Lines and Angles
Angles are everywhere — clocks, doors, swings, even the corner of your book! Drag the angle-maker below and watch the number change. Every book question is solved here too.
Points, lines and rays
| Name | What it is | How we write it |
|---|---|---|
| Point | An exact location. Just a tiny dot — imagined to have no size at all. | A single capital letter, like P |
| Line segment | The shortest path between two fixed points. It has a start AND an end. | AB (with a bar on top) |
| Line | A line segment stretched out forever in both directions. No start, no end. | AB (with arrows on top) |
| Ray | Starts at ONE point and goes on forever in one direction only — like a beam of light from a torch. | AP (with an arrow, starting at A) |
A ray is like a torch beam, a lighthouse beam, or sun rays — they start at one place and go on and on. A line segment is the crease you get when you fold a piece of paper — it has a clear start and end.
How many different lines can you draw through one point? Infinite! You can spin a line around that point forever. But how many lines pass through two fixed points? Only one — that's why "two points determine a unique line."
📘 Figure it Out — Section 2.4 (p.15–16)
The line segments are LM, MP, PQ, QR.
L and R sit on exactly one segment each (they're the two ends). M, P and Q each sit on two segments — they're the "joints" in the middle of the zig-zag.
The rays are TA, TB, TN and NB.
No — T is the starting point of TA, TB and TN, but NOT of NB (that ray starts at N instead)!
(a) Draw ray OP and ray OQ starting from the same point O, pointing in different directions — that shared point O is where they "meet".
(b) Draw line XY and line PQ crossing each other, and label the crossing point M.
(c) Draw a line l with E and F marked on it, and mark D as a separate dot NOT touching the line.
(d) Draw line segment AB, and mark point P somewhere in between A and B, right on the segment.
Five points: D, E, O, B, C
A line: the line through D, E, O (can be named DE or DO or EO, etc.)
Four rays: OC, OB, OE, OD (there are other correct choices too!)
Five line segments: DE, DO, DB, EO, EB (OB and OC also work as extra options)
What is an angle?
An angle is formed by two rays that share the same starting point. That shared point is called the vertex, and the two rays are called the arms.
The size of an angle is the amount of turning (rotation) needed to move one arm onto the other. Not the length of the arms — just how much you had to turn.
🎮 Drag to make your own angle
Some kids think a longer arm means a bigger angle. It does not! Two angles can have very different arm lengths but be exactly the same size, if the amount of turning is the same. Only the turning matters.
📘 Figure it Out — Section 2.5 (p.19–20)
Yes! One of the angles is ∠BDC. Its vertex is D, and its two rays are DB and DC. Try spotting the other angles in the picture the same way — find where two edges meet, that's your vertex!
Draw point S, then two rays leaving S — one through a point labelled T, one through a point labelled R. Since both rays start at S, the angle is ∠TSR (or ∠RST) — vertex S always goes in the middle!
Because in this figure there is more than one angle at point P (for example ∠APB and ∠APC both have their vertex at P). If we just said "∠P", nobody would know which one we meant! Naming it ∠APB (with a point from each arm) makes it exact.
∠RTQ and ∠RTP — both angles share the vertex T and the arm TR, with the other arm going to Q or P.
We get 3 lines: AB, BC, CA.
These 3 points give us 3 angles: ∠ABC (or ∠CBA), ∠BCA (or ∠ACB), and ∠CAB (or ∠BAC) — one angle sitting at each corner of the triangle they form!
We get 6 lines: AB, BC, CD, DA, AC, BD.
Using A, B, C, D we can name 12 angles: ∠BAC, ∠CAD, ∠BAD, ∠ADB, ∠BDC, ∠ADC, ∠DCA, ∠ACB, ∠DCB, ∠CBD, ∠DBA, ∠CBA.
Comparing angles
🔁 Comparing angles by superimposition
Any two angles can be compared by placing one on top of the other — this is called superimposition. The trick: the vertices must overlap exactly. Once you superimpose them, it's instantly clear which angle is bigger, smaller, or if they're exactly equal.
When you superimpose them and the common vertex AND both pairs of arms lie exactly on top of each other, the angles are equal in size. This is because an equal amount of rotation is needed to move one arm onto the other in both cases.
No, it is not! For example, an 89° angle and a 91° angle look almost identical just by eye — you cannot tell them apart without measuring or superimposing them. But some angles (like a very thin one next to a very wide one) are easy to compare just by looking.
📘 Figure it Out — Section 2.6 (p.23)
Fold your paper on a slant. Where the fold line EF crosses the top and bottom edges of the paper, you get four angles: ∠AEF, ∠BEF, ∠DFE, ∠CFE.
By superimposing, you'll find ∠AEF and ∠CFE are larger than ∠BEF and ∠DFE (they're on opposite sides of the fold, so they come in equal pairs). Try folding a few different ways and comparing each time!
(a) ∠AOB is greater. ∠XOY is just one small acute slice, but ∠AOB = ∠AOX + ∠XOY + ∠YOB, so it's built from more turning.
(b) ∠AOB is greater — ∠AOB contains ∠XOB inside it, plus the extra bit ∠AOX.
(c) Neither — they're equal! ∠XOB = ∠XOC in this figure (C sits along the same ray as B from O's point of view in this case).
By just looking, we cannot say for sure — unlike Q2, these two angles are separate figures, not built out of each other. We NEED to either superimpose them (trace one and place it on the other) or measure them with a protractor to know for certain.
🦆 Comparing angles WITHOUT superimposition
Two cranes are arguing about who opens their beak wider — who makes the bigger angle? We could trace and superimpose their angles... but here's a slicker trick using a transparent circle (like a clear plastic sheet with a circle drawn on it)!
Because the circle is the same size every time, matching one arm and comparing where the other arm lands on the circle's edge tells you exactly which angle "ate up" more of the circle — without ever stacking the two original angles on top of each other!
Making rotating arms & the slit game
✂️ Making "rotating arms"
🕳️ Passing through a slit
Now for a fun test of the "amount of turning" idea. Take a piece of cardboard and cut an angle-shaped slit in it, by tracing around one of your rotating arms. Mix up all your rotating arms — can you tell which ones will fit through the slit without unfolding them?
| What happens | What it means |
|---|---|
| Slit angle is greater than the arms' angle | The arms will NOT go through — too narrow, gets stuck |
| Slit angle is less than the arms' angle | The arms will NOT go through — too wide to fit |
| Slit angle is equal to the arms' angle | The arms slide through perfectly! |
Only the pair of arms whose angle exactly matches the slit will pass through. This proves that whether the arms fit depends only on the angle between them — not on how long the straws are (as long as they're shorter than the slit)! Even if you try to "reduce" the angle by pushing the arms in from a funny direction, the angle itself stays exactly the same — only actually rotating the straws changes the angle.
Measuring angles in degrees
We can compare two angles by placing one on top of the other (called superimposition) — but to give an angle an exact number, mathematicians divide a full circle into 360 equal parts. Each part is called one degree, written 1°.
Nobody knows for sure why 360 was chosen, but it goes back to ancient times! The Rigveda — one of the oldest texts belonging to humanity, thousands of years old — actually speaks of a wheel with 360 spokes (Verse 1.164.48). Many ancient calendars (India, Persia, Babylon, Egypt), all more than 3000 years old, were also often based on 360-day years. Babylonian mathematicians also loved using 60s and 360s because of how they counted numbers.
But the most useful, practical reason 360 stuck around: it's the smallest number that divides evenly by almost every number from 1 to 10 (except 7)! That makes it super easy to split a circle into neat, whole-number pieces.
Bonus fact: 360 is also evenly divisible by 12 (the number of months in a year) and by 24 (the number of hours in a day) — which makes 360° even more handy!
| How much of a full turn | Degrees | Name |
|---|---|---|
| A full turn (all the way round) | 360° | Full angle |
| Half a turn | 180° | Straight angle |
| A quarter turn | 90° | Right angle |
Divide 360° by how many parts you want:
Every single one comes out as a whole number! That's exactly why 360 was chosen.
The five kinds of angles
📐 Straight angles & right angles — where they come from
Remember Vidya opening her book cover? When she opens it all the way flat on the table (a half turn), the two "arms" of the angle end up pointing in exactly opposite directions, lying in one straight line. This is called a straight angle — it always measures 180°.
A right angle looks like the shape of the letter "L". Two lines that meet each other at a right angle (90°) are called perpendicular lines. You'll see this word a LOT in geometry — the corner of your notebook page, the edges of a window, and the crossing lines on graph paper are all perpendicular to each other!
A right angle is 1/4 of a full turn (since two right angles = one straight angle = half a turn, one right angle = half of a half turn = a quarter turn). Check: 360° ÷ 4 = 90° ✓
📘 Figure it Out — Section 2.8, right angles (p.29–30)
Most classroom windows are rectangles, so they usually have 4 right angles (one at each corner)! Look around: doors, book covers, desk corners, floor tiles, and the edges of the blackboard are all usually full of right angles too.
A straight angle needs a line that passes exactly THROUGH point A, with two grid points lying on opposite sides in a perfectly straight row (horizontal, vertical, or diagonal through A) — each such straight row through A gives you one way to draw a straight angle.
Take any straight angle line through A (from Q2), then find the grid direction that is exactly perpendicular to it (cuts it into two equal halves). On a normal square grid, if one straight line through A is horizontal, the perpendicular one is vertical — that gives you a right angle. Try each straight-angle direction from Q2 and find its perpendicular partner!
(a) Four right angles. Each one is exactly ¼ of the complete angle (360°) around the point where the two creases cross — because the second crease was made perpendicular to the first, splitting each straight angle into two equal 90° halves.
(b) Make any slanting fold first. Then fold the paper again so that one half of the FIRST crease lands exactly on top of its other half (fold the slanting crease onto itself) — the new crease this makes is automatically perpendicular to the first one!
Acute means "sharp" and obtuse means "blunt" — the names literally describe how the angle looks!
| Type | Size | Looks like |
|---|---|---|
| Acute | Less than 90° | A sharp, narrow "V" |
| Right | Exactly 90° | A perfect corner, like a book |
| Obtuse | Between 90° and 180° | A wide, blunt opening |
| Straight | Exactly 180° | A perfectly flat line |
| Reflex | Between 180° and 360° | The "outside", bigger part of an angle |
Just walk up the number line: small → acute. Exactly 90 → right. Getting wider → obtuse. Exactly flat (180) → straight. Bigger than flat → reflex. They're simply five stops along the journey from 0° to 360°.
📘 Figure it Out — Classifying Angles (p.31)
Window corners and folded perpendicular creases are usually right angles (90°). A book opened flat on the table is a straight angle (180°). A book opened just a little is acute, and opened wide (more than a quarter but less than half-open) is obtuse. Go through each earlier figure and sort it into one of the five types!
Draw several angles under 90° (sharp, narrow V-shapes) pointing up, down, sideways, and tilted — those are all acute. Then draw several between 90° and 180° (wide, blunt openings) in different directions — those are all obtuse. The direction an angle "faces" doesn't change its type, only the amount of turning does!
In an acute angle the two arms open only a little — the corner looks pointy and sharp, like the tip of a sharp pencil. In an obtuse angle the arms open wide — the corner looks blunt and rounded-off, like the tip of a blunt/worn-down pencil. The words describe exactly how "pointy" or "blunt" the angle's corner looks!
Counting carefully in each figure:
Yes — there's a clear pattern! Each new figure adds exactly 9 more acute angles than the one before it (3, then +9 = 12, then +9 = 21, then +9 = 30…). The official NCERT solution also writes this using the count of small "inner" triangles n = 0, 1, 2, 3…, as 9n + 3 acute angles.
⏰ Angles hiding in a clock
The clock face is divided into 12 hour-marks around 360°, so each hour mark = 30° apart.
Just multiply the hour number by 30°!
A door opening makes an angle — the vertex is at the hinge. A swing makes an angle with the tree branch — the wider you start, the faster you go! Scissors and compasses work the same way. Look around your room — angles are everywhere.
Using a protractor
A protractor is a tool shaped like a half-circle, marked from 0° to 180°. It has two rows of numbers — one counting up from left to right, one counting up from right to left. Always check which row matches your angle's starting arm!
If your angle's base arm points to the right (like most angles), read the numbers that go 0→180 left to right. If the base arm points left, use the other row. Mixing them up is the single most common mistake — always double check by asking "is this angle bigger or smaller than a right angle (90°)?" first, as a sanity check.
📘 Figure it Out — the unlabelled protractor (p.35)
On an unlabelled protractor, you count the little 1° marks yourself — long marks every 10°, medium marks every 5° in between.
Yes, using the medium (5°) and long (10°) marks, you can count in jumps of 5 or 10 instead of one-by-one — much faster than counting every single 1° line!
🔢 The labelled protractor — 13 angles from one picture!
A real geometry-box protractor has points marked around its curved edge (say P, Q, R, S, T, U) all measured from the centre O. You don't just get ONE angle from this — you can pair up any two of those points with O to make a brand-new angle!
Reading straight off the protractor scale, here are all 13 angles:
Yes, ∠TOQ is one of them too (same as ∠QOT = 125°)! Notice you can find ANY of these just by subtracting two readings on the same scale — e.g. ∠TOS = reading at T minus reading at S on the same row of numbers — you don't have to count 1° marks one by one.
✏️ How to DRAW a 30° angle, step by step
"Bisect" means to cut exactly in half. An angle bisector is the line that splits one angle into two perfectly equal angles. You can find it by folding paper so one arm lands exactly on the other — the fold crease is the bisector!
Make your own protractor!
Each of these angles = 22.5°.
Why? The straight angle of 180° was folded in half three times in a row (into 8 equal slices), so each tiny angle is
At each folding step, we got half of the previous angle. This process of getting exactly half of a given angle is called bisecting the angle, and the fold line is the angle bisector. Try spotting all the angle bisectors hiding in your handmade protractor!
📘 Figure it Out — using your handmade protractor (p.40–42)
Line up your paper protractor's centre and 0° edge the same way you would with a real one!
This is an open explore-your-classroom activity — try the door, a book cover, the clock hands, a window corner, and compare your paper-protractor reading with what a real protractor gives!
No — your handmade paper protractor only has marks every 22.5°, so it CANNOT give you these exact readings. You'd need a real, finely-marked protractor here.
Use the "leftover of the full circle" trick:
260° — a full turn around a point is always 360°, so whatever angle is NOT marked, subtract it from 360° to find the marked (reflex) one.
This is a hands-on craft activity — fold a square sheet through the 8 steps shown in the book to make a bunny face, then unfold it completely. Draw a line along every crease you find, and use your protractor to measure each angle those crease-lines make. You'll notice lots of 45° and 90° angles, since square-paper folds usually create halves and quarters of a right angle!
🎨 Mind the Mistake, Mend the Mistake!
A student measured six angles (U, V, W, X, Y, Z) with a protractor but made reading mistakes on some of them. Can you spot what went wrong?
The classic mistake is reading the wrong row of numbers — a protractor has an inner scale (counting one way) and an outer scale (counting the other way). If the base arm points right, you must use the row that starts at 0 under that arm and increases towards the other arm — NOT the other scale's number that happens to sit at the same mark! Always sanity-check: does the angle LOOK acute or obtuse? If your reading disagrees with what your eyes tell you, you probably used the wrong scale — go back and re-read using the scale that starts at 0° on the base arm.
🎮 Let's Play a Game! (angle-guessing games)
Split into Team 1 and Team 2. Team 1 secretly draws an angle with a protractor (say 49°) without Team 2 seeing the measurement. Team 2 looks at it and guesses the degree measure WITHOUT a protractor. Team 1 then reveals the true measure.
Team 2's score = the absolute difference between their guess and the real answer. Guess 39° when the real answer is 49°? That's 49° - 39° = 10 points. Five rounds each — lowest total score wins!
This time, Team 1 announces a number out loud (say 34°). A Team 2 player must draw that exact angle on the board without a protractor, while teammates shout "bigger!" or "smaller!" to help. Team 1 then measures it for real.
Same scoring: if the drawn angle measures 25° but the target was 34°, Team 2 scores 34° - 25° = 9 points. Five rounds, lowest score wins!
Where are the angles?
📘 Figure it Out (p.45)
The 12 hour-marks are spread evenly around the centre of the clock (360° total), so each gap between successive numbers is 360° ÷ 12 = 30°. That's why 1 o'clock (1 gap from 12) is exactly 30°.
Try 5, 7, 8, 10, and 11 o'clock too — just multiply the hour number by 30°!
Yes! The vertex is the hinge — the point where the door meets the wall. The two arms are the edge of the door and the edge of the wall/door-frame. The wider the door swings open, the bigger the angle between them.
It's a little trickier to see, but it's there! If we fix the starting/resting position of the swing as one arm, the angle is between that resting position and the highest point she swings up to on one side — the vertex is up at the top where the swing ropes are attached.
Yes, angles describe the slope directly — a bigger angle means a steeper slab. For each angle, one arm is the slanting edge of the slab (this one IS visible), and the other arm is an imaginary horizontal/vertical reference line (this one is NOT visible — you have to imagine it).
Yes! Picture a horizontal line touching the ground/base of the original insect — that's one arm. The same line, now touching the rotated insect's base, is the other arm. The vertex is the point they rotate around. The angle between the two positions of that reference line tells you exactly how far the insect was rotated!
Drawing angles (Section 2.10)
We already saw the 4-step method for drawing a 30° angle. Vidya draws ∠TIN this way: keep IN as the base (reference) arm, then rotate the other arm IT by exactly 30° from it.
📘 Figure it Out (p.49)
Some of the angles you should be able to find: ∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS, ∠CLP — and there are more! Try to spot every vertex where two lines cross.
Make a table with columns "Angle name", "My guess", "Measured value" — the more you practice guessing before measuring, the better your angle-estimating eye gets!
Use the same 4-step method every time: draw a base ray → put the protractor's centre on the start point, 0° lined up with the ray → count round to the target number and mark a dot → join with a ruler. All five of these (110°, 40°, 75°, 112°, 134°) work exactly the same way — only 40° and 75° are acute (under 90°), the rest are obtuse.
Types of angles & their measures (2.11)
We've already met all five types by name. Here's the precise degree-range definition for each, straight from the book:
| Type | Exact degree range |
|---|---|
| Acute | More than 0° and less than 90° |
| Right | Exactly 90° |
| Obtuse | More than 90° and less than 180° |
| Straight | Exactly 180° |
| Reflex | More than 180° and less than 360° |
📘 Figure it Out (p.51–52)
(a) Pick two rays from A that are close together (a small gap) — under 90°, that's acute. (b) Pick two rays with a wider gap — between 90° and 180°, that's obtuse. (c) For reflex, you have to mark the curve on the outside, bigger part between the two rays (over 180°) — the same two rays can show either the small obtuse/acute angle OR the big reflex angle, depending on which side you curve the mark!
Notice ∠PTW and ∠WTP together make a full circle around T: 102° + 258° = 360° — they're the "small side" and "big side" of the exact same two rays!
💡 Let's Explore — the full ∠TER puzzle
∠TER = 80° and ∠REB is a straight angle (so R, E, B are in a line). What's the measure of ∠BET? And — here's the second part many miss — what's the measure of ∠SET, given that OS is perpendicular to REB (so ∠SEB = 90°)?
Part 1 — ∠BET: ∠REB is a straight angle (180°), and ∠TER (80°) is part of it.
Part 2 — ∠SET: Ray ES sits between ray ET and ray ER, and ES is perpendicular to the line REB, which means ∠SER = 90° exactly. Since ∠SER is made up of the smaller ∠SET plus the known ∠TER:
∠BET = 100° and ∠SET = 10° — both use the same trick: find the full angle you know (180° or 90°), then subtract the part you're given!
Every question from the book
Rihan: infinite lines can pass through one point (spin it around). Sheetal: exactly ONE line can pass through two fixed points.
Yes, OB works — the ray is named by its starting point and ANY point along its path, so both OA and OB name the same ray.
But OA and AO are NOT the same — a ray's starting point matters! Ray OA starts at O; ray AO would start at A and go the opposite way.
The pattern adds 9 more acute angles each time (3, 12, 21, then the next would be 30) — this is the same "splitting triangles" question from Section 6 (Kinds of Angles), where we worked out the full rule!
A straight angle is always 180°. ∠TER (80°) is PART of the straight angle ∠REB. So the rest of it, ∠BET, is:
∠BET = 100° — this trick (straight angle minus the known part) works any time one angle sits along a straight line!
Reflex means between 180° and 360°. Check each: 140° (obtuse), 82° (acute), 195° (reflex! it's over 180°), 70° (acute), 35° (acute).
Only 195° is a reflex angle — a full protractor only goes to 180°, so for 195° you measure 180° and then 15° more past it, OR measure 360°−195°=165° from the other side and mark outside.
They always add up to exactly 180° — no matter how the triangle is shaped! Big, small, tall, flat — every triangle's three angles always total 180°. This is one of the most famous facts in all of geometry, and you just discovered it yourself by measuring.
📘 Final Figure it Out (p.53)
Draw each with the base-ray + protractor + ruler method. Remember: (a) 140° obtuse, (b) 82° acute, (c) 195° is a reflex angle (over 180°, so mark the "outside" curve, or measure 180° and add 15° more), (d) 70° acute, (e) 35° acute.
This is a hands-on estimate-then-measure activity using figures from the book — first write down your GUESS for each angle a-f, then check with a real protractor and classify each one. The goal is training your eye to estimate angles accurately, just like in the guessing games!
Draw any closed shape (like a 6-sided polygon) with 6 marked angles A-F: make ∠A, ∠B, ∠C each less than 90° (acute), ∠E exactly 90° (right), and ∠D, ∠F each between 90° and 180° (obtuse). There's no single "correct" shape — any figure meeting those rules works!
An 'M' has 3 corner-angles: two outer "V" shapes and one middle "V" (upside down) where the strokes meet. Draw the two outer strokes each making a 40° angle with the vertical, and the middle dip making a 60° angle at its point — use your protractor at each corner to get the exact measures.
A 'Y' has two upper arms meeting a lower stem. Where the left arm meets the stem, make a 150° angle; where the right arm meets the stem, another 150° angle; and between the two upper arms at the very top, a 60° angle. Check with your protractor: 150° + 60° + 150° = 360°, a full turn around the meeting point — that's why it works out!
24 spokes are spread evenly around the full 360° wheel:
The largest ACUTE angle between any two spokes must be the biggest multiple of 15° that's still under 90°:
Angle between adjacent spokes = 15°. Largest acute angle between two spokes = 75° (that's the gap spanning 5 adjacent 15° slices: 5 × 15° = 75°).
Call the angle x. We need:
So x must be a whole number of degrees with 18° < x < 22.5°:
Check x = 22°: 4×22° = 88° (acute ✓), 5×22° = 110° (obtuse ✓). Check x = 19°: 4×19° = 76° (acute ✓), 5×19° = 95° (obtuse ✓). Both work, and so do 20° and 21°!
Practice like the real exam
Section A · MCQ (1 mark each)
(b) right
(c) 360°
Section B · short answer (2 marks each)
Each hour mark is 30°. At 8 o'clock, the smaller angle between the hands going one way is 8 × 30° = 240° — but that's a reflex angle. Going the shorter way round is 360° - 240° = 120°.
120° (the smaller, obtuse angle)
Section C · longer answer (3 marks each)
A line needs a direction as well as a location. One point only fixes a location — you can still spin a line around it in any direction, so infinite lines pass through it. But once you have TWO points, the direction is also fixed (the line must pass through both), so exactly one straight line can join them.
Section D · 4 marks each
Draw the base ray, mark 55° with the protractor, join with a ruler (same 4-step method from Section 5). To bisect it, fold so one arm lands on the other, or use the protractor again to mark half of 55°.
Each half measures 27.5°.
Section E · case study (4 marks)
A straight angle is always 180° total, split into parts here.
The unknown angle is 35° — this is the same "straight angle minus known parts" trick from Section 6.
You did it! 🎉
🏁 Chapter 2 of 5 · Term 1 Maths · Niyati, Class 6