[{"data":1,"prerenderedAt":1139},["ShallowReactive",2],{"layer:constructing-angles:investigate":3},{"layer":4,"contentHash":1114,"dependencyHashes":1115,"approval":1133,"releaseId":1138},{"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":1109,"reviewStatus":1110,"authoring":1111},1,"constructing-angles","en","investigate","Test it: estimates, radii and angle recipes","Predict, try and check: what really changes an angle, and what never does","Predict and test: does arm length matter, what does a wrong-scale reading look like, how good is your eye, does the compass radius matter, which angles can bisecting and set squares reach, how accurate can a check be, and why bisectors always work.",[13,14,15,16,17],"Explain, with evidence, why arm length and compass radius size do not change an angle, but a changed radius mid-construction does.","Detect a wrong-scale reading using 180 − x and an acute\u002Fobtuse estimate.","Plan ruler-and-compass routes to 15°, 45°, 75°, 105°, 135°, 150° and 165°, and explain why 20° is out of reach.","List every angle the two set squares can make together, and explain why they are all multiples of 15°.","Judge a construction against a ±1° tolerance and test the equidistance property of both bisectors.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Protractor use and basic constructions",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Estimate game, 2 constructions, 2 games, protractor",{"label":38,"value":39},"You need","Geometry box, plain paper",[41,45,51,57,60,78,83,88,91,106,111,114,127,164,167,178,189,201,206,209,222,239,264,269,290,295,298,311,324,336,340,345,350,353,360,373,406,411,423,428,431,436,447,493,496,579,583,594,614,619,622,631,648,651,680,690,695,706,711,714,730,741,760,765,784,793,797,802,805,818,822,835,839,848,858,862,882,887,935,939,945,950,955,1080,1095],{"id":42,"type":43,"markdown":44},"intro","prose","In Discover and Understand you learned the *moves*: line up a protractor, swing a compass, bisect an angle. This layer is different. Here you are the scientist. Every chapter asks a **what happens if…?** or **is it always true?** question, asks you to **commit to a prediction**, and then tests it with a lab, a sheet of paper or a quick calculation.\n\nYou will find out whether long arms make a bigger angle, what a wrong-scale reading looks like, how good your eye really is, whether the compass radius matters, how small an angle you can reach by halving, which angles you can build from 60° and 90°, what your two set squares can make together, how accurate a pencil-and-protractor check can be, and why the bisectors you draw work *every* time.\n\nKeep a geometry box, a few sheets of plain paper and a sharp pencil next to you. Many of the tests take less than a minute to do for real.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how","callout","observation","How to use this layer","Before you open each explanation, **choose an option and say why**. Being surprised is not failure: a wrong prediction you understand afterwards teaches more than a lucky right one. Write your predictions on the side of your paper and tick them off as you go.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch-arms","chapter","Does arm length change the angle?","Chapter 01","1 Arm length",{"id":58,"type":43,"markdown":59},"arms-set","Draw two angles. The first has arms 3 cm long. The second has arms 12 cm long, four times as long. You open them by exactly the same amount. Which is the bigger angle?\n\nMany learners, and quite a few adults, feel that the angle with long arms is \"bigger\". It *looks* bigger: it covers more paper. Before reading on, decide what you think.",{"id":61,"type":62,"prompt":63,"options":64,"explanation":77},"pred-arms","prediction","Angle X has arms 3 cm long. Angle Y has arms 12 cm long. Both are opened by the same amount of turn. Which is true?",[65,68,71,74],{"id":66,"label":67},"a","Y is bigger, because its arms are longer",{"id":69,"label":70},"b","X is bigger, because its arms are closer together",{"id":72,"label":73},"c","They are equal: arm length does not change the angle",{"id":75,"label":76},"d","It depends on which one you measure first","**They are equal.** An angle measures the **amount of turn** from one arm to the other, not the length of the arms or the space between their ends. Imagine the arms as rays: a ray goes on for ever, so a drawn arm is just the part of the ray we chose to draw. Stretching a 3 cm arm to 12 cm draws more of the *same* ray, and the turn between the two rays does not change.\n\nWhat *does* change is the **gap between the arm ends**. At 3 cm the ends of a 40° angle are about 2 cm apart; at 12 cm they are about 8.2 cm apart. The gap grows in proportion to the arm length, but the angle stays 40°.",{"id":79,"type":47,"variant":80,"title":81,"markdown":82},"try-arms","try_it","Test it with two pencils","1. Draw an angle of 40° with a protractor and arms 3 cm long.\n2. Using a ruler, extend both arms to 12 cm without moving the vertex.\n3. Measure again. You should still read 40° (within about 1°).\n4. Now measure the gap between the ends of the arms at 3 cm and at 12 cm. The first gap is about 2.1 cm, the second about 8.2 cm: four times as big, because the arms are four times as long.\n\nThe angle stayed the same while the gap quadrupled. That is the difference between *how far apart the arms are* and *how much they turn*.",{"id":84,"type":47,"variant":85,"title":86,"markdown":87},"mis-arms","misconception","“Longer arms, bigger angle”","This is the most common angle mix-up in school. It comes from judging an angle by the **area** it seems to cover or by the **distance between the arm tips**. Both grow when the arms get longer, but neither is the angle. A tiny clock and Mumbai's Rajabai clock tower show the **same** 90° between the hands at 3 o'clock, even though the tower's hands are metres long.",{"id":89,"type":43,"markdown":90},"arms-useful","There is a useful side to this fact. Because arm length does not change the angle, you are **allowed to extend short arms** before measuring. If an arm is too short to reach the scale of your protractor, lay a ruler along it and extend it lightly in pencil. You are drawing more of the same ray, so the angle you measure is the same angle.\n\nLonger arms also make measurement **more accurate**, as you will discover in Chapter 8: a tiny wobble in where you draw the arm end matters much less when the arm is long.",{"id":92,"type":93,"itemId":94,"prompt":95,"check":96,"hints":101,"feedback":103},"pr-arms","practice","constructing-angles.inv-arm-length","An angle of 72° is drawn with arms 4 cm long. Priya extends both arms to 10 cm. What does the angle measure now, in degrees?",{"kind":97,"answer":98,"tolerance":99,"unit":100},"number",72,0,"°",[102],"Does extending an arm change the amount of turn between the two rays?",{"correct":104,"incorrect":105},"Right: still 72°. Extending the arms draws more of the same rays.","The angle is still 72°. Only the gap between the arm ends grows; the turn between the rays is unchanged.",{"id":107,"type":53,"title":108,"eyebrow":109,"navLabel":110},"ch-scales","Wrong-scale detective","Chapter 02","2 Wrong scale",{"id":112,"type":43,"markdown":113},"scale-set","A protractor carries **two scales** running in opposite directions. On one, 0 is at the right end of the base line; on the other, 0 is at the left end. At every mark, the two numbers **add up to 180**: where one scale says 40, the other says 140. (Which row is printed on the inside differs from protractor to protractor, so go by where the 0 is, not by “inner” or “outer”.)\n\nThe rule you learned is: *line up one arm with the base line, then read the scale whose 0 sits on that arm*. In this chapter you will investigate what happens when someone forgets the rule, and how to catch the mistake.",{"id":115,"type":62,"prompt":116,"options":117,"explanation":126},"pred-scale-1","Kabir measures an angle that is really 35°. He lines up the arm correctly but reads the wrong scale. What number will he write down?",[118,120,122,124],{"id":66,"label":119},"35°",{"id":69,"label":121},"55°",{"id":72,"label":123},"145°",{"id":75,"label":125},"325°","**145°.** The two scales always add to 180, so the wrong scale gives 180 − 35 = 145. Notice that 145° is **obtuse** while the real angle is clearly **acute**, much narrower than a right angle. A one-second estimate (\"this is less than 90°\") would have caught the error immediately.",{"id":128,"type":129,"caption":130,"columns":131,"rows":136},"tbl-scales","table","Reading on the correct scale and on the wrong scale. The two always add to 180°.",[132,133,134,135],"Correct reading","Wrong-scale reading","Correct type","Wrong type",[137,142,145,146,149,152,155,158,161,162,163],[138,139,140,141],"10°","170°","acute","obtuse",[143,144,140,141],"25°","155°",[119,123,140,141],[147,148,140,141],"50°","130°",[150,151,140,141],"65°","115°",[153,154,140,141],"80°","100°",[156,156,157,157],"90°","right",[159,160,141,140],"105°","75°",[148,147,141,140],[144,143,141,140],[139,138,141,140],{"id":165,"type":43,"markdown":166},"scale-pattern","Look down the table and two patterns jump out.\n\n1. **Every wrong reading is 180 minus the right one.** That is simply how the two scales are printed.\n2. **An acute angle always turns into an obtuse reading, and an obtuse angle into an acute one.** Only a right angle survives the mistake: 90 on one scale is 90 on the other.\n\nSo the detective question is always the same: *does my reading have the same type as the angle I can see?* If the angle is clearly narrower than the corner of a page but you wrote a number bigger than 90, you read the wrong scale.",{"id":168,"type":62,"prompt":169,"options":170,"explanation":177},"pred-scale-2","Is it always true that reading the wrong scale can be caught by checking whether the angle is acute or obtuse?",[171,173,175],{"id":66,"label":172},"Yes, for every angle",{"id":69,"label":174},"Yes, except for angles close to 90°",{"id":72,"label":176},"No, it never helps","**Nearly always, but not close to 90°.** An 85° angle read on the wrong scale gives 95°. Both look almost like right angles, so the acute\u002Fobtuse check can fail. For angles within about 10° of 90°, look more carefully: count up from the 0 on the lined-up arm, 10, 20, 30…, and follow that same scale to the other arm. Counting from the correct 0 always works.",{"id":179,"type":180,"title":181,"problem":182,"steps":183},"we-scale","worked_example","Catch the mistake","Meera writes \"The angle PQR is 128°.\" Her drawing shows a narrow angle, clearly smaller than the corner of her notebook. What went wrong, and what is the real measure?",[184,185,186,187,188],"Estimate first: the angle is narrower than a right angle, so it is **acute**, less than 90°.","128° is obtuse, which contradicts the estimate. The likely cause is reading the wrong scale.","The two scales add to 180°, so the other scale shows 180 − 128 = **52°**.","52° is acute, matching the estimate. The real measure is 52°.","Habit to keep: say \"acute\" or \"obtuse\" out loud before you read any number.",{"id":190,"type":93,"itemId":191,"prompt":192,"check":193,"hints":195,"feedback":198},"pr-scale","constructing-angles.inv-wrong-scale","Arjun reads 64° for an angle that is clearly wider than a right angle. If he used the wrong scale, what is the true measure in degrees?",{"kind":97,"answer":194,"tolerance":99,"unit":100},116,[196,197],"The two scales at any mark add to 180°.","An angle wider than a right angle must be more than 90°.",{"correct":199,"incorrect":200},"Yes: 180 − 64 = 116°, which is obtuse, as the picture suggests.","The two readings add to 180°. 180 − 64 = 116°, which is obtuse, matching an angle wider than a right angle.",{"id":202,"type":53,"title":203,"eyebrow":204,"navLabel":205},"ch-eye","How good is your eye?","Chapter 03","3 Estimating",{"id":207,"type":43,"markdown":208},"eye-set","Estimating is not guessing. A good estimate uses **benchmark angles** you already know well: a right angle (the corner of a book), a straight angle (a ruler's edge), half a right angle (45°, a square folded corner to corner) and a third of a right angle (30°, the smallest corner of a 30-60-90 set square).\n\nHow close can you get just by looking? Before playing, predict your own accuracy.",{"id":210,"type":62,"prompt":211,"options":212,"explanation":221},"pred-eye","You will be shown ten random angles and must estimate each without a protractor. How far off do you think your typical estimate will be?",[213,215,217,219],{"id":66,"label":214},"Within 5°",{"id":69,"label":216},"Within 10°",{"id":72,"label":218},"Within 20°",{"id":75,"label":220},"More than 20°","There is no single right answer here: this prediction is about **you**. Play the game below and compare. Many people land within about 10° on acute angles near a benchmark (30°, 45°, 60°) and do worse on angles far from one, such as 110° or 250°. Estimates of reflex angles are often the weakest, because we are used to looking at the smaller opening. Write your prediction down, then check it.",{"id":223,"type":224,"component":225,"componentVersion":5,"config":226,"objective":232,"textAlternative":233,"help":234},"lab-estimate","interactive","angle-lab",{"modes":227,"allowReflex":230,"rounds":231},[228,229],"estimate","classify",true,10,"Estimate ten angles, including reflex ones, before measuring, and see how close your eye gets; classify each angle by type.","This lab draws an angle and asks you to type your estimate in degrees before the exact size is shown. You score more points the closer you get, and your streak grows while you stay near the true value. Reflex angles (bigger than 180°) are included.\n\nA strategy that works: first decide the type. Is it less than 90° (acute), exactly 90° (right), between 90° and 180° (obtuse), 180° (straight) or more than 180° (reflex)? Then compare with the nearest benchmark. For example, an angle a little wider than half a right angle is about 50°. An angle a little past a straight line is about 200°. For a reflex angle, estimate the small angle on the other side first, then subtract from 360°: if the small side looks like 60°, the reflex angle is about 300°.\n\nIn classify mode you name the type of each angle: acute, right, obtuse, straight or reflex.",{"simplerExplanation":235,"hints":236},"First say whether the angle is smaller or bigger than a book corner. Then think \"half a corner is 45°, a third of a corner is 30°\" and adjust.",[237,238],"Decide the type before the number.","For a reflex angle, estimate the small angle on the other side and take it away from 360°.",{"id":240,"type":241,"title":242,"items":243},"steps-estimate","steps","An estimating routine that improves with practice",[244,248,252,256,260],{"title":245,"tag":246,"text":247},"Name the type","0–180 first","Acute, right, obtuse, straight or reflex? This alone rules out most wrong answers.",{"title":249,"tag":250,"text":251},"Find the nearest benchmark","30 · 45 · 60 · 90 · 180","Is it closer to 30°, 45°, 60°, 90° or 180°? Picture a set-square corner or a folded square.",{"title":253,"tag":254,"text":255},"Adjust","a bit more or less","Nudge up or down by 5° or 10°. \"A bit wider than 45°\" might be 50° or 55°.",{"title":257,"tag":258,"text":259},"Reflex? Flip it","360 − small side","Estimate the small angle outside, then subtract from 360°.",{"title":261,"tag":262,"text":263},"Measure and compare","keep score","Write estimate and measurement side by side. Your error shrinks as you practise.",{"id":265,"type":47,"variant":266,"title":267,"markdown":268},"aha-halve","aha","Halving a right angle in your head","Your eye is surprisingly good at spotting **halves**. Imagine a right angle split exactly in two: that is 45°. Split one of those halves again and you get 22.5°. Split a right angle into three equal slices and each is 30°. Most good estimators use this mental halving and slicing rather than trying to \"see\" 37° directly.",{"id":270,"type":93,"itemId":271,"prompt":272,"check":273,"hints":285,"feedback":287},"pr-estimate","constructing-angles.inv-estimate-bench","An angle looks a little narrower than half of a right angle. Which is the best estimate?",{"kind":274,"options":275,"correct":284},"choice",[276,278,280,282],{"id":66,"label":277},"about 20°",{"id":69,"label":279},"about 40°",{"id":72,"label":281},"about 60°",{"id":75,"label":283},"about 110°",[69],[286],"Half a right angle is 45°.",{"correct":288,"incorrect":289},"Right: a little under 45° is about 40°.","Half a right angle is 45°. A little narrower than that is about 40°.",{"id":291,"type":53,"title":292,"eyebrow":293,"navLabel":294},"ch-radius","Does the compass radius matter?","Chapter 04","4 Compass radius",{"id":296,"type":43,"markdown":297},"radius-set","Recall the 60° construction. Draw ray OA. With the compass point on O, draw an arc of any radius that cuts OA at P. **Keeping the same radius**, put the point on P and cut the first arc at Q. Draw ray OQ. Angle QOA is 60°.\n\nTwo things in that recipe are worth testing. First, does the *size* of the radius matter? Second, what happens if the radius accidentally *changes* between the two arcs?",{"id":299,"type":62,"prompt":300,"options":301,"explanation":310},"pred-radius-size","Asha does the 60° construction with a 3 cm radius. Ravi does it with an 8 cm radius. What will their angles measure?",[302,304,306,308],{"id":66,"label":303},"Asha's is smaller, because her arcs are smaller",{"id":69,"label":305},"Both are 60°",{"id":72,"label":307},"Ravi's is exactly 60°, Asha's is not",{"id":75,"label":309},"It depends on the length of OA","**Both are 60°.** Join P and Q. Then OP, OQ and PQ are all equal to the compass radius, so triangle OPQ is **equilateral**, and every angle of an equilateral triangle is 60°. A 3 cm radius makes a small equilateral triangle; an 8 cm radius makes a big one. Both have 60° corners. The size of the radius only changes the size of the triangle, just as arm length did not change the angle in Chapter 1.\n\nA bigger radius is usually **easier to draw accurately**, because the arcs cross more clearly.",{"id":312,"type":62,"prompt":313,"options":314,"explanation":323},"pred-radius-change","Ravi's compass slips. His first arc (centre O) has radius 5 cm, but his second arc (centre P) has radius 6 cm. What happens to his angle?",[315,317,319,321],{"id":66,"label":316},"It is still 60°",{"id":69,"label":318},"It becomes smaller than 60°",{"id":72,"label":320},"It becomes larger than 60°",{"id":75,"label":322},"The arcs no longer meet","**It becomes larger than 60°, about 73.7°.** Now OP = OQ = 5 cm but PQ = 6 cm. The triangle is no longer equilateral: its third side is longer, so the angle at O opens wider. Splitting the triangle down the middle gives two right-angled halves, and the angle at O works out to about 73.7° instead of 60°. A slip of just 1 cm in the compass changed the angle by nearly 14°.\n\nThis is why every construction says *keep the same radius*. A compass with a loose hinge is the most common reason a careful construction comes out wrong.",{"id":325,"type":180,"title":326,"problem":327,"steps":328,"help":334},"we-radius","How far off is a slipped compass?","In the 60° construction, OP = OQ = 5 cm. The compass slipped so that PQ = 6 cm. Estimate angle POQ.",[329,330,331,332,333],"Triangle OPQ has OP = OQ = 5 cm and PQ = 6 cm. It is **isosceles**, not equilateral.","Draw the line from O to the midpoint M of PQ. It cuts the triangle into two equal right-angled triangles, with PM = 3 cm and OP = 5 cm.","In each half, the side opposite the half-angle is 3 and the longest side is 5. The half-angle is the angle whose \"sine\" is 3 ÷ 5 = 0.6, which is about 36.9°. (You can check this in the scale drawing: draw it and measure.)","The whole angle is twice that: about **73.7°**.","Compare: the correct angle is 60°. A 1 cm slip made an error of about 14°, far larger than the ±1° a careful protractor check allows.",{"simplerExplanation":335},"Draw the triangle 5 cm, 5 cm, 6 cm with a ruler and compass and measure the angle between the two 5 cm sides. It comes out close to 74°, not 60°.",{"id":337,"type":47,"variant":80,"title":338,"markdown":339},"try-radius","Two radii, two angles","1. Construct 60° with a 4 cm radius, and again on a new line with a 9 cm radius. Measure both: both about 60°.\n2. Now do it deliberately wrong: first arc 5 cm, then open the compass to 6 cm for the second arc. Measure: about 74°.\n3. Try once more with the second arc at 4 cm. Predict first: will the angle be smaller or larger than 60°?\n\n(For step 3 the triangle is 5, 5, 4, so the angle is smaller: about 47°.)",{"id":341,"type":47,"variant":342,"title":343,"markdown":344},"nuance-radius","nuance","When the radius is allowed to change","Some constructions **do** change the radius between steps. In the angle bisector, for example, the first arc (centre O) can have any radius, and the two arcs from P and Q may use a *different* radius, as long as those two arcs use the **same** radius as each other. Read each recipe carefully: it says which arcs must match. The ones that must match are exactly the ones that make the equal sides in the triangle or kite hidden inside the construction.",{"id":346,"type":53,"title":347,"eyebrow":348,"navLabel":349},"ch-halving","Halving again and again","Chapter 05","5 Halving",{"id":351,"type":43,"markdown":352},"halve-set","The angle bisector splits any angle into two equal halves. If you can make 60°, one bisection gives 30°. Bisect again and you have 15°. Bisect once more: 7.5°. There is no limit to how many times you *could* bisect, at least on perfect paper with a perfect pencil.",{"id":354,"type":355,"component":356,"componentVersion":5,"config":357,"textAlternative":359},"anim-30","animation","compass-construction",{"construction":358},"angle-30","This animation constructs a 30° angle by bisecting a 60° angle, one step at a time.\n\nStep 1: Draw a ray OA.\n\nStep 2: With centre O and any convenient radius, draw a large arc that cuts OA at P.\n\nStep 3: Keeping the same radius, put the compass point on P and draw an arc that cuts the first arc at Q. Joining O to Q would give 60°, because O, P and Q form an equilateral triangle.\n\nStep 4: Now bisect angle QOP. With centre P and a radius more than half of PQ, draw an arc inside the angle.\n\nStep 5: With centre Q and the **same** radius as step 4, draw another arc that crosses the one from step 4 at T.\n\nStep 6: Draw ray OT. It splits the 60° angle into two equal parts, so angle TOA = **30°** (and angle QOT is also 30°).\n\nCheck: a protractor on OA reads 30° at OT, within about 1°.",{"id":361,"type":62,"prompt":362,"options":363,"explanation":372},"pred-halve","Start with 60° and keep bisecting. How many bisections does it take before the angle is smaller than 1°?",[364,366,368,370],{"id":66,"label":365},"3",{"id":69,"label":367},"4",{"id":72,"label":369},"6",{"id":75,"label":371},"60","**6 bisections.** The chain is 60° → 30° → 15° → 7.5° → 3.75° → 1.875° → **0.9375°**. Each halving divides by 2, so after 6 halvings the angle is 60 ÷ 64, just under 1°. Halving shrinks things fast: 60 halvings would give an angle so small no instrument could measure it.",{"id":374,"type":129,"caption":375,"columns":376,"rows":380},"tbl-halve","What repeated bisection of 60° produces",[377,378,379],"Bisections","Angle","Arc between arms at 10 cm radius",[381,385,389,393,396,399,403],[382,383,384],"0","60°","about 10.47 cm",[386,387,388],"1","30°","about 5.24 cm",[390,391,392],"2","15°","about 2.62 cm",[365,394,395],"7.5°","about 1.31 cm",[367,397,398],"3.75°","about 0.65 cm",[400,401,402],"5","1.875°","about 0.33 cm",[369,404,405],"0.9375°","about 0.16 cm",{"id":407,"type":47,"variant":408,"title":409,"markdown":410},"ml-halve","model_limit","Perfect paper, imperfect pencils","In pure geometry you may bisect for ever. On real paper each bisection adds a little error: the pencil line has width, arcs cross at a slant, and the compass point sinks into the paper. After about three or four bisections the angle is so thin that its two arms nearly merge into one thick line, and a protractor can no longer tell 3.75° from 4°. That is a limit of the **tools**, not of the **idea**.",{"id":412,"type":93,"itemId":413,"prompt":414,"check":415,"hints":418,"feedback":420},"pr-halve","constructing-angles.inv-halve-90","Start with a right angle (90°) and bisect three times in a row. What angle do you get, in degrees?",{"kind":97,"answer":416,"tolerance":417,"unit":100},11.25,0.01,[419],"90 ÷ 2 = 45. Keep going.",{"correct":421,"incorrect":422},"Yes: 90 → 45 → 22.5 → 11.25°.","Halve three times: 90 → 45 → 22.5 → 11.25°.",{"id":424,"type":53,"title":425,"eyebrow":426,"navLabel":427},"ch-recipes","Recipes from 60° and 90°","Chapter 06","6 Angle recipes",{"id":429,"type":43,"markdown":430},"recipes-set","With a ruler and compass you have a small set of moves: make 60° (an equilateral triangle), step round again to make 120°, make 90° (bisect between 60° and 120°, or draw a perpendicular), and **bisect** any angle you already have. You can also use a straight line, which gives 180°, and subtract from it.\n\nThe investigation: *which angles can you reach by combining these moves?* Before looking at the table, try to find routes to 45°, 75°, 105°, 135° and 150° on your own.",{"id":432,"type":355,"component":356,"componentVersion":5,"config":433,"textAlternative":435},"anim-45",{"construction":434},"angle-45","This animation constructs a 45° angle by first building 90° and then bisecting it.\n\nStep 1: Draw ray OA. With centre O and a convenient radius, draw a large arc that cuts OA at P.\n\nStep 2: Keeping the same radius, from P cut the arc at Q (the 60° mark), then from Q cut it again at R (the 120° mark).\n\nStep 3: With centres Q and R and equal radius, draw two arcs that cross at S. Draw ray OS. Angle SOA = **90°**, because it lies exactly halfway between 60° and 120°.\n\nStep 4: The first arc crosses OS at U. With centre P and a radius more than half of PU, draw an arc inside the right angle.\n\nStep 5: With centre U and the same radius, draw an arc that crosses the last one at V.\n\nStep 6: Draw ray OV. It bisects the right angle, so angle VOA = **45°**.\n\nCheck with a protractor: 45° on the scale whose 0 lies on OA.",{"id":437,"type":62,"prompt":438,"options":439,"explanation":446},"pred-75","You have constructed 60° (ray OQ) and 90° (ray OS) on the same base ray OA. What do you get if you bisect the angle between OQ and OS?",[440,441,443,444],{"id":66,"label":391},{"id":69,"label":442},"45°",{"id":72,"label":160},{"id":75,"label":445},"150°","**75°.** The angle between OQ and OS is 90 − 60 = 30°. Bisecting it adds half of that, 15°, on top of 60°. The new ray makes 60 + 15 = **75°** with OA. In the same way, bisecting between 90° and 120° gives 90 + 15 = 105°, and bisecting between 90° and 180° gives 90 + 45 = 135°.",{"id":448,"type":129,"caption":449,"columns":450,"rows":453},"tbl-recipes","Routes to common angles using only ruler and compass (every one checked in Python)",[378,451,452],"Route","Arithmetic",[454,456,460,463,466,469,472,475,478,482,485,489],[383,455,371],"Equilateral triangle: two arcs of the same radius",[457,458,459],"120°","Step the same radius round twice","60 + 60",[156,461,462],"Bisect between the 60° and 120° marks","(60 + 120) ÷ 2",[387,464,465],"Bisect 60°","60 ÷ 2",[442,467,468],"Bisect 90°","90 ÷ 2",[391,470,471],"Bisect 30°","30 ÷ 2",[160,473,474],"Bisect between 60° and 90°","(60 + 90) ÷ 2",[159,476,477],"Bisect between 90° and 120°","(90 + 120) ÷ 2",[479,480,481],"135°","Bisect between 90° and 180° (the straight line)","(90 + 180) ÷ 2",[445,483,484],"Bisect between 120° and 180°; or 180° − 30°","(120 + 180) ÷ 2",[486,487,488],"165°","Bisect between 150° and 180°","(150 + 180) ÷ 2",[490,491,492],"22.5°","Bisect 45°","45 ÷ 2",{"id":494,"type":43,"markdown":495},"recipes-pattern","Notice the pattern in the right-hand column. Every recipe uses only three things: **60°**, **adding or subtracting angles that share an arm**, and **halving**. Starting from 60° and 180°, halving gives 30°, 15°, 7.5°…; adding and subtracting mixes them. Everything you reach this way is a whole multiple of 15° or a halving of one.\n\nSo is *every* angle reachable? Try 20°, 40° or 10°. However you combine 60°, 90°, 180° and halving, you never land on them. The sort game below asks you to decide.",{"id":497,"type":224,"component":498,"componentVersion":5,"config":499,"objective":572,"textAlternative":573,"help":574},"lab-sort-constructible","sort-game",{"prompt":500,"bins":501,"items":508,"seconds":99},"Can you make this angle with ruler and compass, using 60°, straight lines and bisecting?",[502,505],{"id":503,"label":504},"yes","Yes, with these moves",{"id":506,"label":507},"no","Not with these moves",[509,512,515,518,521,524,527,530,533,536,539,542,545,548,552,555,559,562,566,569],{"id":510,"label":391,"bin":503,"why":511},"a15","Bisect 60° to get 30°, then bisect 30°.",{"id":513,"label":490,"bin":503,"why":514},"a22","Bisect 90° to get 45°, then bisect 45°.",{"id":516,"label":387,"bin":503,"why":517},"a30","Bisect the 60° angle.",{"id":519,"label":442,"bin":503,"why":520},"a45","Make 90° (between 60° and 120°) and bisect it.",{"id":522,"label":383,"bin":503,"why":523},"a60","The equilateral-triangle construction: two arcs of the same radius.",{"id":525,"label":160,"bin":503,"why":526},"a75","Bisect between the 60° and 90° rays: 60 + 15 = 75.",{"id":528,"label":156,"bin":503,"why":529},"a90","Bisect between 60° and 120°, or draw a perpendicular.",{"id":531,"label":159,"bin":503,"why":532},"a105","Bisect between 90° and 120°: 90 + 15 = 105.",{"id":534,"label":457,"bin":503,"why":535},"a120","Step the same radius round twice: 60 + 60.",{"id":537,"label":479,"bin":503,"why":538},"a135","Bisect between 90° and the straight line: 90 + 45.",{"id":540,"label":445,"bin":503,"why":541},"a150","Straight line minus 30°, or bisect between 120° and 180°.",{"id":543,"label":486,"bin":503,"why":544},"a165","Straight line minus 15°: 180 − 15 = 165.",{"id":546,"label":138,"bin":506,"why":547},"a10","10° would need 30° split into three equal parts. Bisecting only ever halves, so you never reach 10°.",{"id":549,"label":550,"bin":506,"why":551},"a20","20°","20° is a third of 60°, and Wantzel proved in 1837 that trisecting 60° with ruler and compass alone is impossible.",{"id":553,"label":143,"bin":506,"why":554},"a25","Halving 60° and 90° gives 30, 15, 45, 22.5… but never 25. Use a protractor for 25°.",{"id":556,"label":557,"bin":506,"why":558},"a40","40°","40° is twice 20°. If you could make 40°, bisecting would give 20°, which is impossible.",{"id":560,"label":147,"bin":506,"why":561},"a50","No mix of 60°, 90°, 180° and halving lands on 50°. A protractor draws it easily.",{"id":563,"label":564,"bin":506,"why":565},"a70","70°","70° is not a whole multiple of 3°, and no whole-degree angle like that can be constructed exactly.",{"id":567,"label":153,"bin":506,"why":568},"a80","80° would give 20° after bisecting twice (80 → 40 → 20), and 20° is impossible.",{"id":570,"label":154,"bin":506,"why":571},"a100","100° − 90° = 10°, and 10° cannot be constructed, so 100° cannot either.","Sort angles into those you can build with ruler and compass from 60° and bisecting, and those you cannot.","This game shows angle cards one at a time; sort each into \"Yes, with these moves\" or \"Not with these moves\".\n\nThe moves are: construct 60° (equilateral triangle), step round to 120°, make 90°, use the straight angle 180°, add or subtract angles on a shared arm, and bisect.\n\nYes: 15°, 22.5°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, 165°. Each has a route, for example 75° = bisect between 60° and 90°, and 165° = 180° − 15°.\n\nNot with these moves: 10°, 20°, 25°, 40°, 50°, 70°, 80°, 100°. Halving and combining 60° and 90° never lands on them. In fact 20° (a third of 60°) is proved impossible to construct exactly with ruler and compass; so are 10°, 40°, 80° and 100°, which would all lead back to 20° or 10°. These angles are drawn with a protractor instead.",{"simplerExplanation":575,"hints":576},"If you can get there by halving 60°, 90° or 180° and adding the pieces, it goes in Yes. Otherwise, No.",[577,578],"Try writing the angle as 60, 90 or 180 plus or minus halves: 30, 15, 45, 22.5.","20° is a third of 60°, and bisecting only makes halves.",{"id":580,"type":47,"variant":342,"title":581,"markdown":582},"nuance-others","Some angles need cleverer tricks","The sort game only allows 60°, 90° and bisecting. Some other angles, such as **72°** and **36°**, *can* be constructed exactly with ruler and compass, but only by building a regular pentagon, which needs a longer and trickier recipe (you will meet it in Extend). That is why they were left out of the game. There is a complete answer, proved with algebra: an angle of a whole number of degrees is constructible **exactly when that number is a multiple of 3**. So 3°, 6°, 9°… are all reachable, and none of 10°, 20°, 25°, 40°, 50°, 70°, 80° or 100° is. Extend shows how 3° is built.",{"id":584,"type":180,"title":585,"problem":586,"steps":587},"we-105","Plan a route to 105°","Using only ruler and compass, plan how to construct an angle of 105° on ray OA.",[588,589,590,591,592,593],"Write 105 using angles you can already build. 105 = 90 + 15, and 15 is half of the 30° gap between 90° and 120°. So 105 is exactly halfway between **90** and **120**: (90 + 120) ÷ 2 = 105.","Construct 60° and 120° marks on the first arc (centre O) by stepping the radius twice from P: marks Q and R.","Construct 90°: bisect between Q and R to get ray OS.","OS crosses the first arc at U. Bisect the angle between OS and OR: arcs of equal radius from U and R cross at W.","Ray OW makes (90 + 120) ÷ 2 = **105°** with OA.","Check with a protractor: the reading should be 105° within about 1°. It is obtuse, as expected.",{"id":595,"type":93,"itemId":596,"prompt":597,"check":598,"hints":609,"feedback":611},"pr-150","constructing-angles.inv-route-150","Which of these is a correct ruler-and-compass route to 150°?",{"kind":274,"options":599,"correct":608},[600,602,604,606],{"id":66,"label":601},"Bisect 60° and add it to 90°",{"id":69,"label":603},"Construct 30° on the extension of OA beyond O, so the other angle on the straight line is 150°",{"id":72,"label":605},"Bisect 120° twice",{"id":75,"label":607},"Step the compass radius round three times",[69],[610],"Angles on a straight line add to 180°.",{"correct":612,"incorrect":613},"Yes: 180 − 30 = 150°. Option a gives 120°, c gives 30°, d gives 180°.","Angles on a straight line add to 180°, so a 30° angle on one side leaves 150° on the other. Option a gives 90 + 30 = 120°; c gives 30°; d gives 180°.",{"id":615,"type":53,"title":616,"eyebrow":617,"navLabel":618},"ch-setsq","Set-square combinations","Chapter 07","7 Set squares",{"id":620,"type":43,"markdown":621},"setsq-set","A geometry box has two set squares. One has angles **45°, 45°, 90°**. The other has **30°, 60°, 90°**. On their own they give you four angles: 30°, 45°, 60° and 90°. But you can place them **side by side** (adding their angles) or **one on top of the other** (subtracting). How many different angles can you make?",{"id":623,"type":62,"prompt":624,"options":625,"explanation":630},"pred-setsq","Using one corner from each set square, placed side by side or overlapping, which of these angles can you NOT make directly?",[626,627,628,629],{"id":66,"label":160},{"id":69,"label":159},{"id":72,"label":391},{"id":75,"label":147},"**50°.** Side by side: 45 + 30 = 75, 45 + 60 = 105, 45 + 90 = 135, 90 + 30 = 120, 90 + 60 = 150. Overlapping: 45 − 30 = 15, 60 − 45 = 15, 90 − 45 = 45, 90 − 60 = 30. All possible results are multiples of 15°, so 50° is out of reach. Using a straight edge as well, you can also make 180° − 15° = 165°.",{"id":632,"type":129,"caption":633,"columns":634,"rows":639},"tbl-setsq","Every combination of one angle from each set square (computed in Python)",[635,636,637,638],"45° square","30-60-90 square","Side by side (add)","Overlap (subtract)",[640,641,642,643,644,645],[442,387,160,391],[442,383,159,391],[442,156,479,442],[156,387,457,383],[156,383,445,387],[156,156,646,647],"180°","0° (no angle)",{"id":649,"type":43,"markdown":650},"setsq-result","Collect the results and add the straight-angle trick (180° minus any of them), and the full list of angles from 15° to 180° you can draw with set squares is: **15°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, 165°, 180°**. That is every multiple of 15° up to 180°, and nothing else.\n\nThis is not a coincidence. Every set-square angle (30, 45, 60, 90) is a multiple of 15, and adding or subtracting multiples of 15 always gives another multiple of 15.",{"id":652,"type":224,"component":653,"componentVersion":5,"config":654,"objective":674,"textAlternative":675,"help":676},"lab-match-setsq","match-pairs",{"prompt":655,"mode":656,"pairs":657},"Match each angle with a way to make it from the two set squares.","connect",[658,660,662,664,666,668,670,672],{"a":160,"b":659},"30° + 45° side by side",{"a":159,"b":661},"60° + 45° side by side",{"a":391,"b":663},"45° − 30° (overlap the corners)",{"a":457,"b":665},"90° + 30° side by side",{"a":479,"b":667},"90° + 45° side by side",{"a":445,"b":669},"90° + 60° side by side",{"a":486,"b":671},"180° − 15° on a straight line",{"a":646,"b":673},"90° + 90°: the two right angles together","Match angles to set-square combinations that make them, adding corners side by side or subtracting by overlapping.","This game shows eight angles and eight set-square combinations; connect each angle to its combination.\n\nThe correct pairs are: 75° = 30° + 45°; 105° = 60° + 45°; 15° = 45° − 30° (lay the 30° corner over the 45° corner and the uncovered sliver is 15°); 120° = 90° + 30°; 135° = 90° + 45°; 150° = 90° + 60°; 165° = 180° − 15° (a straight line with a 15° angle marked off); 180° = two right angles side by side.\n\nEvery set-square angle is a multiple of 15°, so every combination is too.",{"hints":677},[678,679],"Add the two angles when the set squares sit side by side.","Subtract when one corner lies on top of the other.",{"id":681,"type":180,"title":682,"problem":683,"steps":684},"we-setsq-75","Draw 75° with set squares","Draw an angle of 75° using the two set squares and a ruler.",[685,686,687,688,689],"Draw a straight line with the ruler and mark a point O on it for the vertex.","Place the 45° set square with its 45° corner at O and one edge along the line. Draw along the other edge: that line makes 45° with the base line.","Now place the 30-60-90 set square so its 30° corner is at O and one edge lies along the line you just drew (on the side away from the base line).","Draw along its other edge. The new line makes 45 + 30 = **75°** with the base line.","Check with a protractor: 75° on the scale whose 0 lies on the base line.",{"id":691,"type":47,"variant":692,"title":693,"markdown":694},"careful-setsq","careful","Keep the vertex exactly on O","Set-square combinations only work if **both corners meet at the same point**. If the second set square is shifted even 2 mm along the first line, the lines you draw no longer meet at O, and you get two separate angles instead of one 75° angle. Hold the first set square firmly, and press the second one against it without letting it slide.",{"id":696,"type":93,"itemId":697,"prompt":698,"check":699,"hints":701,"feedback":703},"pr-setsq","constructing-angles.inv-setsq-165","A carpenter needs a 165° angle and has only a straight edge and the two set squares. She marks 15° from one side of a straight line. What angle is left on the other side of her new line, in degrees?",{"kind":97,"answer":700,"tolerance":99,"unit":100},165,[702],"Angles on a straight line add up to 180°.",{"correct":704,"incorrect":705},"Yes: 180 − 15 = 165°.","Angles on a straight line add to 180°, so the other side is 180 − 15 = 165°.",{"id":707,"type":53,"title":708,"eyebrow":709,"navLabel":710},"ch-check","Checking with a protractor","Chapter 08","8 Checking",{"id":712,"type":43,"markdown":713},"check-set","A construction is not finished until it is **checked**. After a 60° construction, lay the protractor on the vertex and read the angle. Will it show exactly 60? Almost never. It might show 59° or 61°. Is that a mistake?\n\nTo answer, find out how big one degree really is on paper. On a circle of radius 5 cm (a typical arc in a construction) a 1° slice has an arc length of only about **0.087 cm**, a little less than **1 mm**. A sharp pencil line is about half a millimetre wide. So the pencil itself can hide half a degree of error.",{"id":715,"type":716,"items":717},"f-arc","formulas",[718,721,724,727],{"expression":719,"caption":720},"arc for 1° = 2 × π × r ÷ 360","The length of arc cut off by one degree on a circle of radius r.",{"expression":722,"caption":723},"r = 3 cm → 0.52 mm","Short arms: one degree is barely half a millimetre.",{"expression":725,"caption":726},"r = 5 cm → 0.87 mm","A typical construction radius: just under a millimetre per degree.",{"expression":728,"caption":729},"r = 10 cm → 1.75 mm","Long arms: nearly 2 mm per degree, much easier to see.",{"id":731,"type":62,"prompt":732,"options":733,"explanation":740},"pred-accuracy","You misplace the end of an arm by 1 mm sideways. Where does that 1 mm matter most?",[734,736,738],{"id":66,"label":735},"When the arm is 3 cm long",{"id":69,"label":737},"When the arm is 15 cm long",{"id":72,"label":739},"It is the same error either way","**On the short arm.** A 1 mm sideways slip at the end of a 3 cm arm swings the arm by about 1.9°. The same slip at the end of a 15 cm arm swings it by only about 0.4°. That is why constructions use **large arcs** and why you should **extend short arms** before measuring: a longer arm turns the same pencil wobble into a smaller angle error.",{"id":742,"type":129,"caption":743,"columns":744,"rows":747},"tbl-err","Angle error caused by a 1 mm sideways slip at the end of an arm",[745,746],"Arm length","Angle error",[748,751,754,757],[749,750],"3 cm","about 1.9°",[752,753],"5 cm","about 1.1°",[755,756],"10 cm","about 0.6°",[758,759],"15 cm","about 0.4°",{"id":761,"type":47,"variant":762,"title":763,"markdown":764},"def-tolerance","definition","Tolerance","A **tolerance** is the amount of error you agree to accept. For school constructions checked with a protractor, a tolerance of **±1°** is sensible: a 60° construction that measures anywhere from 59° to 61° counts as correct. Engineers use the same idea with much tighter tolerances; a machine part might be allowed only ±0.01 mm.",{"id":766,"type":224,"component":767,"componentVersion":5,"config":768,"objective":777,"textAlternative":778,"help":779},"lab-construct","protractor",{"mode":769,"targets":770,"tolerance":5},"construct",[771,772,773,774,775,776],20,75,105,150,200,300,"Draw angles of 20°, 75°, 105°, 150°, 200° and 300° with a virtual protractor to within ±1°, and see how far off each attempt is.","This lab gives you a base ray, a protractor already placed on the vertex, and a target angle. You drag the second arm to the target and press check. A reading within 1° of the target scores full points; the lab tells you how many degrees you were off.\n\nTargets: 20°, 75°, 105°, 150°, 200°, 300°.\n\n20°, 75°, 105° and 150°: read the scale whose 0 lies on the base ray and count up to the target. Estimate first: 20° and 75° are acute; 105° and 150° are obtuse.\n\n200° and 300° are reflex. A semicircular protractor only reaches 180°, so draw them another way: for 200°, go past the straight line by 200 − 180 = 20°; for 300°, draw 360 − 300 = 60° on the other side of the base ray, and the big angle outside is 300°.\n\nNote that 20° cannot be constructed exactly with ruler and compass, but a protractor draws it easily.",{"simplerExplanation":780,"hints":781},"Put the arm on the number, using the scale that starts at 0 on the base ray. For angles bigger than 180°, draw the small angle and take the big one outside.",[782,783],"For 200°, draw 20° beyond the straight line.","For 300°, draw 60° and use the reflex angle outside it.",{"id":785,"type":180,"title":786,"problem":787,"steps":788},"we-check","Is this construction good enough?","Nisha constructs 45° and measures 46°. Sameer constructs 75° and measures 72°. With a tolerance of ±1°, whose construction passes?",[789,790,791,792],"Nisha: the error is 46 − 45 = 1°. That is within ±1°, so her construction **passes**.","Sameer: the error is 75 − 72 = 3°, more than 1°. His construction **does not pass**.","Sameer should look for the cause: a slipped compass, arcs that did not cross clearly, or a bisection done between the wrong marks.","A 3° error on a 75° construction often means one of the arcs was drawn with a changed radius. Redo it, keeping the compass hinge tight.",{"id":794,"type":47,"variant":48,"title":795,"markdown":796},"obs-check","What a protractor check can and cannot tell you","A protractor check tells you **whether** the angle is close enough. It does not tell you **why** it is wrong. When a construction fails the check, go back through the steps: which arcs had to have the same radius? Did they? Did the arcs cross at a clear point, or did they just touch? Arcs meeting at a very shallow slant give a fuzzy crossing point, and that fuzz becomes angle error.",{"id":798,"type":53,"title":799,"eyebrow":800,"navLabel":801},"ch-always","Is it always true?","Chapter 09","9 Always true?",{"id":803,"type":43,"markdown":804},"always-set","Two constructions have a special property that you can test with a divider (or a compass used as a measuring tool).\n\n- **Perpendicular bisector of AB:** every point on it is **equidistant** (equally far) from A and from B.\n- **Angle bisector:** every point on it is equally far from the two **arms** of the angle, where \"distance to an arm\" means the shortest distance, measured along a perpendicular.\n\nAre these true only for the points where the arcs crossed, or for every single point on the line? Predict, then test.",{"id":806,"type":62,"prompt":807,"options":808,"explanation":817},"pred-perp","Draw the perpendicular bisector of a segment AB that is 6 cm long. Pick a point P on it far from AB. Is P equally far from A and from B?",[809,811,813,815],{"id":66,"label":810},"Only if P is where the construction arcs crossed",{"id":69,"label":812},"Yes, for every point on the perpendicular bisector",{"id":72,"label":814},"Only if P is close to AB",{"id":75,"label":816},"Never, except at the midpoint","**Yes, every point.** Put A at 0 and B at 6 cm on a line, so the bisector runs straight up through the 3 cm mark. The point 4 cm above the midpoint is exactly 5 cm from A and 5 cm from B (a 3-4-5 right triangle on each side). The point 2.5 cm *below* the midpoint is also the same distance from both. Folding the paper along the bisector puts A exactly on B, so each point on the fold is carried to itself and its distance to A becomes its distance to B.\n\nThe reverse is also true: any point equally far from A and B lies on the perpendicular bisector.",{"id":819,"type":47,"variant":80,"title":820,"markdown":821},"try-divider","Test it with a divider","1. Draw AB = 6 cm and construct its perpendicular bisector.\n2. Mark three points on the bisector: one near AB, one far above, one below.\n3. For each point, open the divider from the point to A, then swing it to B without changing the opening. The tip should land on B each time.\n4. Now mark a point **off** the bisector and try again. The divider misses B.\n\nYou have tested the claim at six points. It is not a proof, but every test is evidence, and Deepen explains why it must always work.",{"id":823,"type":62,"prompt":824,"options":825,"explanation":834},"pred-bis","Point X lies on the bisector of a 60° angle, 6 cm from the vertex. How far is X from each arm?",[826,828,830,832],{"id":66,"label":827},"3 cm from one arm, 6 cm from the other",{"id":69,"label":829},"3 cm from each arm",{"id":72,"label":831},"6 cm from each arm",{"id":75,"label":833},"It depends which way you measure","**3 cm from each arm.** The bisector makes 30° with each arm. Dropping a perpendicular from X to either arm makes a right-angled triangle with a 30° angle and a 6 cm longest side, and the side opposite 30° is always half the longest side: 3 cm. The two triangles are identical mirror images, so X is 3 cm from both arms. This works for every point on the bisector, not just for X.",{"id":836,"type":47,"variant":80,"title":837,"markdown":838},"try-fold","Paper-folding checks","Paper folding makes the two facts easy to see.\n\n- **Perpendicular:** draw a line, then fold the paper so the line lands exactly on itself. The crease is **perpendicular** to the line. Fold so point A lands on point B and the crease is the **perpendicular bisector** of AB.\n- **Bisector:** draw an angle, then fold so one arm lands exactly on the other. The crease passes through the vertex and is the **angle bisector**.\n\nUnfold and measure with a protractor: the crease makes 90° with the line in the first case, and splits the angle into equal halves in the second.",{"id":840,"type":62,"prompt":841,"options":842,"explanation":847},"pred-fold","You draw an angle of 80° and fold the paper so its arms match. Then you fold again, matching the crease with one arm. What angle does the second crease make with that arm?",[843,844,845,846],{"id":66,"label":557},{"id":69,"label":550},{"id":72,"label":138},{"id":75,"label":153},"**20°.** The first fold bisects 80° into two 40° angles. The second fold bisects one of those 40° angles, leaving 20°. Folding happily makes 20°, even though ruler and compass cannot construct 20° from scratch! The difference is that here the 80° angle was **given** to you, drawn with a protractor. Bisecting a given angle is always possible; creating 20° from nothing is not.",{"id":849,"type":180,"title":850,"problem":851,"steps":852},"we-bis-test","Test the bisector at a second point","An angle of 80° is bisected. Point Y is on the bisector, 10 cm from the vertex. Show that Y is the same distance from both arms.",[853,854,855,856,857],"The bisector makes 80 ÷ 2 = 40° with each arm.","Drop a perpendicular from Y to the first arm. This makes a right-angled triangle with the 40° angle at the vertex and a longest side of 10 cm.","Drop a perpendicular from Y to the second arm. This triangle also has a right angle, a 40° angle at the vertex and the same longest side of 10 cm.","The two triangles have the same angles and the same longest side, so they are identical (congruent). Their sides opposite 40° are equal.","So Y is the same distance from both arms (about 6.4 cm each, if you draw it carefully and measure).",{"id":859,"type":47,"variant":342,"title":860,"markdown":861},"nuance-evidence","Testing is not proving","You tested the bisector claims at a handful of points and they worked each time. That is strong **evidence**, but a mathematician wants a **proof**: an argument showing it must work at *every* point. The folding picture is close to a proof already, because a fold is a perfect mirror. Deepen turns it into a full argument using congruent triangles.",{"id":863,"type":93,"itemId":864,"prompt":865,"check":866,"hints":877,"feedback":879},"pr-equidistant","constructing-angles.inv-equidistant","Point K is 7 cm from A and 7 cm from B. Which must be true?",{"kind":274,"options":867,"correct":876},[868,870,872,874],{"id":66,"label":869},"K is the midpoint of AB",{"id":69,"label":871},"K lies on the perpendicular bisector of AB",{"id":72,"label":873},"K lies on the angle bisector of any angle",{"id":75,"label":875},"AB is 14 cm long",[69],[878],"Which line contains all the points equally far from A and B?",{"correct":880,"incorrect":881},"Yes: every point equidistant from A and B lies on the perpendicular bisector of AB.","The points equally far from A and B are exactly the points of the perpendicular bisector of AB. K need not be the midpoint, and AB need not be 14 cm.",{"id":883,"type":53,"title":884,"eyebrow":885,"navLabel":886},"ch-wrap","Wrap-up: what your tests showed","Chapter 10","10 Wrap-up",{"id":888,"type":889,"title":890,"terms":891},"gloss","glossary","Words from your investigations",[892,896,900,903,907,911,915,919,923,927,931],{"term":893,"meaning":894,"example":895},"Estimate","A sensible approximate value found by reasoning, before measuring exactly.","\"About 50°: a little more than half a right angle.\"",{"term":897,"meaning":898,"example":899},"Benchmark angle","A well-known angle used as a reference when estimating: 30°, 45°, 60°, 90°, 180°, 360°.","A book corner is a 90° benchmark.",{"term":763,"meaning":901,"example":902},"The largest error you agree to accept in a measurement or construction.","±1°: 59° to 61° counts as 60°.",{"term":904,"meaning":905,"example":906},"Accuracy","How close a measured or drawn value is to the true or target value.","A 75° target drawn as 74° is accurate to 1°.",{"term":908,"meaning":909,"example":910},"Equidistant","Equally far from two points or two lines.","Every point on the perpendicular bisector of AB is equidistant from A and B.",{"term":912,"meaning":913,"example":914},"Constructible angle","An angle that can be drawn exactly using only an unmarked straightedge and a compass.","60°, 45°, 75° are constructible; 20° is not.",{"term":916,"meaning":917,"example":918},"Set-square combination","An angle made by placing set-square corners side by side (adding) or overlapping (subtracting).","30° + 45° = 75°.",{"term":920,"meaning":921,"example":922},"Bisect","To split into two equal parts.","Bisecting 90° gives 45°.",{"term":924,"meaning":925,"example":926},"Trisect","To split into three equal parts. Trisecting a general angle is impossible with ruler and compass alone.","Trisecting 60° would give 20°.",{"term":928,"meaning":929,"example":930},"Isosceles triangle","A triangle with two equal sides.","A slipped compass turns the equilateral triangle into an isosceles one.",{"term":932,"meaning":933,"example":934},"Divider","A two-pointed tool used to measure and transfer lengths without a scale.","Checking that a point is equally far from A and B.",{"id":936,"type":937,"prompt":938},"reflect","reflection","Which of your predictions in this layer was furthest from the result? Write what you believed before, what the test showed, and one sentence explaining the result to a younger friend.",{"id":940,"type":941,"conceptId":942,"relation":943,"explanation":944},"conn-angles","connection","angles","helps_understand","Knowing acute, obtuse, straight and reflex angles is exactly what catches wrong-scale readings and makes estimates sensible.",{"id":946,"type":941,"conceptId":947,"relation":948,"explanation":949},"conn-shape","shape-and-space","applied_in","The equilateral triangle hides inside the 60° construction, and accurate angles are needed to draw squares, hexagons and other polygons.",{"id":951,"type":941,"conceptId":952,"relation":953,"explanation":954},"conn-patterns","patterns","related_to","Repeated halving (60, 30, 15, 7.5…) and the multiples of 15° from set squares are number patterns living inside geometry.",{"id":956,"type":956,"title":957,"questions":958},"quiz","Check your investigations",[959,969,982,991,1004,1017,1027,1036,1045,1054,1067],{"itemId":960,"prompt":961,"options":962,"correct":66,"why":968},"constructing-angles.inv-q-arms","An angle of 50° has its arms extended from 5 cm to 15 cm. What does it measure now?",[963,964,965,966],{"id":66,"label":147},{"id":69,"label":445},{"id":72,"label":154},{"id":75,"label":967},"It cannot be measured","Arm length does not change the amount of turn. It is still 50°.",{"itemId":970,"prompt":971,"options":972,"correct":69,"why":981},"constructing-angles.inv-q-scale","A clearly acute angle is recorded as 138°. What is the most likely true measure?",[973,975,977,979],{"id":66,"label":974},"38°",{"id":69,"label":976},"42°",{"id":72,"label":978},"48°",{"id":75,"label":980},"222°","The wrong scale gives 180 − true value. 180 − 138 = 42°, which is acute, as the picture shows.",{"itemId":983,"prompt":984,"options":985,"correct":72,"why":990},"constructing-angles.inv-q-90","Which angle gives the same reading on both protractor scales?",[986,987,988,989],{"id":66,"label":442},{"id":69,"label":383},{"id":72,"label":156},{"id":75,"label":646},"The scales add to 180, so the only reading that equals 180 minus itself is 90°.",{"itemId":992,"prompt":993,"options":994,"correct":69,"why":1003},"constructing-angles.inv-q-radius","In the 60° construction, why must the two arcs have the same radius?",[995,997,999,1001],{"id":66,"label":996},"So the arcs look neat",{"id":69,"label":998},"So triangle OPQ has three equal sides and 60° angles",{"id":72,"label":1000},"So the angle is bigger",{"id":75,"label":1002},"It does not matter","Equal radii make OP = OQ = PQ, an equilateral triangle, whose angles are all 60°. A changed radius gives a different angle.",{"itemId":1005,"prompt":1006,"options":1007,"correct":72,"why":1016},"constructing-angles.inv-q-slip","The first arc has radius 5 cm but the second is drawn with 6 cm. The angle comes out…",[1008,1010,1012,1014],{"id":66,"label":1009},"exactly 60°",{"id":69,"label":1011},"less than 60°",{"id":72,"label":1013},"more than 60°, about 74°",{"id":75,"label":1015},"exactly 90°","The triangle becomes 5, 5, 6. Its third side is longer than the others, so the angle at O opens to about 73.7°.",{"itemId":1018,"prompt":1019,"options":1020,"correct":72,"why":1026},"constructing-angles.inv-q-halve","How many bisections of 60° are needed to get an angle smaller than 1°?",[1021,1022,1023,1024],{"id":66,"label":367},{"id":69,"label":400},{"id":72,"label":369},{"id":75,"label":1025},"8","60 → 30 → 15 → 7.5 → 3.75 → 1.875 → 0.9375: six bisections.",{"itemId":1028,"prompt":1029,"options":1030,"correct":72,"why":1035},"constructing-angles.inv-q-75","Bisecting between the 60° ray and the 90° ray gives…",[1031,1032,1033,1034],{"id":66,"label":391},{"id":69,"label":564},{"id":72,"label":160},{"id":75,"label":153},"(60 + 90) ÷ 2 = 75°.",{"itemId":1037,"prompt":1038,"options":1039,"correct":69,"why":1044},"constructing-angles.inv-q-20","Which angle can NOT be constructed exactly with ruler and compass?",[1040,1041,1042,1043],{"id":66,"label":391},{"id":69,"label":550},{"id":72,"label":479},{"id":75,"label":486},"20° is a third of 60°. Trisecting 60° with ruler and compass alone is impossible. The others come from 60°, 90°, 180° and bisecting.",{"itemId":1046,"prompt":1047,"options":1048,"correct":66,"why":1053},"constructing-angles.inv-q-setsq","Which angle can you make by overlapping the two set squares?",[1049,1050,1051,1052],{"id":66,"label":391},{"id":69,"label":143},{"id":72,"label":147},{"id":75,"label":564},"45° − 30° = 15°. Set-square combinations always give multiples of 15°.",{"itemId":1055,"prompt":1056,"options":1057,"correct":69,"why":1066},"constructing-angles.inv-q-long","Why do careful constructions use long arms and large arcs?",[1058,1060,1062,1064],{"id":66,"label":1059},"They look better",{"id":69,"label":1061},"A small pencil slip makes a smaller angle error",{"id":72,"label":1063},"Large arcs change the angle",{"id":75,"label":1065},"Protractors only read long arms","A 1 mm slip at 3 cm is about 1.9°; at 15 cm it is about 0.4°.",{"itemId":1068,"prompt":1069,"options":1070,"correct":69,"why":1079},"constructing-angles.inv-q-equi","A point on the perpendicular bisector of AB is 9 cm from A. How far is it from B?",[1071,1073,1075,1077],{"id":66,"label":1072},"4.5 cm",{"id":69,"label":1074},"9 cm",{"id":72,"label":1076},"18 cm",{"id":75,"label":1078},"It depends on AB","Every point on the perpendicular bisector is equidistant from A and B.",{"id":1081,"type":1082,"title":1083,"points":1084},"cheat","summary","Cheat sheet",[1085,1086,1087,1088,1089,1090,1091,1092,1093,1094],"**Arm length never changes an angle.** Only the gap between the arm ends grows. Extending short arms before measuring is allowed and makes readings more accurate.","**Wrong scale = 180 − true value.** It turns acute into obtuse and back. Estimate the type first; only angles near 90° slip past this check, so count up from the 0 on the lined-up arm.","**Estimate with benchmarks:** 30°, 45°, 60°, 90°, 180°. For reflex angles, estimate the small side and subtract from 360°.","**Any radius works for the 60° construction, but it must stay the same.** Equal radii make an equilateral triangle. A slip from 5 cm to 6 cm gives about 73.7°.","**Repeated bisection:** 60 → 30 → 15 → 7.5 → 3.75 → 1.875 → 0.9375. Six halvings get below 1°.","**Recipes:** 90 = between 60 and 120; 75 = between 60 and 90; 105 = between 90 and 120; 135 = between 90 and 180; 150 = 180 − 30; 165 = 180 − 15.","**Not constructible:** 10°, 20°, 25°, 40°, 50°, 70°, 80°, 100°. Whole-degree constructible angles are exactly the multiples of 3°; 20° (a third of 60°) is impossible.","**Set squares** (30, 45, 60, 90) combine to every multiple of 15° up to 180°: 15, 75, 105, 120, 135, 150, 165…","**One degree is tiny:** about 0.9 mm of arc at a 5 cm radius. A tolerance of ±1° is fair for school constructions.","**Always true:** every point on the perpendicular bisector is equidistant from A and B; every point on an angle bisector is equidistant from the two arms. Folding shows why.",{"id":1096,"type":1096,"sourceIds":1097},"sources",[1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108],"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-exact-trig-values",[1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108],"needs_review",{"generatedBy":1112,"notes":1113},"claude-code","Draft generated with Python; every angle fact was computed and asserted. Pending owner review.","2eba71276810a05968a096a8132d4ab3af4407d3cee3ce09839054900b4aa551",{"logic:practice":1116,"component:angle-lab@1":1117,"component:compass-construction@1":1118,"component:sort-game@1":1119,"component:match-pairs@1":1120,"component:protractor@1":1121,"source:constructing-angles-britannica-euclid-elements":1122,"source:constructing-angles-mathsisfun-constructions":1123,"source:constructing-angles-mathsisfun-degrees":1124,"source:constructing-angles-mathsisfun-protractor":1125,"source:constructing-angles-ncert-ganita-prakash-6":1126,"source:constructing-angles-ncert-math-6-practical-geometry":1127,"source:constructing-angles-ncert-math-7-practical-geometry":1128,"source:constructing-angles-wikipedia-angle-trisection":1129,"source:constructing-angles-wikipedia-exact-trig-values":1130,"source:constructing-angles-wikipedia-shulba-sutras":1131,"source:constructing-angles-wikipedia-straightedge-compass":1132},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","0eba81381f31d8d78008911eb4c3d62745efd33b4ffadd94aef0851d0a2ad5f3","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","7407db21456592711dd16c6bdad23f042e85ebab9c314b072b17b7065eaf8ae3","ad1a9a6a227fda5d3c1569f37efbe35e448ebaceba8cba872821fd48e2e00ed6","b93faffbd9c4d40f5fce2bc4b2ea0ab5ac64bb8c176f5e2bba3f37444df5e400","216eb0db510461864a47157f14054a39e15b1b0fc461b0fbc77664d9eb28b91d","4d3f50c07f44df57c80455dc39e01b2aa11bb0ee40811fca3b0f12d16e0b3f5e","acad5a4d56d24a5c1ad6e908f3809f2e7b3978f7c2810a9b33cdf82a65c4bb47","4aedaa1be389589b6e840923ef4e92fd15d03eda0b0ba0302d57b7e71bcb3dc7","21119e12648b9efd4cc82b11c59d626f2a53eace3a70552041f26c311b77ba2d","7f00387dc29d17172d25b6aa96420e2544a8bc59edf939af3dce91d515300a1f","fb92459a92df88915d8e44ab46afef7a382dc86bcdfa71f34f9c39dee944ab73","f7559697a2f963f9cb1e02a93fc5697840f583f22605d7483d688664862d70f9","90d6cc756bc1bdd6cde0d5e4ed2000c88c3e2f3a8d91fdaf0ec4ab73af72b434",{"state":1134,"reviewer":1135,"selfReview":230,"reviewedAt":1136,"method":1137},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597004]