[{"data":1,"prerenderedAt":1090},["ShallowReactive",2],{"layer:constructing-angles:discover":3},{"layer":4,"contentHash":1063,"dependencyHashes":1064,"approval":1083,"releaseId":1089},{"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":1058,"reviewStatus":1059,"authoring":1060},1,"constructing-angles","en","discover","Angles you can measure and make","The geometry box, the protractor and the compass trick for an exact 60°","Open the geometry box, learn what a degree is, estimate angles by eye, measure and draw angles with a protractor, and discover how a compass alone can make an exact 60° angle.",[13,14,15,16,17],"Name each tool in the geometry box and say what job it does.","Explain what a degree is and recognise 30°, 45°, 60°, 90°, 120° and 180° by sight.","Estimate an angle before measuring it, and measure and draw angles with a protractor using the correct scale.","Construct an exact 60° angle with a compass and ruler and explain why it works.","Combine set-square angles to make angles such as 75°, 105° and 135°.",30,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 30 minutes",{"label":29,"value":30},"You need","Geometry box, sharp pencil, paper",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Tool sort, estimate game, protractor, 60° build",{"label":38,"value":39},"Big idea","Estimate, then measure",[41,45,51,57,60,93,98,103,106,142,164,169,172,177,252,257,260,276,281,284,287,312,354,414,419,434,439,442,460,474,478,483,486,510,514,517,522,550,564,575,580,596,617,622,625,647,658,662,682,687,690,703,710,714,718,722,727,730,735,738,750,779,810,815,877,881,887,891,896,901,905,1028,1042],{"id":42,"type":43,"markdown":44},"intro-hook","prose","Look at a **patang** (kite) before it goes up on Makar Sankranti. Two thin bamboo sticks cross each other, and the kite flies straight only if they cross at exactly the right angle. Look at a wooden photo frame: four pieces meet at the corners, and if even one cut is a little off, the corner shows an ugly gap. Look at a cricket pitch: the creases are painted at right angles to the pitch, or the umpire's decisions would be unfair.\n\nIn every one of these, someone had to **measure** an angle, or **make** one exactly. This lesson is about how people do that, with the small tin **geometry box** that sits in your school bag.\n\nBy the end you will be able to answer a surprising question: **how do you draw an exact 60° angle with only a compass and a ruler, and no protractor at all?**",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this lesson","Keep your geometry box, a sharp pencil and some plain paper beside you. Each chapter has something to try with your hands or in a lab. Whenever you see a **prediction**, choose an answer before reading on. Guessing first and then checking is how your brain learns fastest.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Why exact angles matter","Chapter 01","1 Why exact angles",{"id":58,"type":43,"markdown":59},"why-exact","An **angle** is made when two straight lines, called **arms**, start from the same point, called the **vertex**. The size of the angle tells you **how much you have to turn** to go from one arm to the other.\n\nYou already know some angles by sight. The corner of your notebook is a **right angle**. A door opened just a little makes a small angle with the wall; a door flung wide open makes a big one. But \"a little\" and \"wide\" are not good enough for a carpenter, a builder or a kite maker. They need to say *exactly* how big an angle is, and then draw or cut it exactly. For that we need a unit, the **degree**, and tools that work with degrees.",{"id":61,"type":62,"caption":63,"columns":64,"rows":68},"table-where","table","Where exact angles show up around you",[65,66,67],"Where","What angle","What goes wrong if it is off",[69,73,77,81,85,89],[70,71,72],"Patang (kite) frame","Sticks crossing at 90°","The kite tilts and spins instead of flying steady",[74,75,76],"Photo frame corner","Two 45° cuts make 90°","A gap appears in the corner",[78,79,80],"Cricket crease","90° to the pitch","Run-outs and no-balls are judged unfairly",[82,83,84],"Staircase","Steps level, railing sloped","People trip; railings feel wrong",[86,87,88],"Honeycomb and tiles","Hexagons with 120° corners","Tiles leave gaps or overlap",[90,91,92],"Road junction","Roads meeting near 90°","Drivers cannot see traffic coming",{"id":94,"type":47,"variant":95,"title":96,"markdown":97},"question-hook","question","The question for this lesson","A protractor is a tool made for measuring angles. A compass is a tool made for drawing circles. **So how can a compass, which only draws round curves, help you make a perfectly exact 60° angle?** Keep this puzzle in your head. You will solve it in Chapter 8.",{"id":99,"type":53,"title":100,"eyebrow":101,"navLabel":102},"ch2","Open the geometry box","Chapter 02","2 The geometry box",{"id":104,"type":43,"markdown":105},"box-intro","Almost every Indian school child owns a small metal or plastic **geometry box** (some people call it an *instrument box*). Open it and you will usually find the same set of tools. Each one has a job. Using the right tool for the job, and using it carefully, is half the secret of good geometry.",{"id":107,"type":62,"caption":108,"columns":109,"rows":113},"table-box","What is inside a geometry box and what each tool is for",[110,111,112],"Tool","What it looks like","Its main job",[114,118,122,126,130,134,138],[115,116,117],"Ruler (scale)","A 15 cm strip marked in cm and mm","Drawing straight lines; measuring lengths",[119,120,121],"Protractor","A half-circle marked 0° to 180°, with two rows of numbers","Measuring angles and drawing angles of a given size",[123,124,125],"Compass","Two legs joined at the top: one sharp point, one holding a pencil","Drawing circles and arcs; copying lengths",[127,128,129],"Divider","Two legs, both with sharp metal points","Comparing and transferring lengths exactly",[131,132,133],"Set square (45°)","A triangle with angles 45°, 45°, 90°","Drawing right angles and 45° angles; parallel lines",[135,136,137],"Set square (30°–60°)","A triangle with angles 30°, 60°, 90°","Drawing 30°, 60° and 90° angles quickly",[139,140,141],"Pencil, eraser, sharpener","The everyday helpers","Thin, sharp lines that can be corrected",{"id":143,"type":144,"tone":145,"items":146},"spec-tools","spec","blue",[147,151,154,157,160],{"label":148,"big":149,"value":150},"Ruler","15 cm","Marked in centimetres (cm) and millimetres (mm). 1 cm = 10 mm.",{"label":119,"big":152,"value":153},"0° – 180°","A half circle. Two scales run in opposite directions.",{"label":123,"big":155,"value":156},"circles","The sharp point stays still; the pencil swings round it.",{"label":127,"big":158,"value":159},"two points","Like a compass with no pencil. Great for copying lengths.",{"label":161,"big":162,"value":163},"Set squares","45°, 30°, 60°","Both have one right angle (90°).",{"id":165,"type":47,"variant":166,"title":167,"markdown":168},"careful-sharp","careful","Sharp points: handle with care","The compass and the divider have **sharp steel points**. Carry them closed, never point them at anyone, and put them back in the box when you are done. Press the point into your paper only as hard as you need to, with a notebook or cardboard underneath, never on a table you are not allowed to mark.",{"id":170,"type":43,"markdown":171},"care-accuracy","Good geometry is careful geometry. A few habits make a huge difference:\n\n- **Sharpen your pencil to a fine point.** A thick line can be half a millimetre wide, and then nobody can tell exactly where it is.\n- **Tighten the compass.** If the hinge is loose, the legs slide apart while you draw, and your circle will not close.\n- **Look at the ruler's zero.** On many rulers, 0 is not at the very end. Measure from the 0 mark, not from the edge.\n- **Keep the protractor clean and flat.** A scratched or bent protractor gives wrong readings.\n- **Draw lightly first.** Construction arcs are helpers, so draw them faintly and keep them. They show your working.",{"id":173,"type":47,"variant":174,"title":175,"markdown":176},"misc-ruler-end","misconception","“Measure from the end of the ruler”","Many rulers have a small blank bit before the 0 mark, and old rulers have worn, chipped ends. If you line up your line with the **end** of the ruler instead of the **0 mark**, every length you measure will be slightly wrong. Always put the 0 mark on the start of the line.",{"id":178,"type":179,"component":180,"componentVersion":5,"config":181,"objective":245,"textAlternative":246,"help":247},"lab-which-tool","interactive","sort-game",{"prompt":182,"bins":183,"items":195,"seconds":244},"Which tool from the geometry box would you pick for each job?",[184,186,188,190,192],{"id":185,"label":148},"ruler",{"id":187,"label":119},"protractor",{"id":189,"label":123},"compass",{"id":191,"label":127},"divider",{"id":193,"label":194},"setsq","Set square",[196,200,204,208,212,216,220,224,228,232,236,240],{"id":197,"label":198,"bin":185,"why":199},"t1","Draw a straight line 7 cm long","A ruler draws straight lines and its cm marks give the length.",{"id":201,"label":202,"bin":187,"why":203},"t2","Find how many degrees a drawn angle is","Measuring degrees is exactly what a protractor is for.",{"id":205,"label":206,"bin":189,"why":207},"t3","Draw a circle of radius 4 cm","Open the compass to 4 cm on the ruler, then swing it round.",{"id":209,"label":210,"bin":191,"why":211},"t4","Check whether two lines on a map are the same length","Open the divider on one line, then lift it onto the other without changing the gap.",{"id":213,"label":214,"bin":193,"why":215},"t5","Draw a quick right angle in the corner of a page","Every set square has a 90° corner ready to trace.",{"id":217,"label":218,"bin":187,"why":219},"t6","Draw an angle of 125°","125° is not on a set square, so use the protractor's scale.",{"id":221,"label":222,"bin":189,"why":223},"t7","Draw a rangoli flower made of arcs","Arcs are pieces of circles, so the compass draws them.",{"id":225,"label":226,"bin":185,"why":227},"t8","Measure the length of your pencil in mm","The small marks between centimetres are millimetres.",{"id":229,"label":230,"bin":193,"why":231},"t9","Draw a 30° angle without reading any numbers","The 30°–60° set square has a ready-made 30° corner.",{"id":233,"label":234,"bin":191,"why":235},"t10","Copy a length from the board onto your paper exactly","A divider carries a length across without needing to read numbers.",{"id":237,"label":238,"bin":189,"why":239},"t11","Mark an exact 60° using arcs only","Equal arcs from a compass make 60°, as you will see in Chapter 8.",{"id":241,"label":242,"bin":193,"why":243},"t12","Draw a 45° line for a paper-folding pattern","The 45° set square has two 45° corners.",0,"Pick the right tool from the geometry box for each everyday drawing or measuring job.","This game shows 12 job cards and five bins: ruler, protractor, compass, divider and set square.\n\n- **Ruler:** draw a 7 cm straight line; measure a pencil in millimetres.\n- **Protractor:** find how many degrees an angle is; draw a 125° angle (125° is not on any set square).\n- **Compass:** draw a circle of radius 4 cm; draw a rangoli flower made of arcs; mark an exact 60° with arcs.\n- **Divider:** check whether two lines on a map are equal; copy a length from the board exactly.\n- **Set square:** draw a quick right angle; draw a 30° angle without reading numbers; draw a 45° line.\n\nThe pattern: rulers for straight lengths, protractors for any number of degrees, compasses for circles and arcs, dividers for carrying lengths, and set squares for the fixed angles 30°, 45°, 60° and 90°.",{"simplerExplanation":248,"hints":249},"Think about the shape of the job. Straight line: ruler. Round curve: compass. Number of degrees: protractor. Same length somewhere else: divider. A ready-made corner: set square.",[250,251],"A divider has no pencil, so it cannot draw. It only carries lengths.","Set squares only give 30°, 45°, 60° and 90° (and sums of them).",{"id":253,"type":53,"title":254,"eyebrow":255,"navLabel":256},"ch3","A degree is a tiny turn","Chapter 03","3 What is a degree?",{"id":258,"type":43,"markdown":259},"degree-intro","Stand up and turn right round until you face the same way again. That is one **full turn**. Mathematicians split a full turn into **360 equal tiny turns**, and each tiny turn is called **one degree**, written **1°**.\n\n- A **full turn** is **360°**.\n- A **half turn** (facing the opposite way) is **180°**. This is a **straight angle**: the two arms make a straight line.\n- A **quarter turn** is **90°**, a **right angle**, like the corner of a page.\n\nOne degree is really small. If you turned only 1° you would hardly notice you had moved.",{"id":261,"type":262,"items":263},"formulas-turns","formulas",[264,267,270,273],{"expression":265,"caption":266},"full turn = 360°","Turning all the way round to face the same way again.",{"expression":268,"caption":269},"half turn = 180°","A straight angle: 360 ÷ 2 = 180.",{"expression":271,"caption":272},"quarter turn = 90°","A right angle: 360 ÷ 4 = 90.",{"expression":274,"caption":275},"one-sixth turn = 60°","360 ÷ 6 = 60. Remember this one: it is the compass angle.",{"id":277,"type":47,"variant":278,"title":279,"markdown":280},"aha-clock","aha","There is a protractor on every clock","A clock face is a full turn of 360° split into 12 hour marks, so the gap between two neighbouring numbers is 360 ÷ 12 = **30°**. At 3 o'clock the hands make 3 × 30 = **90°**. At 2 o'clock they make 60°, and at 4 o'clock 120°. The minute hand turns 360° in 60 minutes, which is **6° every minute**.",{"id":282,"type":43,"markdown":283},"why-360","Why 360 and not 100? Nobody is completely sure, but the idea probably comes from the ancient **Babylonians** of Mesopotamia, more than 4,000 years ago. They counted in groups of 60, and a year is close to 360 days, so the Sun seems to move about one degree across the sky each day.\n\nThere is also a very practical reason 360 has lasted: it **splits evenly in a huge number of ways**. 360 has **24** whole-number divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. So a half, a third, a quarter, a fifth, a sixth, an eighth, a ninth, a tenth and a twelfth of a turn are all whole numbers of degrees. Compare 100, which has only 9 divisors, or 365, which has only 4 (1, 5, 73 and 365).",{"id":285,"type":43,"markdown":286},"degree-small","Just how small is one degree? Imagine two long straight sticks, each 1 metre long, joined at one end. Open them to 1° and the far ends are only about **1.7 cm** apart, roughly the width of your thumb. That is why a sharp pencil and a careful eye matter: a sloppy line can easily be a degree or two off. For school work, being within **1° or 2°** of the true size is counted as accurate.",{"id":288,"type":289,"title":290,"items":291},"timeline-discover","timeline","A short history of exact angles",[292,296,300,304,308],{"time":293,"title":294,"text":295},"~2000 BCE","Counting in sixties","Babylonian astronomers count in groups of 60. Our 360 degrees, and the 60 minutes in an hour, are usually traced back to this habit.",{"time":297,"title":298,"text":299},"800–500 BCE","Ropes and pegs in India","The Sulba Sutras give rules for laying out fire altars with ropes and pegs, including exact right angles.",{"time":301,"title":302,"text":303},"~300 BCE","Euclid's Elements","The Greek book Elements begins by constructing an equilateral triangle with a compass: the 60° trick in this lesson.",{"time":305,"title":306,"text":307},"~150 CE","Degrees in tables","Ptolemy in Alexandria tabulates chords in the Almagest over a circle of 360 degrees, writing the fractions in sixtieths.",{"time":309,"title":310,"text":311},"Today","Geometry box and screens","School children use the same ideas with a protractor and compass; designers use computer drawing programs.",{"id":313,"type":62,"caption":314,"columns":315,"rows":319},"table-benchmarks","Benchmark angles to carry in your head",[316,317,318],"Angle","Name","Where you see it",[320,324,328,331,334,338,342,346,350],[321,322,323],"0°","Zero angle","Both arms lie on top of each other, like closed scissors",[325,326,327],"30°","Acute","Clock hands at 1 o'clock",[329,326,330],"45°","Half a right angle; a square folded corner to corner",[332,326,333],"60°","Each corner of an equilateral triangle; clock at 2 o'clock",[335,336,337],"90°","Right angle","Corner of a page, a door frame, a cricket crease",[339,340,341],"120°","Obtuse","Corners of a honeycomb cell; clock at 4 o'clock",[343,344,345],"180°","Straight angle","A straight line; clock hands at 6 o'clock",[347,348,349],"270°","Reflex","Three quarter turns",[351,352,353],"360°","Complete angle","A full turn",{"id":355,"type":356,"title":357,"terms":358},"glossary-discover","glossary","Words for measuring and making angles",[359,363,366,370,374,378,381,385,388,391,394,396,398,400,403,407,411],{"term":360,"meaning":361,"example":362},"angle","The amount of turn between two arms that start at the same point.","The hands of a clock make an angle.",{"term":364,"meaning":365},"arm","One of the two straight lines (rays) that form an angle.",{"term":367,"meaning":368,"example":369},"vertex","The point where the two arms of an angle meet. Plural: vertices.","The corner point of a page.",{"term":371,"meaning":372,"example":373},"degree (°)","The unit for measuring angles. One full turn is 360 degrees.","A right angle is 90°.",{"term":375,"meaning":376,"example":377},"right angle","An angle of exactly 90°, a quarter turn.","The corner of a notebook.",{"term":379,"meaning":380},"straight angle","An angle of exactly 180°; its arms make a straight line.",{"term":382,"meaning":383,"example":384},"acute angle","An angle more than 0° and less than 90°.","30°, 45°, 60°",{"term":386,"meaning":387,"example":339},"obtuse angle","An angle more than 90° and less than 180°.",{"term":389,"meaning":390,"example":347},"reflex angle","An angle more than 180° and less than 360°.",{"term":392,"meaning":393},"geometry box","A small case holding a ruler, protractor, compass, divider, set squares and pencil.",{"term":187,"meaning":395},"A half-circle (or full-circle) tool marked in degrees, used to measure and draw angles.",{"term":189,"meaning":397},"A tool with a sharp point and a pencil leg, used to draw circles and arcs.",{"term":191,"meaning":399},"A tool with two sharp points, used to compare and copy lengths.",{"term":401,"meaning":402},"set square","A triangle-shaped tool with fixed angles: 45°–45°–90° or 30°–60°–90°.",{"term":404,"meaning":405,"example":406},"arc","A part of a circle.","A rainbow is shaped like an arc.",{"term":408,"meaning":409,"example":410},"estimate","A sensible guess made before measuring, using what you already know.","“It looks a bit less than a right angle, about 80°.”",{"term":412,"meaning":413},"construct","To draw a shape exactly using only a ruler (for straight lines) and a compass.",{"id":415,"type":47,"variant":416,"title":417,"markdown":418},"try-fold","try_it","Make angles by folding paper","Take any scrap of paper with a torn, wavy edge.\n\n1. Fold it once, anywhere. The fold is a straight line.\n2. Fold again so the first fold lies exactly on top of itself. Unfold: the two creases cross at **four right angles** (4 × 90° = 360°).\n3. Fold one right angle in half, corner to corner. You have made **45°**.\n\nYou have just made a right angle and half a right angle with no tools at all. Check them against the corner of your set square.",{"id":420,"type":421,"itemId":422,"prompt":423,"check":424,"hints":428,"feedback":431},"practice-clock","practice","constructing-angles.disc-clock-angle","The clock shows **5 o'clock**. What is the smaller angle between the hour hand and the minute hand, in degrees?",{"kind":425,"answer":426,"tolerance":244,"unit":427},"number",150,"°",[429,430],"Each hour gap on a clock is 360 ÷ 12 = 30°.","From 12 to 5 there are 5 gaps.",{"correct":432,"incorrect":433},"Right: 5 gaps × 30° = 150°, an obtuse angle.","Each hour gap is 30°. From 12 to 5 there are 5 gaps, so 5 × 30° = **150°**.",{"id":435,"type":53,"title":436,"eyebrow":437,"navLabel":438},"ch4","Guess before you measure","Chapter 04","4 Estimate first",{"id":440,"type":43,"markdown":441},"estimate-why","Before a good carpenter picks up a measuring tool, they **look** and **guess**. Guessing first is not cheating. It is a safety net. If your guess says \"about 40°\" and your measurement says 140°, you know at once that something went wrong.\n\nHere is a simple way to estimate any angle:\n\n1. **Compare it with a right angle.** Is it smaller than the corner of a page (acute), bigger (obtuse), or exactly the same?\n2. **Compare with half a right angle (45°).** If it is acute, is it thinner or fatter than half a right angle?\n3. **Use the clock.** Each hour gap is 30°. Picture clock hands to see 30°, 60°, 90°, 120° and 150°.\n4. **Say a number.** Commit to a guess, like \"about 70°\".",{"id":443,"type":444,"prompt":445,"options":446,"explanation":459},"predict-estimate","prediction","Picture an angle that is a bit **less** than a right angle, but clearly **more than half** of a right angle. Which estimate is most sensible?",[447,450,453,456],{"id":448,"label":449},"a","About 20°",{"id":451,"label":452},"b","About 70°",{"id":454,"label":455},"c","About 110°",{"id":457,"label":458},"d","About 160°","**About 70°.** Less than a right angle means under 90°. More than half a right angle means over 45°. So the angle sits somewhere between 45° and 90°, and 70° is the only choice in that range. 110° and 160° are obtuse (more than 90°), and 20° is less than half a right angle.",{"id":461,"type":179,"component":462,"componentVersion":5,"config":463,"objective":467,"textAlternative":468,"help":469},"lab-estimate","angle-lab",{"modes":464,"allowReflex":465,"rounds":466},[408],false,6,"Estimate the size of angles by eye, then see how close you were.","This game shows six angles, one at a time, each between 0° and 180°. For each one you type an estimate in degrees, and the game reveals the true size and how many degrees away you were. The closer you are, the more points you score.\n\nA good way to play: first decide whether the angle is acute (less than 90°) or obtuse (more than 90°). Then compare with 45° (half a right angle) or 135° (a right angle plus half a right angle). Then pick a number. For example, an angle a little wider than half a right angle might be about 50° or 55°. An angle a little more than a right angle might be about 100°.\n\nWith practice, many people get within 10°, and careful estimators within 5°.",{"simplerExplanation":470,"hints":471},"First ask: smaller or bigger than a page corner? Then: thinner or fatter than half of it? Then guess.",[472,473],"A right angle is 90°. Half a right angle is 45°.","Picture clock hands: one hour apart is 30°, two hours apart is 60°.",{"id":475,"type":47,"variant":416,"title":476,"markdown":477},"try-body","Angles with your arms","Stand with both arms stretched straight out in front of you, together. That is 0°. Keep your left arm still and swing your right arm out to the side until it points straight sideways: that is **90°**. Swing it all the way until both arms make one straight line: **180°**. Now ask a friend to call out 30°, 45°, 60° and 120°, and try to show each one. Check with the clock trick.",{"id":479,"type":53,"title":480,"eyebrow":481,"navLabel":482},"ch5","Meet the protractor","Chapter 05","5 Meet the protractor",{"id":484,"type":43,"markdown":485},"protractor-parts","Take the protractor out of your box and look closely. It has three parts that matter:\n\n- The **centre point**: a tiny hole, cross or dot in the middle of the straight edge. This is where the **vertex** of your angle must go.\n- The **base line**: the line on the protractor that runs through the centre point from 0 on one side to 180 on the other. One **arm** of your angle must lie along it.\n- **Two scales**: two rows of numbers from 0 to 180 running round the curved edge. One row counts up from the **right**, the other counts up from the **left**. Which of the two rows is printed on the inside and which on the outside is **not the same on every protractor**, so never go by “inner” or “outer”. Go by where the **0** is.\n\nWhy two scales? So you can measure an angle that opens either way, without turning the protractor upside down.",{"id":487,"type":144,"tone":488,"items":489},"spec-protractor","amber",[490,494,498,502,506],{"label":491,"big":492,"value":493},"Centre point","vertex here","The small mark in the middle of the straight edge. Put the angle's corner exactly on it.",{"label":495,"big":496,"value":497},"Base line","0 to 180","The line through the centre point. Line up one arm of the angle along it.",{"label":499,"big":500,"value":501},"Right-hand scale","0 on the right","The row whose 0 sits at the right-hand end of the base line. Read it when your first arm points right.",{"label":503,"big":504,"value":505},"Left-hand scale","0 on the left","The row whose 0 sits at the left-hand end. Read it when your first arm points left. (Either row may be the inner one.)",{"label":507,"big":508,"value":509},"Small marks","1° each","Between the numbered marks, every little line is one degree.",{"id":511,"type":47,"variant":174,"title":512,"markdown":513},"misc-edge","“The bottom edge of the protractor is the base line”","On many protractors the straight plastic edge is a few millimetres **below** the real base line. If you put your angle's arm along the edge, the vertex is not on the centre point and your reading will be wrong. Look for the printed line that passes through the centre mark and runs to the 0 on each side. That is the base line.",{"id":515,"type":43,"markdown":516},"two-scales-hint","Here is the one rule you need for the two scales:\n\n**Use the scale that shows 0 on the arm you lined up with the base line.**\n\nFollow that scale round, counting up from 0, until you reach the other arm. The number where the other arm crosses is the size of the angle. The other scale at the same spot will show a different number, and the two numbers always add up to **180**. That is why a quick estimate is so useful: it tells you which of the two numbers makes sense.",{"id":518,"type":53,"title":519,"eyebrow":520,"navLabel":521},"ch6","Your first measurement","Chapter 06","6 Measure an angle",{"id":523,"type":524,"title":525,"items":526},"steps-measure","steps","How to measure an angle with a protractor",[527,531,534,538,542,546],{"title":528,"tag":529,"text":530},"Estimate","guess","Look at the angle. Is it acute or obtuse? Make a guess, like “about 50°”.",{"title":532,"tag":367,"text":533},"Place the centre","Put the protractor's centre point exactly on the vertex of the angle.",{"title":535,"tag":536,"text":537},"Line up the base line","one arm","Turn the protractor so that the base line lies exactly along one arm.",{"title":539,"tag":540,"text":541},"Find the 0","choose the scale","Find which scale has 0 on that arm. That is the scale you will read.",{"title":543,"tag":544,"text":545},"Read","other arm","Follow that scale up from 0 to where the other arm crosses. Read the number.",{"title":547,"tag":548,"text":549},"Check","compare","Does the reading match your estimate? If not, you probably read the wrong scale.",{"id":551,"type":552,"title":553,"problem":554,"steps":555,"help":562},"we-measure-50","worked_example","Measuring an angle that opens to the right","An angle has one arm pointing to the right and the other arm pointing up and slightly to the right. Before measuring, you estimate it at \"about 50°\". Where the second arm crosses the protractor, the two scales show **50** and **130**. What is the angle?",[556,557,558,559,560,561],"Estimate: the angle is acute, a bit more than half a right angle, so about 50°.","Place the centre point on the vertex and line up the base line with the arm that points right.","The arm that points right lies on the **0 at the right-hand end** of the base line, so that is the scale to read.","Count up that scale from 0 to the other arm: it crosses at **50**.","The other scale shows 130 at the same spot. Check: 50 + 130 = 180, as always.","The angle is acute, so 130 is impossible. The answer is **50°**, which matches the estimate. ✓",{"simplerExplanation":563},"Two numbers show where the arm crosses: 50 and 130. The angle is small (acute), so it must be 50°.",{"id":565,"type":552,"title":566,"problem":567,"steps":568},"we-measure-left","Measuring an angle that opens to the left","An angle has one arm pointing to the **left** from the vertex, and the other arm pointing up and to the right, so the angle is wide. Your estimate is \"a bit more than a right angle\". The other arm crosses where the scales show **65** and **115**. What is the angle?",[569,570,571,572,573,574],"Estimate: wider than a page corner, so more than 90°.","Centre point on the vertex; base line along the arm that points left.","That arm sits on the **0 at the left-hand end** of the base line, so that is the scale to read.","Count up that scale from 0 until you meet the other arm: **115**.","The other scale shows 65 at the same point, and 65 + 115 = 180.","The angle is obtuse, so the answer is **115°**. ✓",{"id":576,"type":47,"variant":577,"title":578,"markdown":579},"nuance-full-circle","nuance","Full-circle protractors","Some protractors are a whole circle marked 0° to 360°. They are handy for big angles, called **reflex angles** (more than 180°). With a half-circle protractor you can still measure a reflex angle, with a small trick you will meet in the next layer.",{"id":581,"type":179,"component":187,"componentVersion":5,"config":582,"objective":589,"textAlternative":590,"help":591},"lab-measure",{"mode":583,"targets":584,"tolerance":588},"measure",[18,585,586,587,426],45,90,120,2,"Place a virtual protractor on five angles and read the correct scale to measure each one.","This lab shows one angle at a time and a protractor you can move and turn. You put the centre point on the vertex, line up the base line with one arm, then read the scale that starts at 0 on that arm. Answers within 2° of the true value score points.\n\nThe five angles are 30°, 45°, 90°, 120° and 150°.\n\n- **30°**: acute. The other scale shows 150 at the same place, the wrong one.\n- **45°**: half a right angle. The other scale also shows 135 there.\n- **90°**: both scales show 90, because 90 + 90 = 180.\n- **120°**: obtuse. The other scale shows 60, a common trap.\n- **150°**: very wide. The other scale shows 30.\n\nEach time, the true answer is the one that matches your estimate.",{"simplerExplanation":592,"hints":593},"Put the dot on the corner. Put the line on one arm. Start counting from the 0 on that arm.",[594,595],"If the angle looks smaller than a page corner, your answer must be less than 90.","The two numbers at the crossing point always add up to 180.",{"id":597,"type":421,"itemId":598,"prompt":599,"check":600,"hints":611,"feedback":614},"practice-read","constructing-angles.disc-read-scale","You line up one arm on the base line. The other arm crosses the protractor where the two scales show **70** and **110**. The angle is clearly **wider** than a right angle. What is its size?",{"kind":601,"options":602,"correct":610},"choice",[603,605,607,608],{"id":448,"label":604},"70°",{"id":451,"label":606},"110°",{"id":454,"label":343},{"id":457,"label":609},"40°",[451],[612,613],"Wider than a right angle means more than 90°.","Only one of the two numbers is more than 90.",{"correct":615,"incorrect":616},"Right: an obtuse angle must be more than 90°, so it is 110°. The 70 belongs to the other scale.","The angle is wider than a right angle, so it is more than 90°. Of 70 and 110, only **110°** fits. (70 + 110 = 180.)",{"id":618,"type":53,"title":619,"eyebrow":620,"navLabel":621},"ch7","Drawing an angle of a given size","Chapter 07","7 Draw an angle",{"id":623,"type":43,"markdown":624},"draw-intro","Measuring is reading an angle that is already there. **Drawing** is the opposite: you start with a number, like 50°, and make an angle of exactly that size. The protractor does both jobs.",{"id":626,"type":524,"title":627,"items":628},"steps-draw","How to draw an angle of a given size",[629,632,636,638,642,645],{"title":630,"tag":185,"text":631},"Draw one arm","Use a ruler to draw a straight line. Mark one end as the vertex, for example O.",{"title":633,"tag":634,"text":635},"Place the protractor","centre on O","Put the centre point on O and the base line exactly along your arm.",{"title":539,"tag":540,"text":637},"Find the scale with 0 on your arm. Only that scale gives the right angle.",{"title":639,"tag":640,"text":641},"Mark a dot","at the number","Count up that scale to the number you want and make a small dot at the edge.",{"title":643,"tag":185,"text":644},"Join","Remove the protractor and use the ruler to join O to the dot. Label the angle.",{"title":547,"tag":583,"text":646},"Measure your new angle. Does it look like your estimate of the size?",{"id":648,"type":552,"title":649,"problem":650,"steps":651},"we-draw-50","Drawing a 50° angle","Draw an angle of 50° at a point O on a line OA that points to the right.",[652,653,654,655,656,657],"Draw a line OA about 6 cm long with the ruler, with O at the left end.","Put the protractor's centre point on O and the base line along OA.","OA points to the right, so use the scale that has **0 on the right**.","Count up that scale to **50** and make a small dot at the edge of the protractor. Call it B.","Take away the protractor and join O to B with the ruler. ∠AOB = 50°.","Check: 50° is a bit more than half a right angle. Does your angle look like that? ✓",{"id":659,"type":47,"variant":174,"title":660,"markdown":661},"misc-wrong-dot","Putting the dot on the wrong scale","If you want 50° but put your dot at the 50 on the **wrong** scale, you will draw 180° − 50° = **130°**, an obtuse angle. The fix is the estimate: 50° is less than a right angle, so if your drawing looks wider than a page corner, start again.",{"id":663,"type":421,"itemId":664,"prompt":665,"check":666,"hints":677,"feedback":679},"practice-draw","constructing-angles.disc-draw-dot","Riya wants to draw an angle of **30°**. Her first arm points to the **left** from the vertex. Which scale should she use to find 30?",{"kind":601,"options":667,"correct":676},[668,670,672,674],{"id":448,"label":669},"The scale that has 0 on the left",{"id":451,"label":671},"The scale that has 0 on the right",{"id":454,"label":673},"Either scale, it does not matter",{"id":457,"label":675},"Neither: she must turn the paper over",[448],[678],"Which end of the protractor does her arm sit on?",{"correct":680,"incorrect":681},"Yes. Her arm sits on the left, so she counts up from the 0 on the left. Using the other scale would give 150°.","Her arm lies along the left part of the base line, so she must use the scale that starts at 0 on the **left**. The other scale would give her 180° − 30° = 150°.",{"id":683,"type":53,"title":684,"eyebrow":685,"navLabel":686},"ch8","The compass trick: 60° with no protractor","Chapter 08","8 The compass trick",{"id":688,"type":43,"markdown":689},"compass-intro","Now for the puzzle from Chapter 1. Can you make an **exact** 60° angle with only a **compass** and a **ruler**, and no numbers at all?\n\nYes, and it is one of the oldest tricks in mathematics. The ruler is only used to draw straight lines, never to measure. Drawing like this, with only a straight edge and a compass, is called a **construction**.",{"id":691,"type":444,"prompt":692,"options":693,"explanation":702},"predict-compass","You draw a circle with your compass. Without changing how wide the compass is open, you put the point anywhere **on** the circle and draw a mark across it. You move to that mark and repeat, going round and round. How many steps does it take to come back exactly where you started?",[694,696,698,700],{"id":448,"label":695},"4 steps",{"id":451,"label":697},"5 steps",{"id":454,"label":699},"6 steps",{"id":457,"label":701},"It never comes back exactly","**6 steps, exactly.** The compass width is the same as the circle's radius, so each step makes a triangle whose three sides are all the same length, with the centre as one corner. Such a triangle is called **equilateral**, and each of its angles is 60°. Six of those 60° angles fit round the centre: 6 × 60° = 360°, a full turn. Try it with a real compass: this is also how people draw the six-petal flower in rangoli.",{"id":704,"type":705,"component":706,"componentVersion":5,"config":707,"textAlternative":709},"anim-60","animation","compass-construction",{"construction":708},"angle-60","This animation shows how to construct an angle of 60° with only a ruler and compass, one step at a time.\n\n1. **Draw a ray.** Use the ruler to draw a straight ray OA, starting at the point O.\n2. **Draw a big arc.** Put the compass point on O. Open it to any width you like, say 4 cm, and draw a long arc that crosses the ray OA. Call the crossing point P.\n3. **Do not change the compass.** Keep exactly the same width. Move the compass point to P and draw a small arc that cuts the big arc. Call the new crossing point Q.\n4. **Draw the second arm.** Use the ruler to draw a ray from O through Q.\n5. **Done.** The angle AOQ is exactly **60°**.\n\n**Why it works:** OP is a radius of the first arc, and so is OQ. PQ was drawn with the same compass width. So OP, OQ and PQ are all the same length, and triangle OPQ is **equilateral**. Every angle of an equilateral triangle is 60°, so the angle at O is 60°. You can check it with a protractor: it should read 60°.",{"id":711,"type":47,"variant":166,"title":712,"markdown":713},"careful-compass-hold","Holding a compass properly","Hold the compass by the **knob at the top**, not by its legs. Squeezing the legs changes the width without you noticing. Tilt it slightly in the direction you are turning and let it swing in one smooth movement. If the point slips, the arc will not be centred where you think.",{"id":715,"type":47,"variant":278,"title":716,"markdown":717},"aha-any-radius","Any width works, as long as you keep it","It does not matter whether your compass is open 3 cm or 8 cm. A bigger width gives a bigger triangle, but it is still equilateral, so the angle is still 60°. The one thing you must not do is **change the width between the two arcs**. That is why a tight compass hinge matters.",{"id":719,"type":47,"variant":416,"title":720,"markdown":721},"try-flower","Draw a six-petal rangoli flower","1. Draw a circle with your compass, about 4 cm wide.\n2. Keep the same width. Put the point anywhere on the circle and draw an arc from one side of the circle to the other, passing through the centre.\n3. Move the point to where that arc meets the circle and draw another arc the same way.\n4. Keep going round. After six arcs you are back where you started, and a flower with six petals appears.\n\nEach petal tip is 60° round from the next: 6 × 60° = 360°. Colour it in, or use it as the starting pattern for a rangoli.",{"id":723,"type":47,"variant":724,"title":725,"markdown":726},"ex-hexagon","example","Why honeybees would approve","Join the six points where the petals touch the circle and you get a **regular hexagon**, the shape of a honeycomb cell. Each corner of that hexagon is 120°, which is two 60° angles side by side. Floor tiles, nuts and bolts, and pencils often use hexagons for the same reason: they fit together with no gaps.",{"id":728,"type":43,"markdown":729},"more-constructions","Once you can make 60°, a whole family of exact angles opens up. Step round the arc twice from P and you get **120°**. Cut an angle exactly in half, which is called **bisecting** it, and 60° becomes **30°**. A right angle, 90°, can be made in more than one way with a compass, and cutting it in half gives **45°**. You will learn each of these, and why they work, in the next layers.",{"id":731,"type":53,"title":732,"eyebrow":733,"navLabel":734},"ch9","Set squares: ready-made angles","Chapter 09","9 Set squares",{"id":736,"type":43,"markdown":737},"setsquare-intro","Your geometry box also holds two triangles of plastic or metal, called **set squares**. Each one carries a few angles you can trace straight away, with no measuring and no arcs:\n\n- The **45° set square** has angles of **45°, 45° and 90°**.\n- The **30°–60° set square** has angles of **30°, 60° and 90°**.\n\nSo with set squares alone you can draw 30°, 45°, 60° and 90° angles in a second. Engineers and draughtspeople used them all day long before computers.",{"id":739,"type":552,"title":740,"problem":741,"steps":742,"help":748},"we-75","Making 75° with two set squares","Rahul has only his two set squares. How can he draw a 75° angle?",[743,744,745,746,747],"List the angles he has: 30°, 45°, 60° and 90°.","Look for two that add up to 75: 30 + 45 = **75**. ✓","Place the 45° corner of one set square at the vertex and draw along both its edges.","Put the 30° corner of the other set square against one of those lines, on the outside, sharing the same vertex, and draw along its far edge.","The two angles sit side by side, so together they make 30° + 45° = **75°**.",{"simplerExplanation":749},"Put a 30° angle and a 45° angle next to each other at the same corner. 30 + 45 = 75.",{"id":751,"type":62,"caption":752,"columns":753,"rows":755},"table-setsq","Angles you can make with the two set squares",[316,754,547],"How",[756,760,764,768,771,775],[757,758,759],"15°","45° take away 30°","45 − 30 = 15",[761,762,763],"75°","45° next to 30°","45 + 30 = 75",[765,766,767],"105°","45° next to 60°","45 + 60 = 105",[339,769,770],"90° next to 30°, or 60° next to 60°","90 + 30 = 120",[772,773,774],"135°","90° next to 45°","90 + 45 = 135",[776,777,778],"150°","90° next to 60°","90 + 60 = 150",{"id":780,"type":179,"component":781,"componentVersion":5,"config":782,"objective":805,"textAlternative":806,"help":807},"lab-match-tools","match-pairs",{"prompt":783,"mode":784,"pairs":785},"Flip the cards to match each tool or angle to its partner.","memory",[786,788,790,792,795,798,801,803],{"a":119,"b":787},"Measures angles in degrees",{"a":123,"b":789},"Draws circles and arcs",{"a":127,"b":791},"Copies a length exactly",{"a":793,"b":794},"45° set square","Angles 45°, 45° and 90°",{"a":796,"b":797},"30°–60° set square","Angles 30°, 60° and 90°",{"a":799,"b":800},"Two equal compass arcs","An exact 60° angle",{"a":802,"b":761},"30° + 45°",{"a":804,"b":351},"Full turn","Match tools and angle facts from the geometry box in a memory card game.","This is a memory game with 16 face-down cards making 8 pairs. Turn over two cards at a time and keep them if they match.\n\nThe pairs are: protractor ↔ measures angles in degrees; compass ↔ draws circles and arcs; divider ↔ copies a length exactly; 45° set square ↔ angles 45°, 45° and 90°; 30°–60° set square ↔ angles 30°, 60° and 90°; two equal compass arcs ↔ an exact 60° angle; 30° + 45° ↔ 75°; full turn ↔ 360°.\n\nThe fewer turns you need, the better your score.",{"hints":808},[809],"Remember where each card was, even when it does not match.",{"id":811,"type":53,"title":812,"eyebrow":813,"navLabel":814},"ch10","Where people use this","Chapter 10","10 In real life",{"id":816,"type":817,"title":818,"prompt":819,"options":820},"explorer-jobs","explorer","Who measures and makes angles?","Pick a job to see how angles are used in it.",[821,833,844,855,866],{"id":822,"label":823,"chain":824,"badge":829,"note":832},"carpenter","Carpenter",[825,826,827,828],"A frame or a table","Mark the angle","Cut the wood","Corners fit tightly",{"text":830,"tone":831},"Uses a try square and mitre box","yes","A carpenter making a photo frame cuts each end of the wood at **45°**, so two pieces meet to make a 90° corner. A try square (a steel right angle) checks that corners are exactly 90°. For a hexagonal table, the pieces must meet at 120°.",{"id":834,"label":835,"chain":836,"badge":841,"note":843},"architect","Architect",[837,838,839,840],"An idea for a building","Scale drawing","Angles of walls and roofs","Builders follow it",{"text":842,"tone":831},"Draws exact angles","Architects make careful drawings where every angle matters: the slope of a roof so rain runs off in the monsoon, the angle of a staircase so it is safe, the corners of rooms. Today most of this is done on computers, but the ideas are the same as with a protractor and set squares.",{"id":845,"label":846,"chain":847,"badge":852,"note":854},"roads","Road designer",[848,849,850,851],"Traffic needs","Plan the junction","Choose angles","Safe roads",{"text":853,"tone":831},"Plans safe angles","Roads that meet at close to 90° are safer, because drivers can see traffic coming from both sides. Hill roads curve round hairpin bends, and railway tracks join at very small angles so trains can switch smoothly from one track to another.",{"id":856,"label":857,"chain":858,"badge":863,"note":865},"sports","Groundskeeper",[859,860,861,862],"An empty field","Mark a straight line","Mark right angles","Paint the court",{"text":864,"tone":831},"Lays out right angles","To paint a kabaddi court or a football pitch, the corners must be exact right angles. Groundskeepers often use a rope with knots marking 3, 4 and 5 equal lengths: stretched into a triangle, it makes a perfect right angle. People in ancient India used ropes and pegs in a similar way to lay out fire altars.",{"id":867,"label":868,"chain":869,"badge":874,"note":876},"kite","Kite maker",[870,871,872,873],"Two bamboo sticks","Cross them","Tie at 90°","Paper and string",{"text":875,"tone":831},"Needs a right angle","A patang maker ties the straight spine and the bent bow at right angles, with the spine cutting the bow exactly in the middle. If the cross is lopsided, one side of the kite catches more wind and it spins.",{"id":878,"type":47,"variant":724,"title":879,"markdown":880},"ex-rangoli","Rangoli and kolam: angles on the doorstep","Many rangoli and kolam patterns are built on a circle divided into equal parts: 4 parts of 90°, 6 parts of 60°, or 8 parts of 45°. Artists often fold a string or use a round plate as a guide, which is geometry without the geometry box.",{"id":882,"type":883,"conceptId":884,"relation":885,"explanation":886},"conn-angles","connection","angles","helps_understand","Knowing angle types (acute, right, obtuse, straight, reflex) helps you estimate before you measure and spot the wrong-scale mistake.",{"id":888,"type":883,"conceptId":889,"relation":885,"explanation":890},"conn-lines","lines","Constructions are built from lines, rays and segments, so knowing them well makes every step clearer.",{"id":892,"type":883,"conceptId":893,"relation":894,"explanation":895},"conn-shapes","shape-and-space","applied_in","Equilateral triangles, squares and hexagons are all drawn using the exact angles you learn to make here.",{"id":897,"type":53,"title":898,"eyebrow":899,"navLabel":900},"ch11","Check your understanding","Chapter 11","11 Wrap-up",{"id":902,"type":903,"prompt":904},"reflect-discover","reflection","Think of one object at home or at school that needs an exact angle, not just \"about right\". Which tool from your geometry box would you use to make or check it, and what would go wrong if the angle were off by a few degrees?",{"id":906,"type":907,"title":908,"questions":909},"quiz-discover","quiz","Measuring and making angles: quick check",[910,920,929,942,955,968,980,993,1006,1019],{"itemId":911,"prompt":912,"options":913,"correct":454,"why":919},"constructing-angles.disc-q-full-turn","How many degrees are there in one full turn?",[914,915,916,917],{"id":448,"label":335},{"id":451,"label":343},{"id":454,"label":351},{"id":457,"label":918},"100°","A full turn is split into 360 equal parts called degrees.",{"itemId":921,"prompt":922,"options":923,"correct":451,"why":928},"constructing-angles.disc-q-divider","Which tool has two sharp points and no pencil?",[924,925,926,927],{"id":448,"label":123},{"id":451,"label":127},{"id":454,"label":119},{"id":457,"label":194},"A divider has two metal points and is used to compare and copy lengths.",{"itemId":930,"prompt":931,"options":932,"correct":448,"why":941},"constructing-angles.disc-q-centre","When measuring an angle, where must the protractor's centre point go?",[933,935,937,939],{"id":448,"label":934},"On the vertex",{"id":451,"label":936},"At the end of one arm",{"id":454,"label":938},"Anywhere on the angle",{"id":457,"label":940},"Halfway along an arm","The centre point goes exactly on the vertex, where the two arms meet.",{"itemId":943,"prompt":944,"options":945,"correct":448,"why":954},"constructing-angles.disc-q-scale","One arm lies on the base line through the 0 on the right. Which scale should you read?",[946,948,950,952],{"id":448,"label":947},"The scale with 0 on the right",{"id":451,"label":949},"The scale with 0 on the left",{"id":454,"label":951},"The larger number",{"id":457,"label":953},"The smaller number","Always read the scale that has 0 on the arm you lined up. Then check with your estimate.",{"itemId":956,"prompt":957,"options":958,"correct":451,"why":967},"constructing-angles.disc-q-two-numbers","The other arm crosses at 35 on one scale. What does the other scale show there?",[959,961,963,965],{"id":448,"label":960},"35",{"id":451,"label":962},"145",{"id":454,"label":964},"325",{"id":457,"label":966},"55","The two scales always add to 180 at the same point: 180 − 35 = 145.",{"itemId":969,"prompt":970,"options":971,"correct":451,"why":979},"constructing-angles.disc-q-obtuse","An angle looks wider than a page corner. The scales show 65 and 115. What is it?",[972,974,976,977],{"id":448,"label":973},"65°",{"id":451,"label":975},"115°",{"id":454,"label":343},{"id":457,"label":978},"50°","Wider than a right angle means more than 90°, so it is 115°.",{"itemId":981,"prompt":982,"options":983,"correct":451,"why":992},"constructing-angles.disc-q-sixty","In the compass trick, why is angle AOQ exactly 60°?",[984,986,988,990],{"id":448,"label":985},"Because the compass is 6 cm wide",{"id":451,"label":987},"Because triangle OPQ has three equal sides",{"id":454,"label":989},"Because the ruler says so",{"id":457,"label":991},"It is only about 60°","OP, OQ and PQ all equal the compass width, so the triangle is equilateral and each angle is 60°.",{"itemId":994,"prompt":995,"options":996,"correct":454,"why":1005},"constructing-angles.disc-q-steps","Stepping round a circle with the compass kept at the radius, how many steps bring you back to the start?",[997,999,1001,1003],{"id":448,"label":998},"4",{"id":451,"label":1000},"5",{"id":454,"label":1002},"6",{"id":457,"label":1004},"8","Each step turns 60° round the centre, and 6 × 60° = 360°.",{"itemId":1007,"prompt":1008,"options":1009,"correct":448,"why":1018},"constructing-angles.disc-q-setsq","Which two set-square angles can you put side by side to make 105°?",[1010,1012,1014,1016],{"id":448,"label":1011},"45° and 60°",{"id":451,"label":1013},"30° and 45°",{"id":454,"label":1015},"30° and 60°",{"id":457,"label":1017},"45° and 45°","45 + 60 = 105.",{"itemId":1020,"prompt":1021,"options":1022,"correct":454,"why":1027},"constructing-angles.disc-q-clock","What angle do clock hands make at 4 o'clock?",[1023,1024,1025,1026],{"id":448,"label":609},{"id":451,"label":335},{"id":454,"label":339},{"id":457,"label":776},"Each hour gap is 360 ÷ 12 = 30°, so 4 gaps make 4 × 30° = 120°.",{"id":1029,"type":1030,"title":1031,"points":1032},"cheat-discover","summary","Cheat sheet",[1033,1034,1035,1036,1037,1038,1039,1040,1041],"An **angle** is the amount of turn between two **arms** that meet at a **vertex**. It is measured in **degrees (°)**; a full turn is **360°**.","Benchmarks: **90°** right angle (page corner), **180°** straight angle, **45°** half a right angle, **60°** clock at 2 o'clock, **30°** one hour gap on a clock.","The **geometry box**: ruler (lines, lengths), protractor (degrees), compass (circles, arcs), divider (copying lengths), set squares (30°, 45°, 60°, 90°).","**Estimate first.** Decide acute or obtuse, compare with 45° or 90°, then guess a number.","**Measuring:** centre point on the vertex, base line along one arm, read the scale with **0 on that arm**, check with your estimate.","The two scales always add to **180** at the same point, so a reading of 40 on one is 140 on the other.","**Drawing:** draw one arm, put the protractor on it, dot at the number on the right scale, join with a ruler.","**Compass trick:** equal arcs make an equilateral triangle, so the angle is exactly **60°**. Six steps go right round a circle: 6 × 60° = 360°.","Set squares side by side give more angles: 30 + 45 = **75°**, 45 + 60 = **105°**, 90 + 45 = **135°**.",{"id":1043,"type":1044,"sourceIds":1045},"sources-discover","sources",[1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057],"constructing-angles-ncert-math-6-practical-geometry","constructing-angles-ncert-ganita-prakash-6","constructing-angles-ncert-math-7-practical-geometry","constructing-angles-mathsisfun-protractor","constructing-angles-mathsisfun-constructions","constructing-angles-mathsisfun-degrees","constructing-angles-wikipedia-straightedge-compass","constructing-angles-wikipedia-angle-trisection","constructing-angles-wikipedia-shulba-sutras","constructing-angles-britannica-euclid-elements","constructing-angles-wikipedia-degree-angle","constructing-angles-wikipedia-ptolemy-chords",[1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057],"needs_review",{"generatedBy":1061,"notes":1062},"claude-code","Draft generated with Python; every angle fact was computed and asserted. Pending owner review.","434aaa13a1f112d9c3b401e673cd72e55f3e9b451e65bc8591e5f025af9db49c",{"component:sort-game@1":1065,"logic:practice":1066,"component:angle-lab@1":1067,"component:protractor@1":1068,"component:compass-construction@1":1069,"component:match-pairs@1":1070,"source:constructing-angles-britannica-euclid-elements":1071,"source:constructing-angles-mathsisfun-constructions":1072,"source:constructing-angles-mathsisfun-degrees":1073,"source:constructing-angles-mathsisfun-protractor":1074,"source:constructing-angles-ncert-ganita-prakash-6":1075,"source:constructing-angles-ncert-math-6-practical-geometry":1076,"source:constructing-angles-ncert-math-7-practical-geometry":1077,"source:constructing-angles-wikipedia-angle-trisection":1078,"source:constructing-angles-wikipedia-degree-angle":1079,"source:constructing-angles-wikipedia-ptolemy-chords":1080,"source:constructing-angles-wikipedia-shulba-sutras":1081,"source:constructing-angles-wikipedia-straightedge-compass":1082},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","7407db21456592711dd16c6bdad23f042e85ebab9c314b072b17b7065eaf8ae3","0eba81381f31d8d78008911eb4c3d62745efd33b4ffadd94aef0851d0a2ad5f3","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","ad1a9a6a227fda5d3c1569f37efbe35e448ebaceba8cba872821fd48e2e00ed6","b93faffbd9c4d40f5fce2bc4b2ea0ab5ac64bb8c176f5e2bba3f37444df5e400","216eb0db510461864a47157f14054a39e15b1b0fc461b0fbc77664d9eb28b91d","4d3f50c07f44df57c80455dc39e01b2aa11bb0ee40811fca3b0f12d16e0b3f5e","acad5a4d56d24a5c1ad6e908f3809f2e7b3978f7c2810a9b33cdf82a65c4bb47","4aedaa1be389589b6e840923ef4e92fd15d03eda0b0ba0302d57b7e71bcb3dc7","21119e12648b9efd4cc82b11c59d626f2a53eace3a70552041f26c311b77ba2d","7f00387dc29d17172d25b6aa96420e2544a8bc59edf939af3dce91d515300a1f","cc9b994305131f87113cd7c734d7ea0a03ae4904ce15cf134cd8bf3a397c6623","df1e69e29b281b0c0f017bc895cc53331fb534c5eaf4b7afb1ff92d3480ab503","f7559697a2f963f9cb1e02a93fc5697840f583f22605d7483d688664862d70f9","90d6cc756bc1bdd6cde0d5e4ed2000c88c3e2f3a8d91fdaf0ec4ab73af72b434",{"state":1084,"reviewer":1085,"selfReview":1086,"reviewedAt":1087,"method":1088},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597049]