2. Lines and Angles

Rihan can draw countless (infinitely many) lines through one point.

Sheetal can draw only one line through two points.

Line segments: LM, MP, PQ and QR.

Points on exactly one segment: L and R.

Points on two segments: M, P and Q.

Rays: TA, TN (also called TB) and NB.

No. T is the starting point of TA and TN (TB). The ray NB starts at N.

1. Draw two lines OP and OQ that start from the same point O. They meet at O.
2. Draw a ray XY and a line PQ that cross each other. Mark the crossing point as M.
3. Draw a line l. Mark points E and F on it. Mark point D outside the line.
4. Draw a line AB. Mark a point P on the line between A and B.

(a) Five points: D, E, O, B, C

(b) A line: DB (it can also be named DE, EO or OB)

(c) Four rays: OB, OC, OD, OE

(d) Five line segments: DE, DO, DB, EO, OB (OC is also a line segment)

(a) Yes. The ray starts at O and B lies on the ray. So it can be named OB.

(b) No. A ray is named by its starting point first. OA starts at O, but AO would start at A and go the other way.

Yes. In the bicycle, one angle is ∠BDC.

Its vertex is D. Its arms are rays DB and DC.

Angles can also be seen in the window frame, the ladder and the bridge.

Mark a point S. Draw a ray from S through T and another ray from S through R.

The angle formed is ∠TSR (or ∠RST). Its vertex is S.

Three angles have the same vertex P: ∠APB, ∠BPC and ∠APC.

So ∠P does not tell us which angle we mean.

∠QTR (or ∠RTQ) and ∠PTR (or ∠RTP).

We get 3 lines: AB, BC and CA.

We can name 3 angles: ∠ABC, ∠BCA and ∠CAB.

We get 6 lines: AB, AC, AD, BC, BD and CD.

We can name 12 angles (3 at each point):

At A: ∠BAC, ∠CAD, ∠BAD  |  At B: ∠ABC, ∠CBD, ∠ABD

At C: ∠BCA, ∠ACD, ∠BCD  |  At D: ∠ADB, ∠BDC, ∠ADC

No. When two angles are nearly equal, it is hard to compare them just by looking.

Then we need to superimpose them or measure them.

We compare the lengths of two line segments, the sizes of two coins, or two pieces of cloth by placing one over the other.

Let the fold meet the top side at E and the bottom side at F. Four angles are formed: ∠AEF, ∠BEF, ∠DFE and ∠CFE.

If the fold is slanting, two angles are bigger (obtuse) and two are smaller (acute).

If the fold is straight across the paper, all four angles are equal right angles.

(a) ∠AOB is greater. ∠XOY lies inside ∠AOB.

(b) ∠AOB is greater. ∠XOB is a part of ∠AOB.

(c) Both are equal. B and C lie on the same ray, so ∠XOB and ∠XOC are the same angle.

We cannot say just by looking. The two angles only partly overlap.

We have to superimpose them or measure them with a protractor.

The arm OY of the second crane lies outside OB. So the second crane is making the bigger angle.

Yes. Fold the paper so that OB lies exactly on OA. The crease is OC. It divides the straight angle into two equal angles.

A right angle is half of a straight angle. So it is 14 of a full turn (a quarter turn).

Each rectangular window has 4 right angles at its corners.

Other right angles: corners of the blackboard, door, table, book, notebook and floor tiles.

First grid: AB goes to the right. Join A to any grid point on the same row to the left of A. There are 3 ways.

Second grid: AB goes slanting down to the right. Join A to the grid points on the same slanting line, up and to the left of A. There are 2 ways.

First grid: AB is flat (horizontal). Join A to the points straight above it (2 points) or straight below it (3 points). There are 5 ways.

Second grid: AB goes slanting down to the right. Join A to the points on the other slant — up to the right (2 points) or down to the left (2 points). There are 4 ways.

(a) We get 4 right angles.

When we fold, the two angles on each side of the new crease lie exactly on top of each other. So they are equal. Together they make a straight angle (180°). So each angle is 90°.

(b) Steps:

1. Fold the paper in any slanting way and press to get a crease. Unfold it.
2. Fold the paper again so that one part of the crease lies exactly on the other part.
3. Press firmly and unfold. The new crease is perpendicular to the first crease.

In Vidya’s book: Case 1 and Case 2 show acute angles. Case 3 shows a right angle. Case 4 shows an obtuse angle. Case 5 shows a straight angle (book open flat on the table).

The corners of windows and doors are right angles. A pair of scissors opened a little makes an acute angle.

Acute angles: 30°, 45°, 60°, 80° — draw them with the base flat, slanting or upside down.

Obtuse angles: 100°, 120°, 150°, 170° — draw them in different directions too.

An acute angle has a narrow opening, so its corner is pointed and sharp like a needle.

An obtuse angle has a wide opening, so its corner looks blunt.

Figure 1: 3 acute angles.

Figure 2: 12 acute angles (3 at the corners + 3 at each of the 3 middle points).

Figure 3: 21 acute angles.

Next figure: divide the smallest middle triangle again into 4 triangles. It will have 30 acute angles.

Pattern: 3, 12, 21, 30, … Each figure has 9 more acute angles than the previous one.

(a) ∠KAL = 30°

(b) ∠WAL = 50°

(c) ∠TAK = 120°

Yes. We can count the units in 5s and 10s using the medium and long marks.

The outer scale starts at 0 on the left and the inner scale starts at 0 on the right.

So we can measure an angle whether its base arm points to the left or to the right.

OT and OS pass through 20 and 55 on the outer scale.

Yes. ∠TOS = 55° − 20° = 35°

OB passes through 0° on the inner scale. OA passes through 80°.

∠AOB = 80°

Half a turn = 12 of 360° = 180°.

Quarter turn = 14 of 360° = 90°. Also, 12 of 180° = 90°.

One-eighth of a turn = 18 of 360° = 45°.

Another half fold gives an angle of 22.5° (half of 45°).

Each angle = 22.5°

The straight angle of 180° is folded into 8 equal parts. 180° ÷ 8 = 22.5°.

First angle: ∠IHJ = 47°

Second angle: ∠GHK = 23°

Third angle: ∠IHJ = 108°

Do this activity in class. Corners of the board, door and window measure 90°. Measure the angle of an open door or an open book as well.

First angle: ∠IHJ = 42°

Second angle: ∠IHJ = 116°

The paper protractor has marks only for 0°, 22.5°, 45°, 67.5°, 90°, … So it cannot measure these angles exactly.

The marked angle is a reflex angle. Measure the smaller (unmarked) angle first. It is 100°.

Marked angle = 360° − 100° = 260°

(a) 80°   (b) 120°   (c) 60°   (d) 130°   (e) 130°   (f) 60°

∠BXE = 115°

∠CXE = 85°

∠AXB = 180° − 115° = 65°

∠BXC = 115° − 85° = 30°

∠PQR = 45°

∠PQS = 100°

∠PQT = 150°

Do this activity in class. The creases will mostly make angles of 45°, 90° and 135°, because each fold halves a right angle.

In every triangle, the sum of the three angles is 180°.

Conjecture: The sum of the angles of any triangle is always 180°.

(a) The full turn of 360° is divided into 12 equal parts by the 12 numbers. 360° ÷ 12 = 30°. At 1 o’clock, the hands are one part apart.

(b) 2 o’clock = 2 × 30° = 60°; 4 o’clock = 4 × 30° = 120°; 6 o’clock = 6 × 30° = 180°.

(c) 3 o’clock = 90°; 5 o’clock = 150°; 12 o’clock = 0°.

Yes.

Vertex: the hinge line where the door is fixed to the wall.

Arms: the closed position of the door (along the wall/frame) and the open position of the door.

The vertex is the point where the swing ropes are tied at the top.

One arm is the rope when the swing hangs straight down. The other arm is the rope at the starting position.

Yes. The greater the angle, the steeper the slab.

One arm is the slab. The other arm is an imaginary flat (horizontal) line.

The slab is visible. The horizontal line is not visible.

Yes.

The vertex is the point about which the insect turns.

One arm is the horizontal line touching the insect. The other arm is the line along the insect after it turns.

Some of the angles are: ∠BAC, ∠ACD, ∠APL, ∠BPL, ∠PLD, ∠PLS, ∠PRS, ∠BRS, ∠RSL, ∠CLP.

Write your guess and the measured value for each angle in a table. Then find the difference between them.

Steps (example 110°):

1. Draw a ray OA.
2. Place the centre of the protractor on O and the base line along OA.
3. Start from the 0° on OA and mark a point B at 110°.
4. Join OB. ∠AOB = 110°.

Draw the other angles in the same way.

Measure the given angle ∠IHJ with a protractor. It is about 116°.

1. Draw a ray HJ.
2. Place the centre of the protractor on H, with HJ on the 0° line.
3. Mark a point I at 116°.
4. Join HI. ∠IHJ = 116°.

(a) Acute: join A to a point that makes a narrow opening with AB, for example a point one step up and one step to the right of A (first grid).

(b) Obtuse: join A to a point on the other side, for example one step up and one step to the left of A.

(c) Reflex: draw any acute or obtuse angle and mark the outer (bigger) side with a curve.

(a) ∠PTR = 30° — acute

(b) ∠PTQ = 60° — acute

(c) ∠PTW = 102° — obtuse

(d) ∠WTP (marked the other way) = 360° − 102° = 258° — reflex

∠REB is a straight angle = 180°.

∠BET = 180° − 80° = 100°

∠SER = 90°, so ∠SET = 90° − 80° = 10°

Draw (a), (b), (d) and (e) with a protractor as usual.

For (c) 195°: it is a reflex angle. 360° − 195° = 165°. Draw a 165° angle and mark the outer side as the angle. Or draw a straight angle (180°) and add 15° more below the base.

One easy figure is a zig-zag line with six corners.

Using a protractor, make the angles at the corners 40°, 50°, 60° (acute), 90° (right), and 120°, 150° (obtuse).

1. Draw two vertical lines of equal length for the two sides of M.
2. From the top of the left line, draw a slanting line making 40° with it.
3. From the top of the right line, draw a slanting line making 40° with it.
4. Let the two slanting lines meet. The angle between them in the middle is 60°.

1. Draw a vertical line segment for the stem of Y. Mark its top point O.
2. From O, draw a ray making 150° with the stem on the left side.
3. From O, draw another ray making 150° with the stem on the right side.
4. The angle between the two upper arms is 360° − 150° − 150° = 60°.

Angle between two neighbouring spokes = 360° ÷ 24 = 15°

Angles between spokes are 15°, 30°, 45°, 60°, 75°, 90°, …

Largest acute angle = 75° (90° is a right angle, not acute).

4 times the angle is less than 90°. So the angle is less than 22.5°.

5 times the angle is more than 90°. So the angle is more than 18°.

In whole degrees, the angle can be 19°, 20°, 21° or 22°.