Class 6 Mathematics Chapter 2: Lines and Angles

Kabir sharpens his pencil

Kabir marks a dot on his page. His friend Sara says the dot is too fat. He sharpens the pencil and tries again, and this time the mark is thinner. Sharpen it further and it would be thinner still. That shrinking is the whole idea of a point in geometry.

What a point really is

A point tells you one position and nothing else. It carries no length, no breadth and no height. The ink we draw is only a signal to the eye. Picture the tip of a needle or the sharp end of a compass and you are close enough.

Why points get capital letters

Put three dots on a page and you cannot discuss them without names. So each one gets a single capital letter. We might call them H, K and W. Then we can say point H, point K and point W, and everybody knows which dot is meant.

A crease gives you a segment

Fold a sheet of paper and open it out. The crease that appears is perfectly straight. Mark two dots on it and call them K and L. The stretch of crease from K to L, ends included, is a line segment.

Two ends, and the shortest route

You could travel from K to L by a curve, a zigzag, or a wide loop. The straight stretch is the shortest of them all. That is why the segment matters: it is the least distance between its two end points.

Writing the name of a segment

The segment above is called KL. It is equally correct to call it LK. Nothing about the piece of path changes when you read it backwards, so both names describe the same thing.

Refusing to stop

Now imagine that segment stretching. Push it out past K, and push it out past L as well. Keep pushing with no finish in mind. What you are imagining is a line.

A line cannot be finished

Ask yourself whether you could draw a whole line. You could not. Your page ends, the wall ends, the country ends, and the line does not. So we draw a piece of it and put arrowheads on both ends, and the arrowheads carry the message that it goes on.

One point, endless lines

Rihit marks a single dot and starts drawing straight lines through it. He can tilt each new line a little and get another one. There is no end to the supply. One point on its own does not pin down a line.

Two points, exactly one line

Now he marks a second dot. This time only one straight line passes through both dots at once. Two points settle the matter completely, which is why a carpenter snaps a chalk string between two nails.

Half a line is a ray

Switch on a torch in a dark room. The beam begins at the bulb and travels outward. It has a definite beginning and no finish. A ray is drawn the same way: a dot at the start, an arrowhead at the far end.

Naming a ray, starting point first

Suppose a ray begins at V and passes through W. We call it ray VW. The starting letter comes first. Writing WV would describe a ray that begins at W and travels the other way, which is a different thing altogether.

Four pieces, one habit

When you meet a figure, count the ends. Two marked ends is a segment. One marked end with an arrow is a ray. Arrows at both ends is a line. No ends at all, just a single dot, is a point.

Scenario:

A class marks three things on a street map: the stretch of road between two crossings, a torch beam shining down the lane, and the point where two roads meet. Arjun labels all three as lines. Their teacher asks how each one ends. The road stretch has two ends. The beam starts at the torch and carries on out of sight. The crossing has no length at all. The class gives each its proper name: a segment, a ray and a point. Arjun sees that the word line was hiding three different things, and that what distinguishes them is where they stop.

Analysis:

All three looked like lines on the map. Asking where each one ends is what separates a segment from a ray and both from a point.

Activity:

On a map of your area, mark one segment, one ray and one point. Write why each label fits.

Title:

Fold, mark and name

Safety:

Do not shine the torch into anybody's eyes.

What You Notice:

The lines through one dot never run out, and the lines through two dots stop at one. The torch beam has a clear beginning at the bulb and no ending you can point to, which is exactly what makes it a ray.

The word geometry comes from two old Greek words that together mean measuring the earth.

Through a single marked point you could draw lines all day and never run out of new ones.

Surveyors still fix land boundaries by driving in two pegs, because two points settle exactly one straight line.

Telling segment, ray and line apart — Two dots, one dot, no dots — Count the end points in the picture. Two ends means a segment. One end means a ray. No ends at all means a line.

Naming a ray — Start first — The letter of the starting point is written first. Swap the letters and you have named a different ray, pointing the other way.

Vidya opens her book

Vidya lifts the cover of her book a little and stops. Then she lifts it further. Then further still. Nothing about the cover has changed, yet something has clearly grown each time. What has grown is the angle between the cover and the page beneath.

Two rays from one point

Draw a dot and send two rays out from it in different directions. That figure is an angle. The dot they share is the vertex. The two rays are the arms, and they show the two directions the angle sits between.

A small curve marks it

When we want to point at an angle in a busy figure, we draw a short curve near the vertex, crossing from one arm to the other. The curve is not part of the angle. It is a label, saying look here.

Writing the name

Suppose the vertex is Q, with one arm through M and the other through R. We may call the figure angle Q if no confusion is possible. The safer name uses all three letters: angle MQR, or angle RQM.

Why the vertex sits in the middle

In both of those names Q is the middle letter, and that is the rule. If a busy picture has three angles at Q, then angle Q is useless on its own, because it does not say which pair of arms is meant.

Size is an amount of turn

What makes one angle larger than another? Put one arm on top of the other and swing it round. The angle whose arm has to swing further is the larger one. Vidya's cover lifted higher has simply been swung further.

Turning shows up everywhere

Open a pair of scissors and the blades turn about the rivet. Spread a divider and the legs turn about the joint. Push a door and it turns about its hinge. In each case the hinge is the vertex.

The trap of the long arm

A child often says the angle with the longer arms is bigger. It is not. Extending an arm adds ink to the page, but the two directions have not moved apart at all, so the turning between them is unchanged.

Laying one over the other

To compare two angles, trace one on thin paper and lay it over the other. Bring the vertices together and put one arm of each in line. Now look at the free arms. The angle whose free arm lies outside is the larger.

When both free arms match

Sometimes the two free arms fall exactly on one another. Then neither angle is larger, and we say the two are equal. The same amount of turning would carry you from the shared arm to either free arm.

A circle of tracing paper

Two angles drawn far apart are awkward to lay on top of one another. So cut a small circle from tracing paper. Centre it on the first vertex and mark where each arm crosses the edge. Carry that circle to the second angle.

Reading the marks

Place the circle so its centre sits on the second vertex and one arm passes through your first mark. Now see where the second arm falls against your second mark. Inside means smaller, outside means larger, and on it means equal.

Rotating arms from straws

Make your own hinge. Push two paper straws onto the two arms of a paper clip and you can turn one straw against the other. Make several with different openings and arrange them from smallest to largest.

The slit test

Cut an angle shaped slit in a piece of cardboard by tracing one of your straw pairs. Now try every pair against the slit. Only the pair whose opening matches will pass through, however long or short its straws happen to be.

Scenario:

A class compares how wide three things open: a door, a pair of scissors, and a book. Nita says the door opens most because it is biggest. Their teacher asks the class to compare without measuring. They cut a paper wedge and slide it into each opening. The wedge fits furthest into the scissors, so the scissors open widest. Size had nothing to do with it. Nita sees her mistake. She was comparing the objects rather than the openings. An opening is about turn rather than length, so a small pair of scissors can beat a large door.

Analysis:

Nita compared the things instead of the angles. The paper wedge compares openings directly, and the biggest object turns out not to open widest.

Activity:

Cut a paper wedge and compare three openings at home. Rank them, then check whether the biggest object opened widest.

Title:

Straw hinges and a cardboard slit

Safety:

Ask an adult to help with the cutting, and cut away from your hand.

What You Notice:

Only the hinge whose opening matches the slit will go through. Change a hinge to straws of a different length and it still passes, so long as you have not altered the opening. The opening is the angle, and nothing else about the drawing is.

A pair of scissors, a divider and an open book all make an angle whose vertex is the hinge.

An angle cut as a slit in cardboard lets only one pair of rotating arms through, whatever the length of the straws.

An animal opening its jaws wider is simply turning one jaw further from the other.

Naming an angle — Vertex in the middle — Of the three letters in an angle's name, the middle one is always the vertex. Angle MQR and angle RQM both have their corner at Q.

What decides the size — Turn, not length — Ask how far one arm had to swing, not how long the arms are. A longer arm changes the drawing and not the angle.

Vidya flattens the cover

Vidya lays her book on the table and pushes the cover right over. It comes to rest flat, pointing the opposite way from the page. She has stopped turning it, but she could have kept going and brought it all the way round to where it began.

The straight angle

When the two arms of an angle end up pointing in opposite directions, they lie along one straight path. An angle like that is a straight angle. Half of a whole turn has been used to make it.

Cutting a straight angle in two

Take a rectangular sheet and mark a straight angle along one edge, with its corner at O. Can a single ray from O split that angle into two equal pieces? Fold the sheet so that one arm drops exactly onto the other and look at the crease.

What the crease gives you

The crease lands halfway. The fold has proved it: the two angles are laid on one another and match. Each of those equal angles is a right angle, so a straight angle holds exactly two of them.

The L shape, and perpendicular lines

A right angle looks like the corner of a capital L. Only a precise half of a straight angle counts as one, and close is not good enough. Two lines crossing at right angles are called perpendicular.

Smaller and larger than a right angle

Sort a pile of angles by holding each against a right angle. Those that fall short are acute angles; the opening is less than a quarter turn. Those that overshoot but do not reach a straight angle are obtuse angles.

Why comparing is not enough

So far every judgement has needed a second angle to compare against. That is slow, and it will not do if you want to write an angle down or send it to somebody else. We need a number instead.

Borrowing the circle

The circle of tracing paper already carried angles about. Put the idea to work properly. Take the angle at the centre of a circle, which is a whole turn, and cut it into equal parts. Then count how many parts fit inside the angle you care about.

Choosing 360 parts

Mathematicians settled on 360 equal parts. Each part is called one degree and is written as a small raised circle after the number. An angle that holds 40 of these parts measures 40 degrees.

The three measures fall out

A whole turn was cut into 360 parts, so it measures 360 degrees. A straight angle is half of that, giving 180 degrees. Two right angles make a straight angle, so a right angle measures 90 degrees.

A pinch of history

Why 360? The honest answer is that nobody knows for certain. Several old clues survive. The Rigveda, one of the most ancient texts we have, speaks in verse 1.164.48 of a wheel with 360 spokes, and the count clearly mattered long ago.

More clues, still no proof

Old calendars in India, Persia, Babylonia and Egypt all ran on 360 days. Babylonian mathematicians liked counting in sixties, and 60 and 360 sit comfortably together. None of this settles the question, and it is worth saying so plainly.

The practical reason

There is one reason that is easy to check. Try dividing 360 by each number from 1 to 10. Only 7 leaves a remainder. So a turn can be shared into 2, 3, 4, 5, 6, 8, 9 or 10 equal pieces with a whole number of degrees each time.

Building your own protractor

Cut a paper circle and fold it in half, then in half again, and again. The creases mark a whole turn cut into 2, then 4, then 8 equal parts. Number them and you have made a rough protractor of your own.

Scenario:

A class watches the minute hand of a clock. Imran is asked how far it turns in fifteen minutes. He says a lot. Their teacher asks for a number instead. The class works it out. A full turn takes sixty minutes and makes 360 degrees, so one minute is six degrees. Fifteen minutes is ninety degrees, which is a right angle. The children check it on the clock face and the hand sits exactly across. Imran sees that the clock was a protractor all along. Thirty minutes gives a straight angle of 180 degrees, and the quarters of the hour are the special angles.

Analysis:

A lot is not an answer and ninety degrees is. Dividing 360 by 60 turns the clock face into a measuring instrument.

Activity:

Use a clock face to show a right angle and a straight angle with the minute hand. Write the minutes each one takes.

Title:

Fold your own protractor

Safety:

Ask an adult before using scissors, and cut away from your fingers.

What You Notice:

Each fold halves what was there before, so the numbers halve too. The book corner sits on the 90 crease every time. Your paper tool has no small marks on it, which is exactly why a real protractor carries them.

Nobody has ever settled why a turn was cut into 360 parts rather than some other number. Several old reasons are known, and none of them is proved.

The Rigveda, in verse 1.164.48, describes a wheel carrying 360 spokes, so the figure was already familiar thousands of years ago.

Of the numbers from 1 to 10, only 7 refuses to divide 360 exactly, which makes it very easy to share a turn out evenly.

The three fence posts — Quarter 90, half 180, whole 360 — Learn the turns first and the numbers follow. A quarter turn is 90 degrees, a half turn is 180 and the whole turn is 360.

Spotting a right angle — Corner of an L — A right angle has the shape of the capital letter L. If the corner leans open or closed even slightly, it is not a right angle.

The tool in the geometry box

The half circle in your geometry box is a protractor. Its straight edge carries a small mark at the middle, and the curved edge carries rows of tiny lines. The straight angle at that middle mark has been cut into 180 equal unit parts.

Long marks and medium marks

Run your eye along the curved edge. Every tenth unit part carries a long mark. Halfway between two long marks there is a medium one, five parts along. Counting in tens and fives is far quicker than counting single marks.

Why two rows of numbers

There are two rows printed on the edge. One grows as you move left and the other as you move right. Neither row is the correct one. The correct one depends on which arm of your angle you decide to lay along the zero line.

Setting it down, step one

First find the small mark at the middle of the straight edge. That is the centre. Slide the protractor until this centre sits exactly on the vertex of your angle. If the corner is even slightly off, every reading afterwards will be wrong.

Setting it down, step two

Now turn the protractor, keeping the centre where it is, until one arm of the angle runs along the zero line. That arm is your base. The angle is now sitting in the protractor the way it was meant to.

Taking the reading

Look at where the second arm crosses the curved edge. Two numbers meet it, one from each row. Take the one belonging to the row that reads zero on your base arm. That number is the measure of your angle.

A worked reading

Suppose the base arm runs to the right and the row rising from the right shows 60 where the second arm crosses. The angle measures 60 degrees. The other row shows 120 at the same place, and that is the reading of the other angle, not yours.

When no arm sits on zero

Sometimes the figure is crowded and no arm can be brought to zero. Then read both arms on one scale, the same one for both. If they show 25 and 70, subtract, and the angle between them measures 45 degrees.

Three mistakes to look for

A reading goes wrong in only a few ways. The corner may be off the centre mark. The base arm may be a little above or below the zero line. Or the wrong row may have been read, giving a number that is too big.

Checking a reading for sense

Before writing a number down, glance at the angle. Is the opening narrower than a book corner? Then a reading above 90 is certainly wrong. Is it wider? Then a reading below 90 cannot be right either.

Halving an angle by folding

Draw an angle on paper and fold the sheet through the vertex so that one arm lands on the other. Open it out. The crease runs from the vertex into the angle and cuts it into two pieces that matched during the fold.

The bisector

That crease is the bisector of the angle. It leaves a matching half on either side of itself, and it must pass through the vertex. A line that splits the opening but misses the corner does not bisect anything.

Bisecting in numbers

An angle of 80 degrees has a bisector that leaves 40 degrees on each side. An angle of 64 degrees gives two of 32 degrees. Whatever the measure, each half is exactly half of it.

Folding again and again

Start with a straight angle and fold. You get 90 degrees. Fold that and you get 45. Fold once more and you get 22 and a half degrees. Each new crease bisects the angle you had before it.

Scenario:

A class measures the corner of a desk with a protractor. Three children report 88, 90 and 92 degrees. Rohit says two of them must be careless. Their teacher asks all three to measure again, watching how the protractor sits. Two had its centre slightly off the corner point. Measured properly, all three get 90. The class works out why a small slip matters. The centre mark has to sit exactly on the vertex, because moving it shifts both arms at once. Sana adds the second rule they found: read from the scale that starts at the arm, or the answer comes out as its supplement.

Analysis:

Three readings and one corner. Both errors came from how the instrument sat rather than from how anybody read it.

Activity:

Measure four corners in your classroom with a protractor. Write each reading and say which ones should be exactly 90.

Title:

Fold it, then measure it

What You Notice:

Each half comes out at close to half the original reading, and any gap is usually your folding rather than the idea. Measure the second half as well and the two halves agree with each other, which is the quickest proof that the crease really is a bisector.

The two rows of numbers on a protractor always add to 180 at any mark, so 40 on one row sits beside 140 on the other.

Between two long marks on a protractor there is a medium mark, and it sits five degrees along.

Folding an angle in half again and again gives 90, then 45, then 22 and a half degrees.

Setting the protractor — Centre on corner, arm on zero — Two things must be right before you read anything. The centre mark goes on the vertex, and one arm goes along the zero line.

Choosing a scale — Follow your base arm — Look at where your base arm cuts the curved edge. Whichever row reads zero there is the row to use.

Vidya draws to order

So far every angle has been measured after somebody else drew it. Vidya now wants the opposite. She has been told to produce an angle of 115 degrees and she has a ruler, a pencil and a protractor.

Step one, rule the base

She draws a straight arm first and labels its ends K and P. This is the base arm, and K will be the vertex. Choosing the base before touching the protractor keeps the rest of the work simple.

Step two, set the tool

She places the centre mark of the protractor on K and turns the tool until the arm KP runs along the zero line. Nothing is drawn yet. The setting has to be right before any mark is made.

Step three, find the number

She looks for the row of numbers that reads zero where KP crosses the edge. Counting up that row she finds 115 and makes a small dot on the paper just outside the curved edge there. She labels the dot L.

Step four, join and name

She lifts the protractor away and rules a line from K through L. The figure is finished. The angle she has made is angle LKP, and it measures 115 degrees.

Estimate before you measure

A useful habit is to guess first. Look at an angle, write down a guess, then measure it. The gap between the guess and the reading is what teaches you. After a dozen tries most people guess within ten degrees.

Why estimation is worth it

A guess is also a safety net. Suppose your eye says the opening is narrow. Then your protractor says 140 degrees. One of the two must be wrong. Nine times out of ten it turns out to be the row you read, and a second look settles it.

Acute angles

Now to sorting. An acute angle is one whose measure is more than 0 degrees and less than 90 degrees. Openings of 5, 40 and 89 degrees are all acute. An opening of exactly 90 degrees is not.

Obtuse angles

An obtuse angle measures more than 90 degrees and less than 180 degrees. So 91, 130 and 179 degrees are all obtuse. Once more the two ends are shut out: neither 90 nor 180 belongs to this group.

Have we covered everything?

The angles so far stop at a straight line. But an arm can keep turning past that. An opening that has gone beyond a half turn still needs a name, and that is where the last group comes in.

Reflex angles

A reflex angle measures more than 180 degrees and less than 360 degrees. Openings of 200, 250 and 359 degrees are reflex. An exact half turn is not reflex, and neither is a whole turn.

Working out a reflex angle

Two arms at a vertex mark out two angles, a smaller one and a reflex one, and together they make a whole turn. So measure the smaller angle and take it from 360. A smaller angle of 110 degrees leaves a reflex angle of 250 degrees.

The fence posts

Three measures have names of their own and sit in none of the three groups. An angle of 90 degrees is a right angle. One of 180 degrees is a straight angle. One of 360 degrees is a whole angle.

Putting a measure in its place

Given any number from 0 to 360 you can now name its group at once. Below 90 it is acute. Between 90 and 180 it is obtuse. Above 180 it is reflex. On 90, 180 or 360 it has its own name instead.

Scenario:

Anjali plants a straight stick upright in the courtyard of her home in Salem. Early in the morning she marks the tip of its shadow with a pebble. She marks it again at noon, and once more in the late afternoon. By evening three pebbles lie on the ground. She rules a line from the foot of the stick out to each pebble, so that three arms now spread out from one place. Her brother lends her his protractor. She finds that the morning arm and the noon arm are 62 degrees apart. The noon arm and the afternoon arm are 55 degrees apart. She writes both readings in her science notebook beside the times she took them.

Analysis:

One fixed point with several arms gives several angles at once, and the turning of a shadow can be recorded as a measure rather than as a description.

Activity:

Repeat Anjali's experiment in your own yard on a sunny day, taking three marks, and compare your two readings with a friend's.

The word acute once meant sharp and the word obtuse meant blunt, which is exactly how the two openings look.

A reflex angle and the smaller angle beside it always add to a whole turn of 360 degrees.

Estimating a size before measuring it is a skill that improves quickly with a few minutes of practice a day.

The three groups and their fences — Under 90, over 90, over 180 — Acute is under 90. Obtuse is over 90 but under 180. Reflex is over 180 but under 360. The fence posts themselves belong to nobody.

Finding a reflex angle — Whole turn take away — Measure the smaller angle at the vertex, then subtract it from 360. The answer is the reflex angle on the other side.