[{"data":1,"prerenderedAt":1116},["ShallowReactive",2],{"layer:angles:discover":3},{"layer":4,"contentHash":1093,"dependencyHashes":1094,"approval":1110,"releaseId":1115},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1088,"reviewStatus":1089,"authoring":1090},1,"angles","en","discover","Angles are turns","Doors, clocks, scissors and compass directions: meet the angle and learn to name its size","See an angle as a turn and as two arms meeting at a vertex. Measure turns in degrees (full 360°, half 180°, quarter 90°), sort angles into seven types, turn through N, E, S, W, read angles on a clock and meet angle partners.",[13,14,15,16,17],"Describe an angle as an amount of turn and as two rays with a common end point, naming the vertex and arms.","Explain why arm length does not change the size of an angle.","Use full, half and quarter turns (360°, 180°, 90°) and simple fractions of a turn.","Recognise zero, acute, right, obtuse, straight, reflex and complete angles in real objects.","Find angles between clock hands on the hour and describe turns between N, E, S and W.",35,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 35 minutes",{"label":29,"value":30},"Prior knowledge","Straight lines; counting in 30s",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Angle lab ×2, angle pairs, sort game, match game",{"label":38,"value":39},"Units used","degrees (°)",[41,45,51,72,78,81,143,146,161,166,169,174,204,207,212,217,240,245,248,272,317,322,335,350,355,358,362,365,370,373,402,407,424,435,451,456,459,492,496,508,520,593,598,601,625,629,639,656,661,663,702,707,718,738,743,746,755,768,780,785,845,849,853,940,1040,1054,1060,1065,1070,1074],{"id":42,"type":43,"markdown":44},"intro-door","prose","Push open your classroom door and stop halfway. Now push it until it touches the wall. The door did not get longer, and it did not move to a different place: it **turned** around its hinges. How much it turned is what mathematicians call an **angle**.\n\nAngles are hiding everywhere once you start looking. The hands of a clock make one at every moment. A pair of scissors makes one that grows as you open it. A cricketer's bat makes one with the ground, a slice of pizza makes one at its tip, and your elbow makes one every time you lift a glass of water.\n\nIn this layer you will learn to **see** angles, to **compare** them, to give them **names** and to **measure** turns in a special unit called the **degree**. No difficult symbols, just turning, folding and looking.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this layer","Read the chapters in order the first time. When you meet a **prediction**, choose an answer before reading on. When you meet a **lab**, play with it: you learn angles with your hands and eyes, not just by reading. Keep a sheet of paper nearby for folding.",{"id":52,"type":53,"tone":54,"items":55},"plate-spec","spec","blue",[56,60,64,68],{"label":57,"big":58,"value":59},"Big idea","Angle = turn","An angle tells you how much something has turned, or how far apart two straight arms are opened.",{"label":61,"big":62,"value":63},"Unit","degree (°)","A full turn is split into 360 equal small turns. Each one is 1 degree, written 1°.",{"label":65,"big":66,"value":67},"Landmarks","90° · 180° · 360°","Quarter turn, half turn, full turn.",{"label":69,"big":70,"value":71},"Tools you need","paper + eyes","A sheet of paper to fold, a clock face and your own arms.",{"id":73,"type":74,"title":75,"eyebrow":76,"navLabel":77},"ch01","chapter","Turning things: doors, clocks and scissors","Chapter 01","1 Turning things",{"id":79,"type":43,"markdown":80},"turning-things","Look around you and find something that **turns about a fixed point**:\n\n- A **door** turns about its hinges.\n- The **hands of a clock** turn about the centre pin.\n- **Scissors** open about the little screw in the middle.\n- A **ceiling fan** blade turns about the motor at its centre.\n- Your **forearm** turns about your elbow.\n\nIn every case there is a **fixed point** that does not move (the hinge, the pin, the screw, the elbow) and something that swings around it. The angle is the amount of swing.\n\nA door that has opened a little has made a **small angle**. A door flung open against the wall has made a **big angle**. So even before we have numbers, we can already say which of two angles is bigger.",{"id":82,"type":83,"title":84,"prompt":85,"options":86},"explorer-turners","explorer","Things that turn","Pick an everyday object to see where its fixed point is and what makes the angle.",[87,99,110,121,132],{"id":88,"label":89,"chain":90,"badge":95,"note":98},"door","A door",[91,92,93,94],"Fixed point: the hinges","Arm 1: the door frame","Arm 2: the door","Angle: how far it is open",{"text":96,"tone":97},"Angle grows as it opens","yes","A closed door makes no angle with the frame. Open it a little and a thin wedge of space appears. Open it until it touches the wall and the angle is very large, often more than a right angle. Many doors are stopped by a door stopper so they never open wider than a chosen angle.",{"id":100,"label":101,"chain":102,"badge":107,"note":109},"clock","Clock hands",[103,104,105,106],"Fixed point: the centre pin","Arm 1: the hour hand","Arm 2: the minute hand","Angle: the gap between hands",{"text":108,"tone":97},"Changes every minute","At 12 o'clock the two hands lie on top of each other: no gap at all. At 3 o'clock they make a perfect square corner. At 6 o'clock they point in opposite directions and form a straight line. The angle between the hands keeps changing all day.",{"id":111,"label":112,"chain":113,"badge":118,"note":120},"scissors","Scissors",[114,115,116,117],"Fixed point: the screw","Arm 1: one blade","Arm 2: the other blade","Angle: how wide they open",{"text":119,"tone":97},"Two angles at once","Open scissors and look closely: the blades make an angle on one side of the screw, and the handles make an angle on the other side. Those two angles are always the same size. You will find out why in the Understand layer, where they are called vertically opposite angles.",{"id":122,"label":123,"chain":124,"badge":129,"note":131},"fan","Ceiling fan",[125,126,127,128],"Fixed point: the motor","Arm: one blade","It turns round and round","Full turns, many per second",{"text":130,"tone":97},"Keeps turning past a full turn","A fan blade does not stop at a door-sized angle. It keeps going, all the way round and back to where it started, again and again. One trip all the way round is called a full turn. A fan on high speed makes several full turns every second.",{"id":133,"label":134,"chain":135,"badge":140,"note":142},"elbow","Your elbow",[136,137,138,139],"Fixed point: the elbow joint","Arm 1: upper arm","Arm 2: forearm","Angle: how bent your arm is",{"text":141,"tone":97},"Your body measures angles","Hold your arm straight out and your elbow makes a straight line. Bend it to touch your shoulder and the angle becomes small. Doctors and physiotherapists measure elbow and knee angles to see how well a joint is healing after an injury.",{"id":144,"type":43,"markdown":145},"same-turn","Here is a surprising idea. Two doors can be very different: one tall and heavy, one small like a cupboard door. If both are opened by the **same amount of turn**, they have made the **same angle**. The size of the door does not matter. Only the turn matters.\n\nHold that thought: it is the single most important thing to understand about angles, and it trips up many people much older than you.",{"id":147,"type":148,"prompt":149,"options":150,"explanation":160},"predict-long-arms","prediction","Two fans are drawn on the board. Fan A has long blades and Fan B has short blades. The gap between two neighbouring blades is opened by exactly the **same turn** in both fans. Which fan shows the **bigger angle** between two blades?",[151,154,157],{"id":152,"label":153},"a","Fan A, because its blades are longer",{"id":155,"label":156},"b","Fan B, because its blades look more crowded",{"id":158,"label":159},"c","Neither: the angles are the same","**Neither: they are the same angle.** An angle measures **turn**, not length. Longer blades just make the picture bigger; the opening between the blades is the same. A useful trick: imagine shrinking Fan A like a photo on a phone. The blades get shorter, but the opening between them does not change at all.",{"id":162,"type":74,"title":163,"eyebrow":164,"navLabel":165},"ch02","Two arms and a corner","Chapter 02","2 Arms and vertex",{"id":167,"type":43,"markdown":168},"arms-vertex","Now let us draw an angle instead of turning one. Put your pencil on a dot. Draw a straight line out from the dot in one direction. Go back to the same dot and draw another straight line in a different direction.\n\nYou have drawn an **angle**. It has three parts:\n\n- The **vertex** is the corner point where the two lines start (the dot). Say it *VER-tex*. The plural is **vertices**.\n- The two straight lines are the **arms** of the angle.\n- The **opening** between the arms is the angle itself.\n\nEach arm starts at the vertex and goes off in one direction. In geometry, a straight path that starts at a point and goes on forever in one direction is called a **ray** (like a ray of sunlight from the Sun). So an angle is made of **two rays that share the same starting point**. You can learn more about rays in the [lines topic](\u002Ftopics\u002Flines).",{"id":170,"type":47,"variant":171,"title":172,"markdown":173},"def-angle","definition","Angle, in one line","An **angle** is formed when **two rays start from the same point**. The common starting point is the **vertex**, the two rays are the **arms**, and the size of the angle is the **amount of turn** from one arm to the other.",{"id":175,"type":176,"caption":177,"columns":178,"rows":183},"table-parts","table","The parts of an angle, and where to find them on real objects",[179,180,181,182],"Part","What it is","On a clock","On scissors",[184,189,194,199],[185,186,187,188],"Vertex","The corner point where the arms meet","The centre pin","The screw",[190,191,192,193],"Arm 1","One ray starting at the vertex","The hour hand","One blade",[195,196,197,198],"Arm 2","The other ray starting at the vertex","The minute hand","The other blade",[200,201,202,203],"Size of angle","How much you turn from one arm to the other","The gap between the hands","How wide the blades are open",{"id":205,"type":43,"markdown":206},"inside-outside","An angle splits the flat page around it into **two regions**:\n\n- The **inside** of the angle (its **interior**) is the space between the two arms, where the opening is.\n- The **outside** (its **exterior**) is all the rest of the page.\n\nImagine a slice of pizza. The cheesy part between the two cut edges is the **inside** of the angle at the tip. The rest of the pizza, still on the plate, is **outside** that angle. The edges of the slice, the cut lines themselves, are the arms: they are neither inside nor outside.",{"id":208,"type":47,"variant":209,"title":210,"markdown":211},"try-straws","try_it","Make an angle-maker with two straws","You need two drinking straws (or two strips of card) and a pin or a split-pin paper fastener.\n\n1. Push the pin through one end of both straws so they are joined at one point. That point is your **vertex**.\n2. Lay the straws on top of each other. The opening is zero.\n3. Slowly swing one straw away from the other. Watch the opening grow.\n4. Stop when the straws make a square corner like the corner of your notebook. Then keep going until they make a straight line.\n5. Now cut one straw shorter and repeat. The **turn** is the same even though the arm is shorter.\n\nKeep your angle-maker: you will use it again in this layer.",{"id":213,"type":47,"variant":214,"title":215,"markdown":216},"misc-arms","misconception","“Longer arms make a bigger angle”","This is the most common mistake with angles, even among older students. The arms of an angle are rays, and rays go on forever, so the length drawn on the page is just how much the artist chose to show. Two angles with the same **opening** are equal even if one is drawn with arms 1 cm long and the other with arms 10 cm long. To compare two angles, compare the **turns**, not the lines.",{"id":218,"type":219,"itemId":220,"prompt":221,"check":222,"hints":235,"feedback":237},"pr-vertex","practice","angles.disc-vertex","Scissors are open. What is the **vertex** of the angle made by the blades?",{"kind":223,"options":224,"correct":234},"choice",[225,227,229,231],{"id":152,"label":226},"The tip of one blade",{"id":155,"label":228},"The screw that joins the blades",{"id":158,"label":230},"The handle",{"id":232,"label":233},"d","The whole pair of scissors",[155],[236],"The vertex is the fixed point that the arms turn about.",{"correct":238,"incorrect":239},"Yes. The blades turn about the screw, so the screw is the vertex and the blades are the arms.","Look for the point that does not move when the scissors open. Both blades turn about the screw, so that is the vertex.",{"id":241,"type":74,"title":242,"eyebrow":243,"navLabel":244},"ch03","Full turns, half turns and quarter turns","Chapter 03","3 Measuring turns",{"id":246,"type":43,"markdown":247},"full-turn","Stand up, face the classroom door and turn all the way round until you face the door again. You have made a **full turn** (also called a **complete turn**, one **revolution** or one **rotation**).\n\nNow try these:\n\n- Turn until you face the wall **behind** you. That is a **half turn**. Two half turns make a full turn.\n- From the start, turn until you face the wall on your **right**. That is a **quarter turn**. Four quarter turns make a full turn.\n- A **three-quarter turn** takes you to the wall on your **left**, the long way round.\n\nPeople have always needed to measure turns more exactly than \"a bit\" or \"a lot\". Thousands of years ago, astronomers in ancient Babylon (in today's Iraq) are thought to have started splitting a full turn into **360 equal small turns**. We still do the same today. Each small turn is called **one degree**, written **1°**. The little circle ° is the degree sign.",{"id":249,"type":53,"tone":250,"items":251},"spec-turns","amber",[252,256,260,264,268],{"label":253,"big":254,"value":255},"Full turn","360°","All the way round, back to where you started.",{"label":257,"big":258,"value":259},"Half turn","180°","Face the opposite way: 360 ÷ 2 = 180.",{"label":261,"big":262,"value":263},"Quarter turn","90°","A square corner, a right angle: 360 ÷ 4 = 90.",{"label":265,"big":266,"value":267},"Three-quarter turn","270°","Three quarter turns: 3 × 90 = 270.",{"label":269,"big":270,"value":271},"One degree","1°","A tiny turn: 360 of them make a full turn.",{"id":273,"type":176,"caption":274,"columns":275,"rows":280},"table-fractions-turn","Fractions of a full turn in degrees (each worked out as fraction × 360°)",[276,277,278,279],"Turn","Fraction","Degrees","Where you might see it",[281,285,289,293,298,302,307,312],[282,283,254,284],"Full","1","A fan blade going all the way round once",[286,287,266,288],"Three-quarter","¾","Facing north, turning clockwise to face west",[290,291,258,292],"Half","½","An about-turn in the school parade",[294,295,296,297],"Third","⅓","120°","The angle between two blades of a three-blade fan",[299,300,262,301],"Quarter","¼","The corner of a book or a tile",[303,304,305,306],"Sixth","⅙","60°","One slice of a cake cut into 6 equal pieces",[308,309,310,311],"Eighth","⅛","45°","One slice of a pizza cut into 8 equal pieces",[313,314,315,316],"Twelfth","1⁄12","30°","From one number to the next on a clock face",{"id":318,"type":47,"variant":319,"title":320,"markdown":321},"aha-360","aha","Why 360 is such a friendly number","360 can be divided exactly by **2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180**. That means you can cut a full turn into halves, thirds, quarters, fifths, sixths, eighths, tenths or twelfths and always get a whole number of degrees. If a full turn were 100 units, a third of a turn would be 33.333… units, which is messy. The Babylonians chose well.",{"id":323,"type":324,"title":325,"problem":326,"steps":327,"help":332},"we-eighth","worked_example","How many degrees in one slice of pizza?","A round pizza is cut into **8 equal slices**. What is the angle at the tip of each slice?",[328,329,330,331],"All the slices together go all the way round the centre, so together they make a **full turn: 360°**.","The 8 slices are equal, so share 360° equally among 8.","360 ÷ 8 = **45°**.","Check: 8 × 45 = 360. ✓ Each slice has a 45° tip.",{"simplerExplanation":333,"anotherExample":334},"Cut the full turn of 360° into 8 equal parts: 360 ÷ 8 = 45. Each slice is 45°.","A cake cut into 6 equal pieces: 360 ÷ 6 = 60°, so each piece has a 60° tip.",{"id":336,"type":219,"itemId":337,"prompt":338,"check":339,"hints":344,"feedback":347},"pr-thirds","angles.disc-thirds","A ceiling fan has **3 blades**, spaced equally. What is the angle between two neighbouring blades, in degrees?",{"kind":340,"answer":341,"tolerance":342,"unit":343},"number",120,0,"°",[345,346],"Together the three gaps go all the way round.","360 ÷ 3 = ?",{"correct":348,"incorrect":349},"Right: 360 ÷ 3 = 120°. Three gaps of 120° make the full 360°.","The three equal gaps make one full turn of 360°. Share it: 360 ÷ 3 = 120°.",{"id":351,"type":74,"title":352,"eyebrow":353,"navLabel":354},"ch04","The right angle: a square corner","Chapter 04","4 Right angles",{"id":356,"type":43,"markdown":357},"right-angle","A **quarter turn** is so important that it has its own name: a **right angle**. It measures exactly **90°**.\n\nYou see right angles all around you: the corners of your notebook, a window pane, a floor tile, a door frame, a cricket stump standing straight up from flat ground, the plus sign +. Builders care a lot about right angles, because a wall that is not at a right angle to the floor might lean and fall.\n\nIn drawings, a right angle is marked with a **small square** in its corner, instead of the curved arc we use for other angles. When you see that little square, you know the angle is exactly 90° without measuring.",{"id":359,"type":47,"variant":209,"title":360,"markdown":361},"try-fold","Fold your own right-angle checker","Take any scrap of paper, even one with a torn, wavy edge.\n\n1. Fold it once, anywhere. Press the fold flat. The fold is a **straight edge**.\n2. Now fold it again so that the first fold lies exactly on top of itself. Press flat.\n3. The corner where the two folds meet is a **perfect right angle**, 90°.\n\nWhy does it work? The first fold makes a straight line, which is a half turn (180°). The second fold cuts that half turn exactly in half, and half of 180° is 90°.\n\nNow walk around your home and test corners: a book, the TV, a tile, a tabla's top edge against its side. Which corners are exactly right angles? Which are a bit more or a bit less?",{"id":363,"type":43,"markdown":364},"smaller-bigger","Your right-angle checker can do more than find right angles. Hold it against any corner:\n\n- If the corner fits **inside** your checker with space left over, the angle is **smaller than a right angle**.\n- If it fits exactly, it is a **right angle**.\n- If it is wider than your checker, the angle is **bigger than a right angle**.\n\nThis simple test sorts every angle into groups. The groups have names, which you will meet in the next chapter.",{"id":366,"type":74,"title":367,"eyebrow":368,"navLabel":369},"ch05","Sharp and blunt: acute and obtuse","Chapter 05","5 Acute and obtuse",{"id":371,"type":43,"markdown":372},"acute-obtuse","An angle **smaller than a right angle** (more than 0° but less than 90°) is called an **acute angle**. *Acute* comes from a Latin word meaning **sharp**, like the point of a needle. The tip of a pizza slice, the V of a bird's beak and the letter V are acute angles.\n\nAn angle **bigger than a right angle but smaller than a straight line** (more than 90° but less than 180°) is called an **obtuse angle**. *Obtuse* means **blunt**. A laptop screen tipped back to look at, a reclining chair, and the corner of a stop-sign shape are obtuse angles.\n\nWhen the two arms point in exactly **opposite directions** and make a straight line, the angle is a **straight angle**: exactly **180°**, a half turn. A book lying open flat on the table makes a straight angle along its spine.",{"id":374,"type":176,"caption":375,"columns":376,"rows":381},"table-types-simple","Sorting angles by comparing them with a right angle and a straight line",[377,378,379,380],"Name","Size","Compared with a right angle","Everyday example",[382,387,392,397],[383,384,385,386],"Acute","between 0° and 90°","Smaller: fits inside the square corner","The tip of a slice of cake",[388,389,390,391],"Right","exactly 90°","Exactly the same","The corner of a notebook",[393,394,395,396],"Obtuse","between 90° and 180°","Bigger, but not yet a straight line","An open laptop tilted back",[398,399,400,401],"Straight","exactly 180°","Two right angles side by side","A book lying open flat",{"id":403,"type":47,"variant":404,"title":405,"markdown":406},"memory-acute","example","A way to remember","**A-cute** angle is a small, *cute* angle. **Obtuse** sounds big and heavy. And a **right** angle is *right* in the middle between them, standing up straight like a stump in the ground.",{"id":408,"type":409,"component":410,"componentVersion":5,"config":411,"objective":417,"textAlternative":418,"help":419},"lab-explore","interactive","angle-lab",{"modes":412,"allowReflex":415,"rounds":416},[413,414],"explore","classify",false,6,"Drag an arm to make angles from 0° to 180° and watch the name change, then name randomly tilted angles.","One arm points to the right from a corner (the vertex); you turn the other arm. Drag the moving arm, or use the arrow keys (1° per press, 10° with Shift); quick buttons jump to 0°, 90° and 180°. A readout shows the angle in degrees and its type, and the arc is coloured by type.\n\nAt **0°** the arms lie on top of each other: a **zero angle**. From 1° to 89° the name is **acute**. At exactly **90°** a small square appears in the corner: a **right angle**. From 91° to 179° the name is **obtuse**, and at **180°** the arms make a line: a **straight angle**. In this lab the arm cannot go past 180°.\n\nIn classify mode the lab shows angles one at a time, tilted at random, and you name each as zero, acute, right, obtuse or straight. For example, 35° and 72° are acute, 115° and 140° are obtuse. Tip: first compare with a square corner (90°), then with a straight line (180°). Tilting does not change the type.",{"simplerExplanation":420,"hints":421},"Smaller than a square corner: acute. Exactly a square corner: right. Bigger, but not flat yet: obtuse. Flat like a line: straight.",[422,423],"Picture the corner of your notebook. Is the angle smaller or bigger?","Watch the name change exactly at 90° and 180°.",{"id":425,"type":148,"prompt":426,"options":427,"explanation":434},"predict-clock-3","At **3 o'clock**, what kind of angle do the hour hand and minute hand make?",[428,429,431,432],{"id":152,"label":383},{"id":155,"label":430},"Right angle",{"id":158,"label":393},{"id":232,"label":433},"Straight angle","**A right angle.** At 3:00 the minute hand points straight up to 12 and the hour hand points straight across to 3. From 12 to 3 is **a quarter of the way round** the clock, and a quarter turn is 90°.",{"id":436,"type":219,"itemId":437,"prompt":438,"check":439,"hints":446,"feedback":448},"pr-classify-110","angles.disc-classify-110","What type of angle measures **110°**?",{"kind":223,"options":440,"correct":445},[441,442,443,444],{"id":152,"label":383},{"id":155,"label":388},{"id":158,"label":393},{"id":232,"label":398},[158],[447],"Is 110 more or less than 90? Is it less than 180?",{"correct":449,"incorrect":450},"Yes. 110° is more than 90° but less than 180°, so it is obtuse.","110° is bigger than a right angle (90°) but smaller than a straight angle (180°). That makes it obtuse.",{"id":452,"type":74,"title":453,"eyebrow":454,"navLabel":455},"ch06","Beyond a straight line: reflex, complete and zero","Chapter 06","6 Reflex and complete",{"id":457,"type":43,"markdown":458},"reflex","What if you keep turning past a straight line? Open a door all the way round... well, doors cannot, but a clock hand can, and so can your angle-maker.\n\nAn angle **bigger than a straight angle but smaller than a full turn** (more than 180° but less than 360°) is called a **reflex angle**. Reflex angles are the \"long way round\".\n\nHere is the trick. Whenever you draw two arms from a vertex, there are **two** openings: the small one on one side and the big one going round the other side. Together they make a full turn. If the small opening is 60°, the big opening is 360° − 60° = **300°**, which is reflex. Usually when we say \"the angle\", we mean the smaller one, unless someone asks for the reflex angle.\n\n- A **complete angle** (full angle) is exactly **360°**: one full turn, ending where you started.\n- A **zero angle** is exactly **0°**: the two arms lie on top of each other and there is no turn at all.",{"id":460,"type":176,"caption":461,"columns":462,"rows":465},"table-seven-types","All seven types of angle",[463,378,276,464],"Type","Picture it",[466,471,475,477,481,483,488],[467,468,469,470],"Zero","0°","No turn","Clock hands at 12:00, one on top of the other",[383,472,473,474],"more than 0°, less than 90°","Less than a quarter turn","A slice of pizza",[388,389,261,476],"The corner of a page",[393,478,479,480],"more than 90°, less than 180°","Between a quarter and a half turn","A reclining chair",[398,399,257,482],"Clock hands at 6:00",[484,485,486,487],"Reflex","more than 180°, less than 360°","Between a half and a full turn","The outside of a Pac-Man's mouth",[489,490,253,491],"Complete","exactly 360°","A fan blade going round once",{"id":493,"type":47,"variant":319,"title":494,"markdown":495},"aha-pacman","Pac-Man is a reflex angle","Draw a circle with a small wedge cut out, like the video-game character Pac-Man. The mouth is an acute angle. The rest of Pac-Man, the part that is his body, is a **reflex angle** around the same vertex. Mouth + body = a full turn. If his mouth is open 50°, his body angle is 360° − 50° = 310°.",{"id":497,"type":409,"component":410,"componentVersion":5,"config":498,"objective":502,"textAlternative":503,"help":504},"lab-explore-reflex",{"modes":499,"allowReflex":500,"rounds":501},[413,414],true,8,"Turn the arm all the way round from 0° to 360° to meet reflex and complete angles, then name randomly tilted angles.","The same angle-maker, but now the moving arm can turn all the way round. Drag the moving arm, or use the arrow keys (1° per press, 10° with Shift); quick buttons jump to 0°, 90° and 180°, with extra buttons for 270° and 360°.\n\nPast **180°** the name becomes **reflex**: 200°, 270° and 330° are all reflex angles. At 270° the moving arm has made three quarter turns. At exactly **360°** it is back on top of the fixed arm after a full turn: a **complete angle**. The arc shows which opening is being measured; at 300° it is the big opening, and the small opening left over is 360° − 300° = 60°.\n\nIn classify mode, randomly tilted angles appear and you choose from all seven types: zero, acute, right, obtuse, straight, reflex and complete. For a big opening, check: is it more than a straight line? Then it is reflex.",{"simplerExplanation":505,"hints":506},"Past a straight line but not yet all the way round is reflex. All the way round is complete.",[507],"If the angle is more than 180°, check that it is less than 360° before saying reflex.",{"id":509,"type":219,"itemId":510,"prompt":511,"check":512,"hints":514,"feedback":517},"pr-reflex","angles.disc-reflex","The small opening between two arms is **70°**. What is the reflex angle on the other side?",{"kind":340,"answer":513,"tolerance":342,"unit":343},290,[515,516],"Small opening + reflex angle = a full turn.","360 − 70 = ?",{"correct":518,"incorrect":519},"Correct: 360° − 70° = 290°, which is between 180° and 360°, so it is reflex.","The two openings together make a full turn of 360°. So the reflex angle is 360° − 70° = 290°.",{"id":521,"type":409,"component":522,"componentVersion":5,"config":523,"objective":591,"textAlternative":592},"lab-sort-types","sort-game",{"prompt":524,"bins":525,"items":535,"seconds":342},"Sort each real-life angle into its type.",[526,528,530,532,534],{"id":527,"label":383},"acute",{"id":529,"label":388},"right",{"id":531,"label":393},"obtuse",{"id":533,"label":398},"straight",{"id":457,"label":484},[536,540,544,547,551,555,559,563,567,571,575,579,583,587],{"id":537,"label":538,"bin":527,"why":539},"i1","The tip of a slice of pizza cut into 8 equal pieces","360° ÷ 8 = 45°, which is less than 90°.",{"id":541,"label":542,"bin":529,"why":543},"i2","The corner of a carrom board","A carrom board is a square, so every corner is 90°.",{"id":545,"label":482,"bin":533,"why":546},"i3","The hands point to 12 and 6, opposite each other: a half turn, 180°.",{"id":548,"label":549,"bin":529,"why":550},"i4","Clock hands at 3:00","12 to 3 is a quarter of the way round: 90°.",{"id":552,"label":553,"bin":527,"why":554},"i5","Clock hands at 1:00","12 to 1 is one twelfth of a full turn: 30°.",{"id":556,"label":557,"bin":531,"why":558},"i6","Clock hands at 5:00 (the smaller angle)","5 hour-marks × 30° = 150°, between 90° and 180°.",{"id":560,"label":561,"bin":531,"why":562},"i7","A laptop screen tipped back to about 110°","110° is more than a right angle but less than a straight line.",{"id":564,"label":565,"bin":533,"why":566},"i8","A book lying open flat on the desk","The two covers point in opposite directions: 180°.",{"id":568,"label":569,"bin":457,"why":570},"i9","Pac-Man's body when his mouth is open 60°","360° − 60° = 300°, between 180° and 360°.",{"id":572,"label":573,"bin":527,"why":574},"i10","The letter V","The two strokes of a V meet at a narrow, sharp angle.",{"id":576,"label":577,"bin":529,"why":578},"i11","A cricket stump standing on flat ground","The stump stands straight up, at 90° to the ground.",{"id":580,"label":581,"bin":457,"why":582},"i12","Turning from facing north to facing west, clockwise","North → east → south → west clockwise is three quarter turns: 270°.",{"id":584,"label":585,"bin":531,"why":586},"i13","The gap between two blades of a three-blade fan","360° ÷ 3 = 120°, between 90° and 180°.",{"id":588,"label":589,"bin":527,"why":590},"i14","A ramp for wheelchairs meeting the ground","Ramps are gentle, only a few degrees steep: acute.","Sort 14 everyday angles into acute, right, obtuse, straight and reflex.","A sorting game with five bins: acute, right, obtuse, straight and reflex. Each card describes a real angle.\n\nAcute: a pizza slice cut into 8 (45°), clock hands at 1:00 (30°), the letter V, and a wheelchair ramp meeting the ground (a few degrees).\n\nRight: the corner of a carrom board, clock hands at 3:00 and a cricket stump standing on flat ground (all 90°).\n\nObtuse: clock hands at 5:00 (150°), a laptop screen tipped back to about 110°, and the gap between blades of a three-blade fan (120°).\n\nStraight: clock hands at 6:00 and a book lying open flat (both 180°).\n\nReflex: Pac-Man's body when his mouth is open 60° (300°) and turning clockwise from north to west (270°).\n\nStrategy: compare with a square corner (90°) and a straight line (180°) first.",{"id":594,"type":74,"title":595,"eyebrow":596,"navLabel":597},"ch07","Turning on the spot: directions","Chapter 07","7 Directions",{"id":599,"type":43,"markdown":600},"directions","Maps and compasses use four main directions: **North (N)**, **East (E)**, **South (S)** and **West (W)**. If you stand facing north at sunrise, the Sun rises roughly on your right, in the east.\n\nThe four directions are spaced **a quarter turn apart**:\n\n- From N to E is a quarter turn: **90°**.\n- From N to S is a half turn: **180°**.\n- From N all the way round to N again is a full turn: **360°**.\n\nThe direction of turning matters too. Turning the same way as the hands of a clock is called **clockwise**. Turning the other way is **anticlockwise** (in some countries, *counterclockwise*). Facing north, a quarter turn **clockwise** brings you to **east**; a quarter turn **anticlockwise** brings you to **west**.",{"id":602,"type":176,"caption":603,"columns":604,"rows":608},"table-directions","Turning from North",[605,276,606,607],"Start facing","Direction of turn","End facing",[609,613,616,619,620,621,622,623],[610,262,611,612],"North","Clockwise","East",[610,262,614,615],"Anticlockwise","West",[610,258,617,618],"Either way","South",[610,266,611,615],[612,258,617,615],[618,262,611,615],[615,262,611,610],[610,254,617,624],"North (back to start)",{"id":626,"type":47,"variant":209,"title":627,"markdown":628},"try-parade","School parade game","Play this with friends, taking turns as the leader.\n\nThe leader calls out turns: \"**Right turn!**\" (a quarter turn clockwise, 90°), \"**Left turn!**\" (a quarter turn anticlockwise, 90°), \"**About turn!**\" (a half turn, 180°). Everyone turns on the spot.\n\nAfter a few calls, the leader asks: \"Which direction are you facing now?\" Anyone facing the wrong way sits down. Hard mode: the leader calls turns in degrees, such as “Turn 270° clockwise!”",{"id":630,"type":324,"title":631,"problem":632,"steps":633},"we-turns","Where am I facing now?","Riya faces **north**. She turns **90° clockwise**, then **180°**, then **90° anticlockwise**. Which way is she facing now?",[634,635,636,637,638],"Start: north.","90° clockwise from north → **east**.","180° (either way) from east → **west**.","90° anticlockwise from west → **south** (anticlockwise goes W → S → E → N).","She is facing **south**. Check by adding the turns: clockwise +90, then +180, then −90 gives 180° altogether, a half turn from north, which is south. ✓",{"id":640,"type":219,"itemId":641,"prompt":642,"check":643,"hints":650,"feedback":653},"pr-direction","angles.disc-direction","Arjun faces **east** and turns **90° anticlockwise**. Which direction does he face now?",{"kind":223,"options":644,"correct":649},[645,646,647,648],{"id":152,"label":610},{"id":155,"label":618},{"id":158,"label":615},{"id":232,"label":612},[152],[651,652],"Anticlockwise goes the opposite way to clock hands: N → W → S → E → N.","So from east, the next direction anticlockwise is…?",{"correct":654,"incorrect":655},"Yes: a quarter turn anticlockwise from east brings you to north.","Clockwise from north you meet east. So going back the other way (anticlockwise) from east, you reach north.",{"id":657,"type":74,"title":658,"eyebrow":659,"navLabel":660},"ch08","Angles on a clock face","Chapter 08","8 Clock angles",{"id":100,"type":43,"markdown":662},"A clock is a wonderful angle machine. The 12 numbers are spread evenly all the way round, so the gap from one number to the next is:\n\n**360° ÷ 12 = 30°**.\n\nThis gives you an easy way to find the angle between the hands **on the hour**: count the numbers between the hands and multiply by 30.\n\n- At **1:00** the hands are 1 number apart: 1 × 30° = **30°** (acute).\n- At **2:00**: 2 × 30° = **60°** (acute).\n- At **3:00**: 3 × 30° = **90°** (right angle).\n- At **4:00**: 4 × 30° = **120°** (obtuse).\n- At **6:00**: 6 × 30° = **180°** (straight angle).\n- At **12:00**: the hands overlap: **0°** (zero angle).\n\nThe minute hand is quicker. It goes all the way round (360°) in one hour, so in **15 minutes** it turns a quarter turn, 90°, and in **30 minutes** a half turn, 180°.",{"id":664,"type":176,"caption":665,"columns":666,"rows":670},"table-clock-hours","The smaller angle between the hands at each hour (count the gaps, multiply by 30°)",[667,668,669,463],"Time","Gaps between hands","Angle",[671,674,676,679,682,685,689,692,694,696,698,700],[672,673,468,467],"12:00","0",[675,283,315,383],"1:00",[677,678,305,383],"2:00","2",[680,681,262,388],"3:00","3",[683,684,296,393],"4:00","4",[686,687,688,393],"5:00","5","150°",[690,691,258,398],"6:00","6",[693,687,688,393],"7:00",[695,684,296,393],"8:00",[697,681,262,388],"9:00",[699,678,305,383],"10:00",[701,283,315,383],"11:00",{"id":703,"type":47,"variant":704,"title":705,"markdown":706},"careful-clock","careful","Only exactly on the hour","The \"count the gaps and multiply by 30\" rule works **only when the minute hand points exactly at 12**. At 3:30, for example, the hour hand has moved half-way from 3 to 4, so the angle is **not** a simple count. You will learn how to handle times like that in the deeper layers.",{"id":708,"type":219,"itemId":709,"prompt":710,"check":711,"hints":712,"feedback":715},"pr-clock-4","angles.disc-clock-4","What is the smaller angle between the hands of a clock at **4:00**?",{"kind":340,"answer":341,"tolerance":342,"unit":343},[713,714],"Count the numbers from 12 to 4.","Each gap is 30°.",{"correct":716,"incorrect":717},"Right: 4 gaps × 30° = 120°, an obtuse angle.","From 12 to 4 there are 4 gaps. Each gap is 360° ÷ 12 = 30°, so the angle is 4 × 30° = 120°.",{"id":719,"type":409,"component":720,"componentVersion":5,"config":721,"objective":736,"textAlternative":737},"lab-match-turns","match-pairs",{"prompt":722,"mode":723,"pairs":724},"Match each turn or time to its angle in degrees.","connect",[725,726,727,728,729,731,732,734],{"a":261,"b":262},{"a":257,"b":258},{"a":253,"b":254},{"a":265,"b":266},{"a":730,"b":305},"Clock hands at 2:00",{"a":553,"b":315},{"a":733,"b":310},"One slice of a pizza cut into 8",{"a":735,"b":296},"Clock hands at 4:00","Connect each turn, clock time or pizza slice to the right number of degrees.","A matching game with eight pairs. Quarter turn ↔ 90° (360 ÷ 4). Half turn ↔ 180° (360 ÷ 2). Full turn ↔ 360°. Three-quarter turn ↔ 270° (3 × 90). Clock hands at 2:00 ↔ 60° (2 gaps × 30°). Clock hands at 1:00 ↔ 30°. One slice of a pizza cut into 8 ↔ 45° (360 ÷ 8). Clock hands at 4:00 ↔ 120° (4 × 30°). Every answer comes from sharing out a full turn of 360°.",{"id":739,"type":74,"title":740,"eyebrow":741,"navLabel":742},"ch09","Angles that team up","Chapter 09","9 Angle partners",{"id":744,"type":43,"markdown":745},"partners","Sometimes two angles sit side by side and together make something special.\n\n- Two angles that **together make a right angle** (90°), like 30° and 60°, are called **complementary** angles. Think of them as two pieces that complete a square corner.\n- Two angles that **together make a straight angle** (180°), like 120° and 60°, are called **supplementary** angles.\n\nLook at a road junction where a straight road meets a side road. The side road splits the straight line into two angles. However the side road is tilted, the two angles **always add up to 180°**, because together they make the straight line. That means if you know one of them, you can work out the other without measuring!",{"id":747,"type":324,"title":748,"problem":749,"steps":750},"we-missing-straight","The missing angle on a straight road","A lane meets a straight main road. One angle between the lane and the road is **130°**. What is the other angle on the same side of the road?",[751,752,753,754],"The two angles sit side by side along the straight road, so together they make a **straight angle: 180°**.","Missing angle = 180° − 130°.","= **50°**.","Check: 130° + 50° = 180°. ✓",{"id":756,"type":409,"component":757,"componentVersion":5,"config":758,"objective":762,"textAlternative":763,"help":764},"lab-linear-intro","angle-pairs",{"scene":759,"initialAngle":760,"challenges":761},"linear-pair",130,3,"Tilt a ray standing on a straight line and see that the two angles always add up to 180°; find three missing angles.","A straight line with a ray standing on it makes two angles side by side, labelled ∠a and ∠b. It starts at 130° and 50°. A live table shows both sizes as you drag the ray, and a button highlights the pair.\n\nWhen one angle grows, the other shrinks by the same amount: 120° and 60°, 90° and 90°, 45° and 135°. They always add up to **180°**, because together they make the straight line.\n\nIn the three challenges, the lab tells you one angle and asks for the other; type the number of degrees. If ∠a is 70°, then ∠b = 180° − 70° = 110°. After each answer the lab explains the reason.",{"simplerExplanation":765,"hints":766},"Two angles side by side on a straight line always make 180°. Missing one = 180 minus the one you know.",[767],"Watch the live table as you drag, and add the two sizes.",{"id":769,"type":219,"itemId":770,"prompt":771,"check":772,"hints":774,"feedback":777},"pr-complement","angles.disc-complement","Two angles together make a **right angle**. One of them is **35°**. What is the other?",{"kind":340,"answer":773,"tolerance":342,"unit":343},55,[775,776],"A right angle is 90°.","90 − 35 = ?",{"correct":778,"incorrect":779},"Yes: 90° − 35° = 55°. 35° and 55° are complementary.","Together the two angles make 90°. So the other angle is 90° − 35° = 55°.",{"id":781,"type":74,"title":782,"eyebrow":783,"navLabel":784},"ch10","Angles everywhere","Chapter 10","10 Real-life angles",{"id":786,"type":83,"title":787,"prompt":788,"options":789},"explorer-real","Where angles do real work","Pick one to see why the angle matters.",[790,801,812,823,834],{"id":791,"label":792,"chain":793,"badge":798,"note":800},"ramp","Wheelchair ramp",[794,795,796,797],"Ground","Ramp surface","Small acute angle","Easy to push up",{"text":799,"tone":97},"Gentle angles help","A ramp for wheelchairs, prams and trolleys must be gentle. India's Harmonised Guidelines for Universal Accessibility say a ramp must be no steeper than 1 unit up for every 12 units along the ground — an angle of about 4.8°, less than 5° — and prefer a gentler 1 in 15 where there is room. A steeper ramp would be too hard to push up and dangerous to roll down.",{"id":802,"label":803,"chain":804,"badge":809,"note":811},"ladder","Ladder on a wall",[805,806,807,808],"Wall and ground","Ladder leans","About 76° with the ground","Safe to climb",{"text":810,"tone":97},"Too steep or too flat is unsafe","If a ladder leans too flat it can slide out at the bottom; too steep and it can tip backwards. Safety advice says to put the foot one unit out from the wall for every four units up. Worked out exactly that is about 76° with the ground, which safety guides round off to “about 75°”.",{"id":813,"label":814,"chain":815,"badge":820,"note":822},"roof","Roofs",[816,817,818,819],"Rain or snow falls","Roof slopes at an angle","Water runs off","House stays dry",{"text":821,"tone":97},"Steeper where it rains more","Houses in very rainy or snowy places, like the hills of Kerala or Himachal Pradesh, often have steep sloping roofs so water and snow slide off quickly. Many houses in drier plains have flat roofs, useful for drying grain or sleeping on hot summer nights.",{"id":824,"label":825,"chain":826,"badge":831,"note":833},"bowling","Cricket bowling",[827,828,829,830],"Bowler's arm","Elbow angle","Must stay nearly straight","Legal delivery",{"text":832,"tone":97},"Rules use angles","Cricket's laws say a bowler must not throw the ball. The ICC calls an action illegal if the elbow straightens by more than 15° between the arm reaching the horizontal and the ball being released. Special cameras measure the elbow angle when an action is tested.",{"id":835,"label":836,"chain":837,"badge":842,"note":844},"pie","Pie charts",[838,839,840,841],"A whole group","360° for the whole circle","Each slice's angle","Shows its share",{"text":843,"tone":97},"Angles show data","In a pie chart, the full circle (360°) stands for everyone. If half the class chose cricket as their favourite sport, cricket's slice gets half of 360°, which is 180°. You will use this in the data-handling topic.",{"id":846,"type":47,"variant":209,"title":847,"markdown":848},"try-hunt","Angle hunt at home","Make a table with five columns: acute, right, obtuse, straight, reflex. Hunt for at least two examples of each at home or at school. Some ideas: the hands of a clock at different times, a partly open door, a folding chair, a coat hanger, the letter shapes on a signboard, the corner of a rangoli pattern.\n\nWhich type was hardest to find? (Most people find reflex angles the hardest. Here is a hint: look for the \"long way round\" angle outside a slice of cake.)",{"id":850,"type":851,"prompt":852},"reflect-turn","reflection","Explain to a younger child why a door opened a little and a big gate opened the same amount make the **same** angle. Use the word *turn* in your answer.",{"id":854,"type":855,"title":856,"terms":857},"glossary-discover","glossary","Angle words",[858,862,866,870,874,876,880,884,888,892,896,900,904,908,912,916,920,924,928,932,936],{"term":859,"meaning":860,"example":861},"angle","The amount of turn between two rays that start from the same point.","The hands of a clock make an angle.",{"term":863,"meaning":864,"example":865},"vertex","The common starting point of the two arms of an angle; the corner point. Plural: vertices.","The screw of a pair of scissors.",{"term":867,"meaning":868,"example":869},"arm","One of the two rays that form an angle.","Each hand of a clock is an arm.",{"term":871,"meaning":872,"example":873},"ray","A straight path that starts at a point and goes on forever in one direction.","A beam of light from a torch.",{"term":62,"meaning":875,"example":775},"The unit for measuring angles. A full turn is 360 degrees.",{"term":877,"meaning":878,"example":879},"full turn","Turning all the way round to face the same way again: 360°. Also called a complete turn or revolution.","A fan blade going round once.",{"term":881,"meaning":882,"example":883},"half turn","Turning to face the opposite way: 180°.","An about-turn in a parade.",{"term":885,"meaning":886,"example":887},"quarter turn","One quarter of a full turn: 90°.","Turning from facing north to facing east.",{"term":889,"meaning":890,"example":891},"zero angle","An angle of 0°: the arms lie on top of each other.","Clock hands at 12:00.",{"term":893,"meaning":894,"example":895},"acute angle","An angle more than 0° and less than 90°.","A slice of pizza.",{"term":897,"meaning":898,"example":899},"right angle","An angle of exactly 90°, a square corner. Marked with a small square.","The corner of a page.",{"term":901,"meaning":902,"example":903},"obtuse angle","An angle more than 90° and less than 180°.","A reclining chair.",{"term":905,"meaning":906,"example":907},"straight angle","An angle of exactly 180°: the arms make a straight line.","Clock hands at 6:00.",{"term":909,"meaning":910,"example":911},"reflex angle","An angle more than 180° and less than 360°.","The body of Pac-Man.",{"term":913,"meaning":914,"example":915},"complete angle","An angle of exactly 360°, one full turn.","A wheel turning round once.",{"term":917,"meaning":918,"example":919},"clockwise","Turning in the same direction as the hands of a clock.","North to east is a clockwise quarter turn.",{"term":921,"meaning":922,"example":923},"anticlockwise","Turning in the opposite direction to the hands of a clock.","North to west is an anticlockwise quarter turn.",{"term":925,"meaning":926,"example":927},"interior of an angle","The region between the two arms of an angle.","The cheesy part of a pizza slice.",{"term":929,"meaning":930,"example":931},"exterior of an angle","All of the flat surface outside the two arms.","The rest of the pizza, outside the slice.",{"term":933,"meaning":934,"example":935},"complementary angles","Two angles that add up to 90°.","30° and 60°",{"term":937,"meaning":938,"example":939},"supplementary angles","Two angles that add up to 180°.","120° and 60°",{"id":941,"type":942,"title":943,"questions":944},"quiz-discover","quiz","Check your turning",[945,958,968,978,987,998,1007,1020,1031],{"itemId":946,"prompt":947,"options":948,"correct":155,"why":957},"angles.disc-q-turn","What does the size of an angle measure?",[949,951,953,955],{"id":152,"label":950},"The length of its arms",{"id":155,"label":952},"The amount of turn between its arms",{"id":158,"label":954},"How thick the lines are",{"id":232,"label":956},"How far the vertex is from the edge of the page","An angle measures **turn**. Arm length has nothing to do with it.",{"itemId":959,"prompt":960,"options":961,"correct":158,"why":967},"angles.disc-q-half","How many degrees is a half turn?",[962,963,965,966],{"id":152,"label":262},{"id":155,"label":964},"100°",{"id":158,"label":258},{"id":232,"label":254},"A full turn is 360°, so half of it is 360 ÷ 2 = 180°.",{"itemId":969,"prompt":970,"options":971,"correct":158,"why":977},"angles.disc-q-acute","Which of these angles is acute?",[972,974,975,976],{"id":152,"label":973},"95°",{"id":155,"label":262},{"id":158,"label":310},{"id":232,"label":258},"Acute means more than 0° and less than 90°. Only 45° fits.",{"itemId":979,"prompt":980,"options":981,"correct":232,"why":986},"angles.disc-q-clock6","What angle do the clock hands make at 6:00?",[982,983,984,985],{"id":152,"label":305},{"id":155,"label":262},{"id":158,"label":296},{"id":232,"label":258},"The hands point at 12 and 6, opposite each other, so they make a straight angle: 6 × 30° = 180°.",{"itemId":988,"prompt":989,"options":990,"correct":158,"why":997},"angles.disc-q-reflex","Which angle is reflex?",[991,993,994,996],{"id":152,"label":992},"170°",{"id":155,"label":258},{"id":158,"label":995},"250°",{"id":232,"label":254},"Reflex angles are more than 180° and less than 360°. 250° fits; 180° is straight and 360° is complete.",{"itemId":999,"prompt":1000,"options":1001,"correct":155,"why":1006},"angles.disc-q-dir","You face north and turn 180°. Which way do you face?",[1002,1003,1004,1005],{"id":152,"label":612},{"id":155,"label":618},{"id":158,"label":615},{"id":232,"label":610},"A half turn from north faces the opposite direction: south.",{"itemId":1008,"prompt":1009,"options":1010,"correct":158,"why":1019},"angles.disc-q-arms","Angle P has arms 2 cm long. Angle Q has the same opening but arms 8 cm long. Which is true?",[1011,1013,1015,1017],{"id":152,"label":1012},"Q is 4 times bigger",{"id":155,"label":1014},"P is bigger",{"id":158,"label":1016},"They are equal",{"id":232,"label":1018},"You cannot compare them","Same opening means same turn, so the angles are equal. Arm length does not matter.",{"itemId":1021,"prompt":1022,"options":1023,"correct":152,"why":1030},"angles.disc-q-supp","Two angles side by side make a straight line. One is 100°. What is the other?",[1024,1026,1027,1028],{"id":152,"label":1025},"80°",{"id":155,"label":262},{"id":158,"label":964},{"id":232,"label":1029},"260°","Angles along a straight line add to 180°, so the other is 180° − 100° = 80°.",{"itemId":1032,"prompt":1033,"options":1034,"correct":158,"why":1039},"angles.disc-q-fan","A fan has 4 equally spaced blades. What is the angle between neighbouring blades?",[1035,1036,1037,1038],{"id":152,"label":310},{"id":155,"label":305},{"id":158,"label":262},{"id":232,"label":296},"360° ÷ 4 = 90°, a right angle.",{"id":1041,"type":1042,"title":1043,"points":1044},"cheat-sheet","summary","Cheat sheet",[1045,1046,1047,1048,1049,1050,1051,1052,1053],"An **angle** is the **amount of turn** between two rays (arms) that start from one point (the **vertex**).","**Arm length does not matter.** Only the opening, the turn, decides the size of an angle.","We measure turns in **degrees (°)**. Full turn **360°**, half turn **180°**, quarter turn **90°**.","**Zero** 0° · **acute** between 0° and 90° · **right** 90° · **obtuse** between 90° and 180° · **straight** 180° · **reflex** between 180° and 360° · **complete** 360°.","A **right angle** is a square corner, marked with a small square. Fold paper twice to make a right-angle checker.","Two arms make two openings: small + reflex = 360°.","**Directions** N, E, S, W are 90° apart. **Clockwise** = the way clock hands go; **anticlockwise** = the other way.","On a clock, each number is **30°** from the next. On the hour: angle = gaps × 30°.","Two angles making a **right angle** are **complementary**; two making a **straight line** are **supplementary**.",{"id":1055,"type":1056,"conceptId":1057,"relation":1058,"explanation":1059},"conn-lines","connection","lines","helps_understand","Angles are built from rays. The lines topic explains lines, rays and line segments, and what happens when lines cross.",{"id":1061,"type":1056,"conceptId":1062,"relation":1063,"explanation":1064},"conn-construct","constructing-angles","applied_in","To measure an angle exactly or draw one of a chosen size, you use a protractor or a compass. That is the measuring and constructing angles topic.",{"id":1066,"type":1056,"conceptId":1067,"relation":1068,"explanation":1069},"conn-shapes","shape-and-space","related_to","Every corner of a triangle, square or other shape is an angle. A square has four right angles.",{"id":1071,"type":1056,"conceptId":1072,"relation":1063,"explanation":1073},"conn-data","data-handling","Pie charts share out 360° among the slices, so each slice's angle shows its share of the data.",{"id":1075,"type":1076,"sourceIds":1077},"sources-discover","sources",[1078,1079,1080,1081,1082,1083,1084,1085,1086,1087],"angles-ncert-class7-lines-angles","angles-mathsisfun-angles","angles-khan-angle-basics","angles-wiki-degree","angles-wiki-clock-angle","angles-ncert-class6-lines-angles","angles-mathsisfun-complementary","angles-mohua-accessibility-ramps","angles-niosh-ladder","angles-icc-illegal-bowling",[1078,1079,1080,1081,1082,1083,1084,1085,1086,1087],"needs_review",{"generatedBy":1091,"notes":1092},"claude-code","Draft generated with Python generator scripts; every angle computed and asserted. Pending owner review.","cdc1cd07de230e14c8c93c13223f31b2c9699d243ff9a6f59e92303dcc9c8132",{"logic:practice":1095,"component:angle-lab@1":1096,"component:sort-game@1":1097,"component:match-pairs@1":1098,"component:angle-pairs@1":1099,"source:angles-icc-illegal-bowling":1100,"source:angles-khan-angle-basics":1101,"source:angles-mathsisfun-angles":1102,"source:angles-mathsisfun-complementary":1103,"source:angles-mohua-accessibility-ramps":1104,"source:angles-ncert-class6-lines-angles":1105,"source:angles-ncert-class7-lines-angles":1106,"source:angles-niosh-ladder":1107,"source:angles-wiki-clock-angle":1108,"source:angles-wiki-degree":1109},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","cbb509e9a575a1b2ef133804e3450487de14952c1df934747fe56ff5d10d847e","c4487affbd8ca1c7cbfd54d9293933b40e3c24201a82aaeb13cce9bb41d81a26","68741673c0dc6c96760fde888327818c508788257b5306bad48ab148615541a4","2dab8c7c8736bbad61c4a7223c297c3b250b20399d5b5544c02434cab39a0c9c","7656284145acec3d6ead6ded762ed0eef2bb3137446c527350b7b999562580df","26708f3ab1210e6438207954ba955d647d14dc15ef1ddc44511e1608d988e5a7","36960f0a34d1e3f73475a665e4b12c1140dfbbeac1c1b2772bf71a8d3b9522a5","548a49fd09ac1b7cec7c74b038e1a87c5d3add389d6f27376ac054da17d20972","dd85dfd2c7ee127db700b8721af5975437c999a7625b759ffb988d5f7af0c2e2","141f5516f8832cc0eea67b34a5726cd12ed7309036b178da74f753ca6e4ae706","79b5ead855ebdf6f15ad061e87052809bfff58b83c8dfb9f7a1f592debdf62e9",{"state":1111,"reviewer":1112,"selfReview":500,"reviewedAt":1113,"method":1114},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597035]