∠BAD, ∠CAD.
3. The same happens at each of the other points. At B: ∠ABC, ∠ABD, ∠CBD. At C: ∠ACB, ∠ACD, ∠BCD. At D: ∠ADB, ∠ADC, ∠BDC.
4. That is 3 angles at each of the 4 points, so 4 × 3 = 12 angles. Mark each one with a small curve at its vertex.

Answer6 lines: AB, AC, AD, BC, BD, CD. 12 angles: ∠BAC, ∠BAD, ∠CAD, ∠ABC, ∠ABD, ∠CBD, ∠ACB, ∠ACD, ∠BCD, ∠ADB, ∠ADC, ∠BDC.

Watch this explained “Counting angles: three points, then four”, 10:12 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 23One way to think about it

1. Fold the sheet so that the crease runs at a slant from the top side to the bottom side. Open it and draw a line along the crease.
2. Name the corners A and B on the top side (A on the left) and D and C on the bottom side (D on the left). Name the points where the crease meets the top and bottom sides P and Q.
3. The fold makes four angles with the sides: ∠APQ and ∠BPQ at P, and ∠DQP and ∠CQP at Q. The two angles at P together make the straight side AB, and the two at Q together make the straight side DC.
4. Compare them by superimposition: trace one angle and place it on another, vertex on vertex and one arm on one arm. ∠APQ matches ∠CQP exactly, and ∠BPQ matches ∠DQP. Unless the crease goes straight across, one of these pairs is smaller than the other.
5. Fold more sheets at different slants and compare in the same way. The more slanted the crease, the smaller its two small angles and the larger its two large angles. A crease that goes straight across makes all four angles equal.

In shortThe crease makes four angles with the sides: ∠APQ = ∠CQP and ∠BPQ = ∠DQP, and one pair is smaller than the other unless the crease goes straight across. The largest and smallest angles depend on how you fold: a more slanted crease gives a smaller smallest angle and a larger largest angle, so different students will have different answers.

Watch this explained “Superimposition: vertices first, then an arm”, 1:46 into Comparing two angles without measuring either of them · हिंदी में देखें

Open NCERT p. 23Matches NCERT’s answer

a ∠AOB or ∠XOY

1. In the figure, both ray OX and ray OY lie strictly between ray OB and ray OA.
2. So the whole of ∠XOY sits inside ∠AOB, which is the larger span from OB all the way to OA.

Answer∠AOB is greater, because both arms of ∠XOY lie inside ∠AOB.

b ∠AOB or ∠XOB

1. Ray OX lies strictly between ray OB and ray OA.
2. So ∠XOB is only the part of ∠AOB between OB and OX.

Answer∠AOB is greater, because ∠XOB is only part of it.

c ∠XOB or ∠XOC

1. B and C are two different points marked on the very same ray from O.
2. The second arm of ∠XOB and of ∠XOC is therefore the same ray, so nothing about the angle has changed — only which point on that ray was used to name it.

Answer∠XOB and ∠XOC are equal — they are the same angle.

Watch this explained “The arm-length trap: ∠XOB against ∠XOC”, 5:34 into Comparing two angles without measuring either of them · हिंदी में देखें

Open NCERT p. 23Checked by computer

1. Both angles have their vertex at O. Going round from OB, the rays come in the order OB, OY, OA, OX.
2. So ∠AOB is made of two parts, ∠AOY and ∠YOB, and ∠XOY is made of two parts, ∠AOY and ∠XOA.
3. The part ∠AOY belongs to both angles, so only the other parts need comparing: ∠YOB and ∠XOA.
4. ∠YOB is a thin angle and ∠XOA is a wide one. If you trace ∠YOB and superimpose it on ∠XOA (vertex on O, one arm along OA), it fits well inside.
5. So ∠XOA is greater than ∠YOB, and adding the same part ∠AOY to both keeps that order.
6. The answer key at the back of the book says you cannot tell by looking and must superimpose or measure; doing that on the printed figure gives ∠XOY about 111° and ∠AOB about 55°, so ∠XOY is greater.

Answer∠XOY is greater. Both angles contain ∠AOY, and the rest of ∠XOY (that is, ∠XOA) is bigger than the rest of ∠AOB (that is, ∠YOB).

Watch this explained “Reading the answer off the overlap”, 2:48 into Comparing two angles without measuring either of them · हिंदी में देखें

Open NCERT p. 29One way to think about it

1. Each window pane is usually a rectangle, and a rectangle's four corners are all right angles, so count 4 right angles per pane.
2. Multiply by the number of panes in a window, and again by the number of windows, for the windows' total.
3. Look around for other rectangular things — the door frame, the blackboard, floor tiles, the corners of a desk — each corner of these is a right angle too.

In shortEvery rectangular window pane has 4 right angles, so the total depends on how many panes and windows the classroom has; right angles also show up at the corners of the door, the blackboard, desks, and floor tiles.

Watch this explained “A right angle is a quarter turn”, 7:08 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 30Checked by computerAnswers can differ: one example

1. Number the dots in each grid by column (1 to 7 from the left) and row (1 to 6 from the top). In the left grid A is at (4, 3) and B at (7, 3). In the right grid A is at (3, 3) and B at (5, 5).
2. A straight angle at A with arm AB needs the other arm to point exactly opposite to AB. So the new dot must lie on the line BA, continued beyond A.
3. Left grid: AB goes 3 steps to the right along a row, so go left from A along the same row. The dots there are (3, 3), (2, 3) and (1, 3).
4. Right grid: AB goes 2 steps right and 2 steps down, so go diagonally up and to the left from A. The dots there are (2, 2) and (1, 1).

AnswerLeft grid: join A to any of the 3 dots to its left in the same row, (3, 3), (2, 3) or (1, 3). Right grid: join A to either of the 2 dots on the diagonal up and to the left, (2, 2) or (1, 1). In each grid these dots all lie on one line through A, so they all give the same straight angle with AB.

Watch this explained “Straight and right angles on a dot grid”, 8:56 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 30Checked by computerAnswers can differ: one example

1. Use the same numbering (column from the left, row from the top). Left grid: A (4, 3) and B (7, 3). Right grid: A (3, 3) and B (5, 5).
2. A right angle at A is half of the straight angle made by continuing BA past A, so the new arm must be perpendicular to AB.
3. Left grid: AB runs along a row, so the perpendicular runs along A's column. The dots are (4, 1) and (4, 2) above A, and (4, 4), (4, 5) and (4, 6) below A.
4. Right grid: AB goes 2 right and 2 down. Swap the steps and change one sign to get the perpendicular: 1 right and 1 up, or 1 left and 1 down. The dots are (4, 2) and (5, 1), and (2, 4) and (1, 5).

AnswerLeft grid: join A to (4, 1), (4, 2), (4, 4), (4, 5) or (4, 6), straight up or down from A, which is 5 ways. Right grid: join A to (4, 2), (5, 1), (2, 4) or (1, 5), along the diagonal that crosses AB, which is 4 ways. Each of these makes a right angle with AB.

Watch this explained “Straight and right angles on a dot grid”, 8:56 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 31Matches NCERT’s answer

a How many right angles do you have now?

1. The second crease is made by folding the paper so that the slanting crease falls exactly on itself, one part of it on the other part.
2. The new crease crosses the slanting crease at a point, and 4 angles are formed around that point.
3. On each side of the new crease, the slanting crease makes a straight angle, and the new crease splits it into two angles. The fold laid one of these angles exactly on the other, vertex on vertex and arm on arm, so they are equal. Each is therefore half of a straight angle, which is a right angle.
4. The same fold also laid the two angles on the other side of the slanting crease on top of each other, so those are right angles too.

Answer4 right angles. The fold laid one part of the slanting crease exactly on the other, so the two angles on each side of the new crease cover each other exactly, and two equal halves of a straight angle are right angles.

b

1. Fold the paper anywhere at a slant and press it flat. Open it out: this is the slanting crease.
2. Fold the paper again so that the slanting crease lands exactly on top of itself, one part of the crease on the other, and press flat.
3. Open the paper. The new crease is perpendicular to the slanting crease.

AnswerMake one slanting fold. Then fold again so that the first crease lies exactly on top of itself. The second crease is perpendicular to the first.

Watch this explained “Perpendicular — and why it doesn't mean vertical”, 7:56 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 31One way to think about it

1. Compare each angle with a right angle. Smaller than a right angle is acute. Exactly a right angle is a right angle. Bigger than a right angle but smaller than a straight angle is obtuse. Arms in one straight line make a straight angle.
2. Fig. 2.11: ∠AOB, with A, O and B on one line, is a straight angle.
3. In the next figure (page 28), where ray OC divides the straight angle AOB: ∠COB is acute and ∠AOC is obtuse.
4. In the paper-folding figure on page 28, the crease through O divides the straight angle AOB into two right angles.
5. In the dot-grid questions just before this one, joining A to a dot exactly opposite B gives a straight angle, and joining A to a dot in the perpendicular direction gives a right angle.
6. In the picture of three groups of angles, the first group are acute, the second are right angles and the third are obtuse.

In shortFor example: ∠AOB in Fig. 2.11 is a straight angle; in the figure after it, ∠COB is acute and ∠AOC is obtuse; the crease in the paper-folding figure makes two right angles. Many other angles in the chapter's figures can be sorted the same way, so answers will differ.

Watch this explained “Acute, right, obtuse — three groups, no numbers”, 10:21 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 31Checked by computerAnswers can differ: one example

1. For an acute angle, draw two rays from a point turned less than a quarter turn apart, for example about 40 degrees.
2. For an obtuse angle, draw two rays turned more than a quarter turn but less than a half turn apart, for example about 120 degrees.
3. Draw each one again tilted at a different orientation on the page, since an angle's size does not depend on which way it is facing.

AnswerAn example acute angle of about 40° and an example obtuse angle of about 120°, each drawn facing a few different ways.

Watch this explained “Acute, right, obtuse — three groups, no numbers”, 10:21 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 32One way to think about it

1. An angle smaller than a right angle comes to a narrow, pointed corner, which looks like a sharp point — matching the ordinary meaning of 'acute'.
2. An angle bigger than a right angle opens out into a wide, spread corner, which looks blunt rather than pointed — matching the ordinary meaning of 'obtuse'.

In shortThey are not arbitrary labels: a small (acute) angle really does look sharp, like a pointed corner, and a large (obtuse) angle really does look blunt, like a wide, flattened corner, so the everyday words describe exactly how each type of angle looks.

Watch this explained “Acute, right, obtuse — three groups, no numbers”, 10:21 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 32Checked by computer

1. Every small triangle in these figures has the same shape as the big one, so each of its corners is an acute angle. Angles made by two arms along one straight line, or wider than a corner of the big triangle, are not acute.
2. First figure: one triangle, with 1 acute angle at each corner, so 3.
3. Second figure: the 3 corners of the big triangle still have 1 acute angle each, which is 3. Each of the 3 midpoints where the inner triangle touches a side has 4 rays leaving it: two along the side and two along the inner triangle. Between neighbouring rays there are 3 acute angles there (for example, at the midpoint D of side AB with inner corners E and F: ∠ADF, ∠FDE and ∠EDB). That is 3 × 3 = 9 more, so 3 + 9 = 12.
4. Third figure: the small triangle drawn inside the middle triangle adds 3 new points, each again with 4 rays and 3 acute angles, so 9 more, and nothing changes at the old points. That gives 12 + 9 = 21.
5. Next figure: join the midpoints of the newest small triangle. This adds 3 more points and 9 more acute angles, so 21 + 9 = 30.
6. Pattern: 3, 12, 21, 30, … goes up by 9 each time, so the nth figure has 3 + 9 × (n − 1) = 9n − 6 acute angles.

AnswerThe figures have 3, 12 and 21 acute angles, and the next figure has 30. The numbers go up by 9 each time, so the nth figure has 9n − 6.

Watch this explained “Acute, right, obtuse — three groups, no numbers”, 10:21 into Straight and right angles as the landmarks of a full turn · हिंदी में देखें

Open NCERT p. 35Matches NCERT’s answer

a ∠KAL

1. The vertex A sits on the centre of the protractor, so count the 1-degree units between arms AK and AL.
2. Counting gives 30 units, as the book shows.

Answer∠KAL = 30°

b ∠WAL

1. Count along the scale from the bottom end of the protractor's straight edge, using the long marks (every 10) and the medium marks (every 5): ray AT crosses at 10, ray AW at 50, ray AL at 100 and ray AK at 130.
2. ∠WAL is the gap between AW and AL: 100 − 50 = 50.

Answer∠WAL = 50°

c ∠TAK

1. From the same readings, ray AT is at 10 and ray AK is at 130.
2. ∠TAK is the gap between them: 130 − 10 = 120.

Answer∠TAK = 120°

Watch this explained “Reading by counting, one unit at a time”, 1:05 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 40Checked by computer

1. Put the centre of the protractor on the vertex H and line up one arm with the 0° line. Read the number where the other arm crosses the same scale.
2. First diagram: ∠IHJ is less than a right angle. The arm HI crosses the scale at about 47°.
3. Second diagram: ∠GHK is the narrow angle at H between the arms going down to G and to K. It reads about 23°.
4. Third diagram: ∠IHJ opens wider than a right angle. It reads about 108°.
5. These are the values in NCERT's answer key. On the printed page the three angles measure about 47°, 23½° and 109°, so a reading within a degree of these is also correct. A protractor cannot be read more finely than that.

Answerfirst diagram ∠IHJ = 47°, second diagram ∠GHK = 23°, third diagram ∠IHJ = 108° (NCERT's answer key; on the printed page they measure about 47°, 23½° and 109°, so a reading within a degree is fine).

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 40One way to think about it

1. Pick a real corner in your classroom, for example the corner of a book, a window, or a door standing open.
2. Place the protractor's centre exactly on the corner (the vertex of that angle).
3. Line up the zero mark of the protractor with one edge of the corner.
4. Read off the degree where the other edge crosses the scale — that is the angle's measure.
5. Repeat for a few different corners and compare: some will be near 90°, some smaller, some larger.

In shortDifferent corners in the room give different readings — a book's corner is close to 90°, a partly opened door gives a smaller or larger angle depending on how far it is pushed.

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 41Checked by computerReads two ways: both answers shown

1. With a standard protractor, centre on H: the first angle, ∠IHJ, measures 42°.
2. The second angle, ∠IHJ, is marked on its opening between HI and HJ (not the reflex side). It measures 116°.
3. Your paper protractor was folded in half three times, so its creases are only at 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°.
4. 42° is not on a crease: it lies between 22.5° and 45°. 116° is not on a crease either: it lies between 112.5° and 135°.
5. The last sentence of the question can be read two ways.
6. Read as "can the paper protractor give these measures?" (NCERT's answer key uses this reading): No. It can only tell you which two creases the arm falls between.
7. Read as "can it be used at all?": Yes, roughly. Both angles are less than 180°, so they fit on the paper protractor, and it shows that 42° is a little less than 45° and 116° is a little more than 112.5°.

AnswerFirst angle ∠IHJ = 42°, second angle ∠IHJ = 116°. Read as "can the paper protractor give these measures?" (NCERT's reading): no, neither angle falls on one of its creases. Read as "can it be used at all?": yes, but only to say roughly how big each angle is.

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 41Checked by computer

1. The curve is drawn round the outside of the vertex, so the marked angle is the big one, more than 180°. A protractor only goes up to 180°, so it cannot measure this angle in one go.
2. Instead, measure the unmarked angle between the two arms. It is about 100°.
3. The marked angle and the unmarked angle together make one full turn, 360°.
4. So the marked angle = 360° − 100° = 260°.
5. Another way: extend one arm backwards through the vertex. That makes a straight angle, 180°. The part left over between the extended arm and the other arm is 180° − 100° = 80°. So the marked angle = 180° + 80° = 260°.
6. 260° is the value in NCERT's answer key. On the printed page the unmarked angle measures about 101½°, so measuring the page itself gives about 258° or 259°; that is also a correct reading.

AnswerMeasure the unmarked angle, 100°, and take it away from a full turn: marked angle = 360° − 100° = 260° (NCERT's answer key; the printed figure gives about 258°–259°).

Watch this explained “Bigger than the scale: subtract from 360°”, 10:11 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 41Checked by computer

1. For each angle, put the centre of the protractor on the vertex and line up one arm with 0°. Read the other arm on the same scale (the one that starts at 0° on your arm).
2. (a) is less than a right angle: 80°. (b) is more than a right angle: 120°.
3. (c) is less than a right angle: 60°. (d) is more than a right angle: 130°.
4. (e) is more than a right angle: 130°. (f) is less than a right angle: 60°.
5. These are the values in NCERT's answer key. On the printed page the angles measure about 79°, 121°, 59°, 129°, 128° and 61°, so a reading within 2° of the key is also correct.

Answer(a) 80° (b) 120° (c) 60° (d) 130° (e) 130° (f) 60° (NCERT's answer key; on the printed page they measure about 79°, 121°, 59°, 129°, 128° and 61°).

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 42Matches NCERT’s answer

1. A, X and E lie on one straight line, and the protractor drawn in the figure is centred at X.
2. Reading where each ray crosses the printed scale: ∠AXB = 65°, ∠BXC = 30°, ∠CXE = 85°.
3. ∠BXE is the two middle pieces together: ∠BXC + ∠CXE = 30° + 85° = 115°.
4. Check: ∠AXB + ∠BXE should fill the whole straight line AXE — 65° + 115° = 180°, which matches.

Answer∠BXE = 115°, ∠CXE = 85°, ∠AXB = 65°, ∠BXC = 30°.

Watch this explained “Read both arms and subtract”, 4:23 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 42Checked by computer

1. Put the centre of the protractor on Q and line up the arm QP with 0°.
2. Read the other arms on the same scale: QR crosses at 45°, so ∠PQR = 45°.
3. QS crosses at 100°, so ∠PQS = 100°.
4. QT crosses at 150°, so ∠PQT = 150°.
5. These are the values in NCERT's answer key. On the printed page ∠PQT measures about 151°, so a reading of 150° or 151° is correct.

Answer∠PQR = 45°, ∠PQS = 100°, ∠PQT = 150° (NCERT's answer key; on the printed page ∠PQT measures about 151°, so 150° or 151° is fine).

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 43One way to think about it

1. Fold the square paper as shown in the 8 pictures, then unfold it fully and flatten it.
2. Draw over every crease with a pencil and ruler so each becomes a straight line.
3. Wherever two lines (or a line and an edge of the paper) meet, put the protractor's centre on that point, line the zero line up with one line and read where the other crosses.
4. Write each measure next to its angle. The first fold is along a diagonal of the square, so that crease makes 45° with the sides of the square — a good check on your measuring.

In shortThere is no single list of answers: it depends on your folds. You will find, for example, 45° where the diagonal crease meets a side of the square; measure the other angles the same way.

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 43Checked by computer

(a)

1. Measure each corner with the protractor: about 45° at A, 65° at B, 70° at C.
2. Add: 45° + 65° + 70° = 180°.

Answer45°, 65°, 70°; sum 180°.

(b)

1. Measure each corner: about 54° at A, 63° at B, 63° at C.
2. Add: 54° + 63° + 63° = 180°.

Answer54°, 63°, 63°; sum 180°.

(c)

1. Measure each corner: about 31° at A, 52° at B, 97° at C (the angle at C is a little more than a right angle).
2. Add: 31° + 52° + 97° = 180°.

Answer31°, 52°, 97°; sum 180°.

Conjecture

1. All three triangles, of very different shapes, give the same total.
2. Your own measurements may be off by a degree or two, so a total of 179° or 181° still points to the same thing.

AnswerThe three angles of any triangle add up to 180°.

Watch this explained “Making the subtraction vanish”, 6:16 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 45Checked by computerAnswers can differ: one example

(a) the angle between the hands is 30°. Why?

1. A clock face is one full turn, 360°, cut into 12 equal hour-divisions.
2. So one hour-division is 360° ÷ 12 = 30°.
3. At 1 o'clock the minute hand points at 12 and the hour hand has moved 1 division, so the angle between them is 1 × 30° = 30°.

Answer30° per hour-division, which is why the 1 o'clock angle is 30°.

(b) What will be the angle at 2 o'clock?

1. At 2 o'clock the hour hand has moved 2 divisions from 12: 2 × 30° = 60°.
2. At 4 o'clock it has moved 4 divisions: 4 × 30° = 120°.
3. At 6 o'clock it has moved 6 divisions: 6 × 30° = 180° — the hands point in exactly opposite directions.

Answer60° at 2 o'clock, 120° at 4 o'clock, 180° at 6 o'clock.

(c) Explore other angles made by the hands

1. Pick any hour H from 1 to 12 and multiply by 30° to get the hour hand's turn from 12.
2. If that is more than 180°, the hands' actual angle is 360° minus it, since the angle between two hands is never taken bigger than a straight angle.
3. For 5 o'clock: 5 × 30° = 150°, which is already under 180°, so the hands are 150° apart.

AnswerAt 5 o'clock, the hands make an angle of 150°.

Watch this explained “Angles you have to find first — clock, door, ramp, swing”, 12:36 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 45Matches NCERT’s answer

1. Look at the door from above, as in the picture. The door turns about its hinge, and the hinge stays where it is.
2. So the hinge is the vertex of the angle.
3. One arm is the door in its open position. The other arm is the line of the wall or doorway — where the door lies when it is shut. This arm is not drawn; you imagine it.
4. The wider the door is opened, the bigger the angle between these two arms.

AnswerYes. The vertex is the hinge; the arms are the open door and the line of the wall or doorway where the shut door would lie.

Watch this explained “Angles you have to find first — clock, door, ramp, swing”, 12:36 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 45Checked by computer

1. The rope is tied to the branch at a fixed point, and swings away from its resting, straight-down position.
2. That tie-point on the branch is the vertex.
3. One arm is the rope's resting position (hanging straight down); the other arm is the rope's position when Vidya starts swinging.

AnswerYes — the vertex is where the rope meets the branch, and the two arms are the rope at rest and the rope pulled back to start the swing.

Watch this explained “Angles you have to find first — clock, door, ramp, swing”, 12:36 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 46Checked by computer

1. A slanting slab makes an angle with the horizontal ground.
2. This angle determines the steepness (slope) of the slab.
3. A greater angle means a steeper slope.
4. One arm is the surface of the slab (visible).
5. The other arm is the horizontal reference (not visible but implied).
6. The angle between them describes how much the slab is slanted.

AnswerYes, slopes can be described using angles. One arm is the visible slab surface, and the other arm is the invisible horizontal reference.

Watch this explained “Angles you have to find first — clock, door, ramp, swing”, 12:36 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 46Checked by computer

1. In each pair, the insect has been turned about the point where it touches the horizontal line; that point does not move.
2. So that point is the vertex of the angle.
3. One arm is the horizontal line the insect rests on before it turns. The other arm is the line along the insect's base after it turns.
4. The angle between these arms tells how much the insect has turned. Here the base goes from lying flat to standing upright — about a right angle, a quarter turn.

AnswerYes. The vertex is the point where the insect touches the line; the arms are the horizontal line (its base before turning) and the line along its base after turning. The size of the angle is the amount of rotation.

Watch this explained “Initial position, final position, and the sweep”, 7:07 into An angle is an amount of turn, not a pair of drawn arms · हिंदी में देखें

Open NCERT p. 49Checked by computerReads two ways: both answers shown

1. The figure has two long lines (the top one through A and B, the bottom one through C and D), the line segment AC joining them, and two slanting lines. One slanting line crosses the top line at P and the bottom line at L; the other crosses the top line at R and the bottom line at S.
2. Angles are made wherever these meet: at A, C, P, R, L and S. The question does not say exactly which angles to count, so here are the ways of reading it.
3. Reading 1 (NCERT's answer key uses this one): the angles between the two long lines. There are two at each of the six points. At A: ∠CAP and the angle between AC and the top line to the left. At C: ∠ACD and the angle between CA and the bottom line to the left. At P: ∠APL and ∠RPL. At R: ∠PRS and ∠BRS. At L: ∠CLP and ∠SLP. At S: ∠LSR and the angle between SR and the bottom line to the right. That makes 12 angles.
4. NCERT's key lists ∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS and ∠CLP, then says "Try more!". ∠DLP and ∠CLP are the same angle, because D and C both lie on the bottom line to the left of L.
5. Reading 2: every angle at the six points. The slanting lines carry on above the top line and below the bottom line, so P, R, L and S each have two more angles outside the band. AC stops at A and C, so those have only two each. 4 + 4 + 4 + 4 + 2 + 2 = 20 angles.
6. Reading 3: the slanting lines have arrows, so they are lines, not line segments, and go on for ever. Extended upwards they meet above the figure and make 4 more angles there: 20 + 4 = 24 angles.
7. Now measure, with the centre of the protractor on each point. At A: ∠CAP = 107° and the angle on the left = 73°. At C: ∠ACD = 73° and the angle on the left = 107°.
8. At P: ∠APL = 82° and ∠RPL = 98°. At L: ∠CLP = 98° and ∠SLP = 82°. The extra angles above P and below L are 98° and 82° again.
9. At R: ∠PRS = 102° and ∠BRS = 78°. At S: ∠LSR = 78° and the angle on the right = 102°. The extra angles above R and below S are 78° and 102° again.
10. Where the slanting lines meet if extended: about 20° and 160°.
11. Check: the two angles side by side on a straight line always add to 180°: 107° + 73°, 82° + 98°, 102° + 78° and 20° + 160°. Put your guesses beside these measures in your table to see how close you were.

AnswerRead as the angles between the two long lines (NCERT's reading): 12 angles. ∠CAP = 107°, the angle at A on the left = 73°, ∠ACD = 73°, the angle at C on the left = 107°, ∠APL = 82°, ∠RPL = 98°, ∠CLP (= ∠DLP) = 98°, ∠SLP = 82°, ∠PRS = 102°, ∠BRS = 78°, ∠LSR = 78°, the angle at S on the right = 102°.
Read as every angle at the six points: 20 angles (the 8 extra ones, above P and R and below L and S, are 98°, 82°, 78° and 102° again).
Read as also counting where the slanting lines meet if extended: 24 angles (the 4 new ones are about 20° and 160°).

Watch this explained “The guessing games, and how they are scored”, 11:20 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 50One way to think about it

1. Draw a ray OA with a ruler; O will be the vertex.
2. Put the protractor's centre exactly on O with the zero line along OA.
3. Use the row of numbers that reads 0 on OA. Count up that row to the measure you want and make a small dot there.
4. Remove the protractor and join O to the dot. The angle between OA and the new ray is the required angle.
5. Check by eye: 40° and 75° should be narrower than a right angle; 110°, 112° and 134° wider. If an obtuse one looks narrow, you used the wrong row.

In shortRepeat these steps for each of 110°, 40°, 75°, 112° and 134°.

Watch this explained “Drawing to order: four steps to 30°”, 9:06 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 50One way to think about it

1. The curve is drawn at H, so H is the vertex; the arms are HI and HJ. The angle is ∠IHJ.
2. Put the protractor's centre on H with the zero line along HJ. On the row that starts at 0 on HJ, read where HI crosses: about 116°.
3. Draw a new ray OA on your page.
4. Put the protractor's centre on O with the zero line along OA, mark 116° on the row that starts at 0 on OA, and join O to the mark.
5. Measure your new angle to check that it is 116°.

In shortThe given angle ∠IHJ is about 116°; draw a 116° angle using the steps above.

Watch this explained “Drawing to order: four steps to 30°”, 9:06 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 51Checked by computerAnswers can differ: one example

(a) An acute angle

1. In the left-hand grid, A is the left end of the red segment, as in the worked example. Count in dot steps from A, with right and up positive.
2. Join A to the dot one step right and one step up, (1, 1).
3. This line makes 45° with the red segment — less than 90°, so the angle marked between them is acute.

AnswerJoin A to (1, 1): an acute angle of 45°.

(b) An obtuse angle

1. Join A to the dot one step left and one step up, (−1, 1).
2. The angle from the red segment round to this line is 135° — more than 90° but less than 180°, so it is obtuse.

AnswerJoin A to (−1, 1): an obtuse angle of 135°.

(c) A reflex angle

1. Join A to the dot one step right and one step down, (1, −1).
2. The small angle between the red segment and this line is 45°. Draw the curve the long way round instead, over the top and round the left.
3. That angle is 360° − 45° = 315°, more than 180°, so it is reflex.

AnswerJoin A to (1, −1) and mark the long way round: a reflex angle of 315°.

Right-hand grids

1. There the red segment slants down to the right; take A at its upper end.
2. Acute: join A to the dot one step to the right (45°).
3. Obtuse: join A to the dot one step to the left (135°).
4. Reflex: join A to the dot one step to the right and mark the curve the long way round (360° − 45° = 315°).

AnswerMany other dots work as well — any choice is right if the angle you mark with the curve is of the asked type.

Watch this explained “Building each type on a dot grid”, 7:51 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 52Checked by computer

1. Put the centre of the protractor on T and line up the arm TP with 0°.
2. (a) TR crosses the scale at 30°, so ∠PTR = 30°. It is less than 90°, so it is acute.
3. (b) TQ crosses at 60°, so ∠PTQ = 60°. It is less than 90°, so it is acute.
4. (c) TW crosses at 102°, so ∠PTW = 102°. It is between 90° and 180°, so it is obtuse.
5. (d) ∠WTP is marked by the curve that goes the long way round T, so it is the reflex angle. A protractor cannot reach it directly: ∠WTP = 360° − 102° = 258°. It is more than 180°, so it is reflex.
6. These are the values in NCERT's answer key. On the printed page ∠PTW measures about 103°, so ∠WTP comes to about 257°; readings of 103° and 257° are also correct, and every type stays the same.

Answer(a) ∠PTR = 30°, acute (b) ∠PTQ = 60°, acute (c) ∠PTW = 102°, obtuse (d) ∠WTP = 360° − 102° = 258°, reflex (NCERT's answer key; on the printed page (c) and (d) measure about 103° and 257°).

Watch this explained “One picture, two angles: why the curve is drawn”, 6:17 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 53One way to think about it

1. For 140°, 82°, 70° and 35°: draw a ray OA, put the protractor's centre on O with the zero line along OA, mark the measure on the row that starts at 0 on OA, and join O to the mark.
2. Check by eye: 140° is wider than a right angle; 82°, 70° and 35° are narrower.
3. 195° is more than the 180° a protractor shows. Its partner angle is 360° − 195° = 165°. Draw OA, mark 165° and join O to the mark to get ray OB.
4. The small angle AOB is 165°; the angle going the long way round from OA to OB is 360° − 165° = 195°. Mark that one with a curve drawn round the outside.
5. (Another way: extend OA backwards past O to make a straight angle, 180°, then draw 15° more beyond it: 180° + 15° = 195°.)

In shortDraw 140°, 82°, 70° and 35° directly with the protractor. For 195°, draw 165° and mark the angle on the other side: 360° − 165° = 195°.

Watch this explained “Drawing to order: four steps to 30°”, 9:06 into Reading and drawing an angle with a protractor · हिंदी में देखें

Open NCERT p. 53Checked by computer

(a)

1. Estimate first: about half a right angle. Then measure with the centre on the vertex and one arm on zero.
2. Readings within about a degree of these are fine, because printed lines have some thickness.

AnswerAbout 45° — acute.

(b)

1. Estimate: nearly a straight angle. Measure.

AnswerAbout 166° — obtuse.

(c)

1. Estimate: a bit more than a right angle. Measure.

AnswerAbout 120° — obtuse.

(d)

1. Estimate: about a third of a right angle. Measure.

AnswerAbout 33° — acute.

(e)

1. Estimate: close to a right angle. Measure: it is a little more than 90°.

AnswerAbout 97° — obtuse.

(f)

1. The loop is drawn round the outside of the vertex, so the marked angle is the reflex one, not the thin gap between the arms.
2. Measure the thin gap: about 11°. Then 360° − 11° = 349°.

AnswerAbout 349° — reflex.

Watch this explained “The whole angle — and the two ends nothing claims”, 5:16 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 53One way to think about it

1. A single closed shape with six corners will not work: the corners of any six-sided shape add up to 720°, but three acute angles, a right angle and two obtuse angles add up to less than 3 × 90° + 90° + 2 × 180° = 720°.
2. Instead, draw an open zig-zag line that bends six times. At each bend, set the angle between the two pieces with your protractor.
3. Make the bends, for example, 30°, 45° and 60° (acute), 90° (right), and 120° and 150° (obtuse). Keep the pieces short enough that they do not cross.
4. Mark each of the six angles with a small curve on the side you measured, and check each with the protractor.

In shortMany figures work. One example: a zig-zag line with six bends of 30°, 45°, 60°, 90°, 120° and 150° — three acute, one right and two obtuse.

Watch this explained “The whole angle — and the two ends nothing claims”, 5:16 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 53One way to think about it

1. A capital M has three corners: the two top corners (on the sides) and the valley in the middle. The 40° angles go at the two top corners and the 60° angle at the middle valley.
2. Start with the middle: mark a point V and draw two equal lines up from V with 60° between them (measure it with the protractor). Call their top ends P and Q.
3. At P, put the protractor's centre on P with the zero line along PV, mark 40° on the outer side, and draw the left stroke of the M down from P through that mark. Do the same at Q for the right stroke.
4. Make the two outer strokes the same length. They will lean slightly outwards (10° from upright), which is fine.
5. Measure all three angles to check: 40°, 60°, 40°. They are at three different corners, so they do not need to add up to anything in particular.

In shortDraw the middle V at 60° first, then add each outer stroke at 40° to the line it meets at the top corner. The M has angles 40°, 60° and 40°.

Watch this explained “Drawing an M and a Y — and which one has to add to 360°”, 10:19 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 53One way to think about it

1. In a Y, all three strokes meet at the same central point, so the three angles between them go all the way round that point.
2. Check first that the three wanted angles add to a full turn: 150° + 60° + 150° = 360° — they do, so the letter can be drawn.
3. Draw one arm of the Y, then use the protractor at the shared point to mark the next arm 150° round from it, and the third arm 60° further round from that (which lands 150° short of closing the full turn back to the first arm).

In shortDraw the Y's three strokes meeting at one point with successive angles of 150°, 60° and 150° between them, which correctly add up to 360°.

Watch this explained “Drawing an M and a Y — and which one has to add to 360°”, 10:19 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 54Matches NCERT’s answer

1. The 24 spokes are evenly spaced around one full turn, 360°.
2. So the angle between two neighbouring spokes is 360° ÷ 24 = 15°.
3. Every angle between any two spokes is a whole multiple of 15°: 15°, 30°, 45°, 60°, 75°, 90°, ...
4. The largest of these that is still acute (strictly less than 90°) is 75°, since the next one, 90°, is a right angle, not acute.

Answer15° between adjacent spokes; the largest acute angle between two spokes is 75°.

Watch this explained “A 24-spoke wheel, and a puzzle whose answer is a range”, 11:41 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें

Open NCERT p. 54Checked by computerAnswers can differ: one example

1. Call the angle x°. It is acute, and so are 2x, 3x and 4x. The strongest of these is 4x < 90, so x < 22.5.
2. 5x is obtuse: 90 < 5x < 180, so 18 < x < 36.
3. Both must be true: 18 < x < 22.5.
4. Check with 20°: 40°, 60° and 80° are acute, and 5 × 20° = 100° is obtuse.

AnswerAny measure between 18° and 22.5°, not including 18° or 22.5° themselves. In whole degrees: 19°, 20°, 21° or 22°.

Watch this explained “A 24-spoke wheel, and a puzzle whose answer is a range”, 11:41 into Acute, obtuse and reflex: one classification covering every angle · हिंदी में देखें