[{"data":1,"prerenderedAt":1095},["ShallowReactive",2],{"layer:angles:investigate":3},{"layer":4,"contentHash":1075,"dependencyHashes":1076,"approval":1089,"releaseId":1094},{"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":1070,"reviewStatus":1071,"authoring":1072},1,"angles","en","investigate","Is it always true? Testing angle ideas","Predict, test with labs and numbers, hunt counterexamples and find the reasons behind angle patterns","Investigate angle estimation, sums of angle types, complement and supplement patterns, linear pairs and their bisectors, crossing lines, clock-hand puzzles, turning walks around shapes and the tear-the-corners experiment, sorting claims into always, sometimes and never.",[13,14,15,16,17],"Decide whether claims about angles are always, sometimes or never true, using counterexamples and reasons.","Discover and explain patterns: supplement − complement = 90°, perpendicular bisectors of a linear pair, three lines through a point.","Use the minute hand's gain of 5.5° per minute to predict when clock hands overlap or form right angles.","Relate turning angles to inside angles of regular polygons and explain which shapes tile a floor.","Solve multi-step missing-angle problems with a reason for each step and check them by a second route.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Angle types and pairs (Understand)",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Angle lab, angle pairs ×3, sort game, match game",{"label":38,"value":39},"Key habit","Always \u002F sometimes \u002F never?",[41,45,51,80,86,101,116,126,131,134,167,246,269,274,279,282,321,324,334,343,357,362,373,385,388,399,411,416,421,424,431,456,459,470,474,477,486,497,509,514,574,577,590,601,615,620,625,628,657,667,672,677,680,691,725,737,741,746,750,753,782,787,797,802,805,815,823,832,836,847,877,881,917,1026,1040,1046,1051,1055,1059],{"id":42,"type":43,"markdown":44},"intro","prose","Mathematicians do not just learn facts; they **test** them. They ask \"what happens if…?\", make a guess, try examples, and then ask the most important question of all: **is it always true?**\n\nIn this layer you are the investigator. Every chapter starts with a question. You will predict, test with labs and numbers, look for patterns and counterexamples, and write down what you found. Some findings will be **always** true, some only **sometimes**, and some **never**. Learning to tell these apart is the heart of mathematical thinking.\n\nKeep a notebook open. For each investigation write: *my prediction*, *what I tried*, *what I found*, *always \u002F sometimes \u002F never?*",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-investigate","callout","observation","The three words that matter","- **Always true:** true for every example you could ever try. One example is not enough to show this; you need a **reason**.\n- **Sometimes true:** true for some examples and false for others. Show one of each.\n- **Never true:** false for every example. Again, you need a reason, not just a few tries.\n\nA single example that breaks a claim is called a **counterexample**. One counterexample is enough to show a claim is not *always* true.",{"id":52,"type":53,"title":54,"items":55},"steps-investigate","steps","The investigation cycle",[56,60,64,68,72,76],{"title":57,"tag":58,"text":59},"Ask","what if…?","Start with a question you do not know the answer to.",{"title":61,"tag":62,"text":63},"Predict","commit first","Write down your guess before testing, so you can learn from surprises.",{"title":65,"tag":66,"text":67},"Test","labs and numbers","Try many examples, including edge cases such as 0°, 90°, 180° and equal angles.",{"title":69,"tag":70,"text":71},"Look for a pattern","tables help","Put results in a table and look for what stays the same.",{"title":73,"tag":74,"text":75},"Explain","find a reason","Find a reason that works for every case, not just the ones you tried.",{"title":77,"tag":78,"text":79},"Decide","always \u002F sometimes \u002F never","Classify the claim, with a counterexample or a reason.",{"id":81,"type":82,"title":83,"eyebrow":84,"navLabel":85},"ch01","chapter","How good is your eye?","Chapter 01","1 Estimation",{"id":87,"type":88,"prompt":89,"options":90,"explanation":100},"predict-arms","prediction","Two angles are drawn on the board. Angle P has arms 3 cm long; angle Q has arms 12 cm long. Most people think Q looks bigger. If you measured them, what would you expect?",[91,94,97],{"id":92,"label":93},"a","Q is bigger, because its arms are 4 times longer",{"id":95,"label":96},"b","They could be equal: the arm length tells you nothing about the size",{"id":98,"label":99},"c","P is bigger, because short arms make a sharper angle","**They could be equal.** Arm length has no effect on the size of an angle, which is the amount of turn. Your eye is easily fooled: long arms enclose a larger *area* of page, which feels \"bigger\". Test yourself in the lab below. Researchers who study visual illusions have found that people misjudge angles in many ways, for example often overestimating small acute angles.",{"id":102,"type":103,"component":104,"componentVersion":5,"config":105,"objective":110,"textAlternative":111,"help":112},"lab-estimate","interactive","angle-lab",{"modes":106,"allowReflex":108,"rounds":109},[107],"estimate",true,10,"Estimate ten randomly tilted angles, including reflex ones, score points for closeness and track whether you guess high or low.","The estimate game shows ten angles one after another, each tilted at random. You type an estimate in degrees; the lab then shows the true value, draws your guess as a dashed arm marked “you”, and awards more points the closer you were.\n\nAs an investigation, record each estimate and the true value in two columns and find the difference (estimate − true). If most differences are positive you tend to overestimate; if negative, you underestimate. Many people guess small acute angles too big and underestimate obtuse angles near 180°. Does the tilt of the angle make it harder?\n\nTips that improve scores: compare with 90° and 180° first, use 45° and 135° as halfway marks, and for reflex angles estimate the small opening and subtract from 360°. An average error under 10° is a very good eye.",{"hints":113},[114,115],"Decide the type first.","Rotate the page in your head so one arm is horizontal.",{"id":117,"type":118,"title":119,"problem":120,"steps":121},"we-error","worked_example","Measuring your own accuracy","Meera estimated four angles: she said 50°, 100°, 160°, 300°. The true sizes were 42°, 110°, 171° and 290°. What was her average error, and does she guess high or low?",[122,123,124,125],"Differences (estimate − true): 50 − 42 = **+8**, 100 − 110 = **−10**, 160 − 171 = **−11**, 300 − 290 = **+10**.","Sizes of the errors: 8, 10, 11, 10. Average error = (8 + 10 + 11 + 10) ÷ 4 = 39 ÷ 4 = **9.75°**.","Signs: two too high, two too low. She overestimated the acute and reflex angles and underestimated the obtuse ones.","Conclusion: an average error of under 10° is very good. To improve, she should check her obtuse estimates against 135° and 180°.",{"id":127,"type":82,"title":128,"eyebrow":129,"navLabel":130},"ch02","Adding angle types: always, sometimes, never?","Chapter 02","2 Adding types",{"id":132,"type":43,"markdown":133},"adding-types","What type of angle do you get if you **add** two angles of known types? Try these investigations before reading on.\n\n**Acute + acute.** 20° + 30° = 50° (acute). 50° + 60° = 110° (obtuse). 45° + 45° = 90° (right). So the sum of two acute angles is **sometimes** acute, sometimes right, sometimes obtuse. It is **never** straight or bigger, because each is less than 90°, so the sum is less than 180°.\n\n**Acute + right.** 10° + 90° = 100°. 89° + 90° = 179°. The sum is always more than 90° and less than 180°, so it is **always obtuse**.\n\n**Obtuse + obtuse.** 91° + 91° = 182°. 179° + 179° = 358°. Each is more than 90°, so the sum is more than 180°; each is less than 180°, so the sum is less than 360°. The sum is **always reflex**.",{"id":135,"type":136,"caption":137,"columns":138,"rows":143},"table-types-sums","table","What can the sum of two angle types be? (strict types, no zero angles)",[139,140,141,142],"Sum of","Smallest possible sum is just above","Largest possible sum is just below","Result",[144,149,153,157,161,164],[145,146,147,148],"acute + acute","0°","180°","Sometimes acute, right or obtuse",[150,151,147,152],"acute + right","90°","Always obtuse",[154,155,155,156],"right + right","180° exactly","Always straight",[158,151,159,160],"acute + obtuse","270°","Sometimes obtuse, straight or reflex; never acute or right",[162,147,159,163],"right + obtuse","Always reflex",[165,147,166,163],"obtuse + obtuse","360°",{"id":168,"type":103,"component":169,"componentVersion":5,"config":170,"objective":244,"textAlternative":245},"lab-sort-asn","sort-game",{"prompt":171,"bins":172,"items":182,"seconds":243},"Is each statement always true, sometimes true or never true?",[173,176,179],{"id":174,"label":175},"always","Always true",{"id":177,"label":178},"sometimes","Sometimes true",{"id":180,"label":181},"never","Never true",[183,187,191,195,199,203,207,211,215,219,223,227,231,235,239],{"id":184,"label":185,"bin":177,"why":186},"s1","The sum of two acute angles is acute.","20° + 30° = 50° is acute, but 50° + 60° = 110° is obtuse.",{"id":188,"label":189,"bin":174,"why":190},"s2","The sum of two obtuse angles is reflex.","Each is between 90° and 180°, so the sum is between 180° and 360°.",{"id":192,"label":193,"bin":174,"why":194},"s3","An acute angle plus a right angle is obtuse.","The sum is more than 90° and less than 180°.",{"id":196,"label":197,"bin":180,"why":198},"s4","Two acute angles are supplementary.","Their sum is less than 90° + 90° = 180°.",{"id":200,"label":201,"bin":180,"why":202},"s5","Two obtuse angles are supplementary.","Their sum is more than 90° + 90° = 180°.",{"id":204,"label":205,"bin":177,"why":206},"s6","An angle and its supplement are both right angles.","Only when the angle is exactly 90°.",{"id":208,"label":209,"bin":174,"why":210},"s7","The complement of an acute angle is acute.","If x is between 0° and 90°, so is 90° − x.",{"id":212,"label":213,"bin":174,"why":214},"s8","The supplement of an obtuse angle is acute.","If x is between 90° and 180°, then 180° − x is between 0° and 90°.",{"id":216,"label":217,"bin":177,"why":218},"s9","A linear pair contains an obtuse angle.","120° and 60° do; 90° and 90° do not.",{"id":220,"label":221,"bin":174,"why":222},"s10","Vertically opposite angles are equal.","Each is 180° minus the same neighbouring angle.",{"id":224,"label":225,"bin":177,"why":226},"s11","Adjacent angles add up to 180°.","Only when their outer arms form a straight line (a linear pair).",{"id":228,"label":229,"bin":180,"why":230},"s12","A reflex angle has a complement.","A complement must be 90° minus the angle, which would be negative.",{"id":232,"label":233,"bin":177,"why":234},"s13","Three angles around a point are all obtuse.","120° + 120° + 120° = 360° works; 100° + 100° + 160° also works; but 30° + 150° + 180° does not.",{"id":236,"label":237,"bin":180,"why":238},"s14","Four angles around a point are all obtuse.","Four angles each more than 90° add to more than 360°.",{"id":240,"label":241,"bin":177,"why":242},"s15","The difference of two obtuse angles is acute.","170° − 100° = 70° is acute, but 120° − 120° = 0°, which is a zero angle, not acute.",0,"Decide whether 15 claims about angles are always, sometimes or never true.","A sorting game with three bins: always, sometimes and never true.\n\nAlways: two obtuse angles sum to a reflex angle; acute + right is obtuse; the complement of an acute angle is acute; the supplement of an obtuse angle is acute; vertically opposite angles are equal.\n\nSometimes: two acute angles sum to an acute angle (20° + 30° yes, 50° + 60° no); an angle and its supplement are both right (only for 90°); a linear pair contains an obtuse angle (not 90° and 90°); adjacent angles add to 180° (only a linear pair); three angles around a point are all obtuse (120°, 120°, 120° yes); the difference of two obtuse angles is acute (equal angles give 0°).\n\nNever: two acute angles are supplementary; two obtuse angles are supplementary; a reflex angle has a complement; four angles around a point are all obtuse (sum over 360°).",{"id":247,"type":248,"itemId":249,"prompt":250,"check":251,"hints":264,"feedback":266},"pr-acute-obtuse","practice","angles.inv-acute-obtuse","An acute angle and an obtuse angle are added. Which of these results is **impossible**?",{"kind":252,"options":253,"correct":263},"choice",[254,256,258,260],{"id":92,"label":255},"An obtuse angle",{"id":95,"label":257},"A straight angle",{"id":98,"label":259},"A reflex angle",{"id":261,"label":262},"d","A right angle",[261],[265],"The obtuse angle alone is already more than 90°.",{"correct":267,"incorrect":268},"Correct: the sum is always more than 90°, so it can never be exactly a right angle. Examples: 10° + 100° = 110° (obtuse), 60° + 120° = 180° (straight), 80° + 170° = 250° (reflex).","The obtuse angle is already more than 90°, so adding any acute angle gives more than 90°. A right angle (exactly 90°) is impossible. The other three all happen: 10° + 100° = 110°, 60° + 120° = 180°, 80° + 170° = 250°.",{"id":270,"type":47,"variant":271,"title":272,"markdown":273},"nuance-sometimes","nuance","Watch the edge cases","Many \"always\" claims break at a boundary. \"The difference of two obtuse angles is acute\" works for 170° and 100° but fails for 120° and 120°, where the difference is 0°, a zero angle. When you test a claim, deliberately try the edges: equal angles, 0°, 90°, 180° and 360°. That is where counterexamples hide.",{"id":275,"type":82,"title":276,"eyebrow":277,"navLabel":278},"ch03","Patterns in complements and supplements","Chapter 03","3 Comp. patterns",{"id":280,"type":43,"markdown":281},"cs-pattern","Choose any acute angle. Write down its complement and its supplement. Now subtract: supplement − complement. Try a few before reading on.",{"id":283,"type":136,"caption":284,"columns":285,"rows":290},"table-cs-pattern","Supplement minus complement, for several acute angles",[286,287,288,289],"Angle x","Complement","Supplement","Supplement − complement",[291,296,301,306,311,316],[292,293,294,295],"10°","80°","170°","170 − 80 = **90°**",[297,298,299,300],"25°","65°","155°","155 − 65 = **90°**",[302,303,304,305],"40°","50°","140°","140 − 50 = **90°**",[307,308,309,310],"55°","35°","125°","125 − 35 = **90°**",[312,313,314,315],"70°","20°","110°","110 − 20 = **90°**",[317,318,319,320],"85°","5°","95°","95 − 5 = **90°**",{"id":322,"type":43,"markdown":323},"cs-why","Every row gives **90°**. Is that always true, or did we just pick lucky angles? Here is a reason that works for every acute angle x:\n\n- supplement − complement = (180° − x) − (90° − x)\n- = 180° − x − 90° + x\n- = **90°**, because the −x and +x cancel.\n\nSo it is **always** true. The reason is more convincing than a thousand examples, because it covers every possible x at once. In words: the supplement is always exactly one right angle bigger than the complement.",{"id":325,"type":88,"prompt":326,"options":327,"explanation":333},"predict-comp-comp","What is the complement of the complement of 35°?",[328,329,330,331],{"id":92,"label":308},{"id":95,"label":307},{"id":98,"label":309},{"id":261,"label":332},"145°","**35°.** The complement of 35° is 55°, and the complement of 55° is 35° again. In general, 90° − (90° − x) = x. Taking the complement twice brings you back to where you started, just like turning around twice.",{"id":335,"type":103,"component":336,"componentVersion":5,"config":337,"objective":341,"textAlternative":342},"lab-complementary","angle-pairs",{"scene":338,"initialAngle":339,"challenges":340},"complementary",70,6,"Investigate how a complement shrinks as the angle grows, then solve six missing-complement challenges.","A right angle split by a movable ray into complementary angles ∠a and ∠b, starting at 70° and 20°. The live table shows both sizes.\n\nInvestigation: drag so ∠a goes up in 10° steps: 10°, 20°, 30° … 80°. Record ∠b each time: 80°, 70°, 60° … 10°. Every time ∠a rises by 10°, ∠b falls by exactly 10°, and the total stays 90°. Which angle equals its own complement? (45°.)\n\nIn the six challenges the lab gives one part and asks for the other; type the number of degrees. For example, if ∠b is 35°, then ∠a = 90° − 35° = 55°. Harder “puzzle” versions, such as an angle 30° more than its complement, are in the practice question below the lab.",{"id":344,"type":248,"itemId":345,"prompt":346,"check":347,"hints":351,"feedback":354},"pr-cs-diff","angles.inv-cs-diff","The **supplement** of an angle is **4 times** its **complement**. Find the angle.",{"kind":348,"answer":349,"tolerance":243,"unit":350},"number",60,"°",[352,353],"Supplement − complement is always 90°.","If the complement is c, the supplement is 4c, and 4c − c = 90°.",{"correct":355,"incorrect":356},"Yes: 3c = 90° so c = 30°. The angle is 90° − 30° = 60°. Check: its supplement 120° is 4 × 30°.","Let the complement be c. The supplement is always c + 90°, and here it is also 4c. So 4c = c + 90°, 3c = 90°, c = 30°, and the angle is 60°.",{"id":358,"type":82,"title":359,"eyebrow":360,"navLabel":361},"ch04","Linear pairs under the microscope","Chapter 04","4 Linear pairs",{"id":363,"type":88,"prompt":364,"options":365,"explanation":372},"predict-lp-obtuse","Two angles form a linear pair. Can **both** of them be acute?",[366,368,370],{"id":92,"label":367},"Yes, for example 80° and 80°",{"id":95,"label":369},"No: at least one must be 90° or more",{"id":98,"label":371},"Only if the line is not really straight","**No.** A linear pair adds up to 180°. If both were less than 90°, the sum would be less than 180°. So at least one angle is 90° or more. And if one angle is exactly 90°, so is the other. (80° + 80° = 160°, which is not a straight line.) In the same way, both cannot be obtuse, because then the sum would be more than 180°.",{"id":374,"type":103,"component":336,"componentVersion":5,"config":375,"objective":379,"textAlternative":380,"help":381},"lab-linear",{"scene":376,"initialAngle":377,"challenges":378},"linear-pair",40,8,"Test claims about linear pairs by dragging, then solve eight missing-angle challenges.","A straight line with a ray standing on it, making the linear pair ∠a and ∠b, starting at 40° and 140°. The live table shows both sizes and a button highlights the pair.\n\nInvestigate these claims by dragging: “Whenever one angle is acute, the other is obtuse” (true, except at 90° and 90°). “Both angles can be obtuse” (never: the sum would pass 180°). “The two angles can be equal” (only at 90° each).\n\nThe eight challenges each give one angle and ask for the other; type the number of degrees. If ∠b = 115°, then ∠a = 180° − 115° = 65°. Ratio and algebra puzzles about linear pairs are in the practice questions of this chapter.",{"hints":382},[383,384],"Subtract the given angle from 180°.","If the given angle is acute, your answer must be obtuse.",{"id":386,"type":43,"markdown":387},"bisector-invest","Here is a beautiful investigation. Take a linear pair, say 50° and 130°. Cut each angle exactly in half with a ray (a ray that halves an angle is called its **bisector**). What is the angle between the two bisectors?\n\n- Half of 50° is 25°; half of 130° is 65°. The angle between the bisectors is 25° + 65° = **90°**.\n- Try 20° and 160°: 10° + 80° = **90°**.\n- Try 90° and 90°: 45° + 45° = **90°**.\n\nIt seems to be always 90°. **Why?** The two angles add to 180°, so their halves add to half of 180°, which is 90°. So the bisectors of a linear pair are **always perpendicular**. You can construct bisectors with a compass in the [constructing angles topic](\u002Ftopics\u002Fconstructing-angles) and check this on paper.",{"id":389,"type":248,"itemId":390,"prompt":391,"check":392,"hints":394,"feedback":396},"pr-lp-ratio","angles.inv-lp-ratio","The angles of a linear pair are in the ratio **4 : 5**. Find the **smaller** angle.",{"kind":348,"answer":393,"tolerance":243,"unit":350},80,[395],"4 + 5 = 9 equal parts make 180°.",{"correct":397,"incorrect":398},"Right: one part is 180° ÷ 9 = 20°, so the angles are 80° and 100°.","The two angles together make 180°, shared in 9 equal parts: 180 ÷ 9 = 20°. The smaller angle is 4 × 20° = 80°.",{"id":400,"type":118,"title":401,"problem":402,"steps":403,"help":409},"we-lp-puzzles","Three linear-pair puzzles","The angles of a linear pair are ∠a and ∠b. Find both angles when (i) ∠a is **5 times** ∠b; (ii) ∠a : ∠b = **2 : 3**; (iii) ∠a is **30° more** than ∠b.",[404,405,406,407,408],"Every part uses the same fact: **∠a + ∠b = 180°** (linear pair). The trick is to write both angles using one unknown.","(i) Let ∠b = x, so ∠a = 5x. Then 6x = 180°, x = 30°. The angles are **150° and 30°**.","(ii) 2 + 3 = 5 equal parts make 180°, so one part is 36°. The angles are **72° and 108°**.","(iii) Let ∠b = x, so ∠a = x + 30. Then 2x + 30 = 180, 2x = 150, x = 75. The angles are **105° and 75°**.","Check each pair adds to 180°: 150 + 30, 72 + 108, 105 + 75. ✓ Notice that in every case one angle is acute and the other obtuse, as the investigation predicted.",{"simplerExplanation":410},"Turn the words into “parts” of 180°. Five times means 6 parts; 2 : 3 means 5 parts.",{"id":412,"type":47,"variant":413,"title":414,"markdown":415},"misc-lp-both","misconception","“If the two angles look equal, they are both 90°”","In a linear pair, the angles are both 90° only if they are **exactly** equal, and you can only know that from given facts or measurement, not from a sketch. A ray that leans just 2° makes angles of 88° and 92°, which look the same on paper. Builders use a set-square or a plumb line, not their eyes, to check that a wall stands at 90° to the floor.",{"id":417,"type":82,"title":418,"eyebrow":419,"navLabel":420},"ch05","Crossing lines: how little do you need to know?","Chapter 05","5 Crossing lines",{"id":422,"type":43,"markdown":423},"crossing","When two lines cross, four angles appear. How many of them do you need to be told before you can work out all four?\n\nTry it: if one angle is 65°, the angle opposite is 65° (vertically opposite), and the two others are each 180° − 65° = 115° (linear pairs). **One angle is enough.**\n\nWhat if the lines cross at a right angle? Then all four angles are 90°. That is the only way to have four equal angles at a crossing: four equal angles adding to 360° must each be 90°.",{"id":425,"type":103,"component":336,"componentVersion":5,"config":426,"objective":429,"textAlternative":430},"lab-intersecting",{"scene":427,"initialAngle":428,"challenges":378},"intersecting",65,"Rotate one line of an X, test which angles stay equal, and find missing angles from a single clue.","Two straight lines crossing at O make four angles, ∠a, ∠b, ∠c, ∠d going round, starting at 65°, 115°, 65°, 115°. The live table updates as you rotate one line; buttons highlight vertically opposite pairs and linear pairs.\n\nInvestigate: which angles change together? Opposite angles always stay equal; neighbours always add to 180°; all four always add to 360°. When one angle reaches 90°, all four are 90°. So one angle is enough to find the other three.\n\nThe eight challenges give one angle and ask for another, either opposite or next to it; type the number of degrees. If ∠b = 130°, then ∠d = 130° and ∠a = 50°. Algebra versions, such as vertically opposite angles (3x + 10)° and (5x − 30)°, are in the worked examples and practice.",{"id":432,"type":136,"caption":433,"columns":434,"rows":439},"table-many-lines","n straight lines all passing through one point (no two lines the same)",[435,436,437,438],"Lines","Angles around the point","Vertically opposite pairs","Angles you must know",[440,444,447,450,452,454],[441,442,443,443],"1","2 (two straight angles)","0",[445,446,445,441],"2","4",[448,449,448,445],"3","6",[446,451,446,448],"8",[453,33,453,446],"5",[449,455,449,453],"12",{"id":457,"type":43,"markdown":458},"many-lines-pattern","The table shows a pattern: **n lines through a point make 2n angles**, arranged in **n pairs** of vertically opposite angles. The n angles on one side of any of the lines lie along that line, so they add to 180°, which is why you only need to be told **n − 1** of them. For two lines, one clue is enough; for three lines, two clues; for six lines, five clues.",{"id":460,"type":248,"itemId":461,"prompt":462,"check":463,"hints":464,"feedback":467},"pr-vo-alg","angles.inv-vo-alg","Two lines cross. A pair of **vertically opposite** angles measure **(3x + 10)°** and **(5x − 30)°**. Find the size of each of these two angles.",{"kind":348,"answer":339,"tolerance":243,"unit":350},[465,466],"Vertically opposite angles are equal, so set 3x + 10 = 5x − 30.","Solve for x, then substitute back.",{"correct":468,"incorrect":469},"Correct: 2x = 40, so x = 20 and each angle is 3 × 20 + 10 = 70°. The other two angles at the crossing are 110° each.","Vertically opposite angles are equal: 3x + 10 = 5x − 30. Subtract 3x and add 30: 40 = 2x, so x = 20. Each angle is 3 × 20 + 10 = 70° (check: 5 × 20 − 30 = 70).",{"id":471,"type":47,"variant":413,"title":472,"markdown":473},"misc-examples-prove","“I tried five examples, so it is always true”","Examples can build confidence, but they can never prove an “always” claim, because there are infinitely many angles you did not try. A famous warning: someone might test “n² + n + 41 is always prime” for n = 0, 1, 2, … 39 and find it works every time, yet it fails at n = 40. For crossing lines, the reason (both opposite angles are 180° minus the same neighbour) is what makes the claim safe.",{"id":475,"type":43,"markdown":476},"three-lines","Now make it harder. What if **three** lines pass through the **same point**? Draw it: you get **six** angles around the point. Predict before reading on: how many angles do you need to know to find all six?\n\nLabel them a, b, c, d, e, f going round. Each line gives a pair of vertically opposite angles, so **d = a, e = b, f = c**. The six angles add up to 360°, so 2a + 2b + 2c = 360°, which means **a + b + c = 180°**. (You can also see this directly: a, b and c sit side by side along one straight line.)\n\nSo you need to know **two** of a, b, c; the third comes from the straight line, and the other three come from vertically opposite angles.",{"id":478,"type":118,"title":479,"problem":480,"steps":481},"we-three-lines","Three lines through one point","Three lines meet at O, making six angles. Going round, the first two are **40°** and **75°**. Find all six.",[482,483,484,485],"Call the six angles a, b, c, d, e, f going round. a = 40°, b = 75° (given).","a, b and c lie along one straight line, so c = 180° − 40° − 75° = **65°** (angles on a line).","d = a = 40°, e = b = 75°, f = c = 65° (vertically opposite angles).","Check: 40 + 75 + 65 + 40 + 75 + 65 = 360°. ✓",{"id":487,"type":248,"itemId":488,"prompt":489,"check":490,"hints":491,"feedback":494},"pr-three-lines","angles.inv-three-lines","Three lines meet at a point, making six angles. Two neighbouring angles are **50°** and **60°**. What is the **largest** of the six angles?",{"kind":348,"answer":339,"tolerance":243,"unit":350},[492,493],"Three neighbouring angles lie along a straight line.","The other three angles repeat these by vertically opposite pairs.",{"correct":495,"incorrect":496},"Yes: the third angle is 180° − 50° − 60° = 70°, and the six angles are 50°, 60°, 70°, 50°, 60°, 70°.","Three neighbouring angles make a straight line: 180° − 50° − 60° = 70°. The other three are copies (vertically opposite). The largest is 70°.",{"id":498,"type":248,"itemId":499,"prompt":500,"check":501,"hints":503,"feedback":506},"pr-y-junction","angles.inv-y-junction","Three straight roads meet at a roundabout-free Y-junction, like three rays from one point. Two of the angles between the roads are **110°** and **125°**. What is the third angle?",{"kind":348,"answer":502,"tolerance":243,"unit":350},125,[504,505],"The three angles fill all the space around the point.","Angles around a point add to 360°.",{"correct":507,"incorrect":508},"Right: 360° − 110° − 125° = 125°. Two of the three angles are equal, so the junction is symmetric about one road.","Three rays from one point make three angles that add to 360°. The third is 360° − 110° − 125° = 125°. (Careful: these are three rays, not three lines, so there are no vertically opposite pairs here.)",{"id":510,"type":82,"title":511,"eyebrow":512,"navLabel":513},"ch06","Clock detective","Chapter 06","6 Clock detective",{"id":515,"type":136,"caption":516,"columns":517,"rows":522},"table-clock-3","The angle between the hands, every 5 minutes from 3:00 to 4:00 (positions measured clockwise from 12)",[518,519,520,521],"Time","Minute hand","Hour hand","Smaller angle",[523,525,530,533,537,541,546,550,554,558,562,567,572],[524,146,151,151],"3:00",[526,527,528,529],"3:05","30°","92.5°","62.5°",[531,532,319,308],"3:10","60°",[534,151,535,536],"3:15","97.5°","7.5°",[538,539,540,313],"3:20","120°","100°",[542,543,544,545],"3:25","150°","102.5°","47.5°",[547,147,548,549],"3:30","105°","75°",[551,552,553,544],"3:35","210°","107.5°",[555,556,314,557],"3:40","240°","130°",[559,159,560,561],"3:45","112.5°","157.5°",[563,564,565,566],"3:50","300°","115°","175°",[568,569,570,571],"3:55","330°","117.5°","147.5°",[573,146,539,539],"4:00",{"id":575,"type":43,"markdown":576},"clock-invest","Look down the last column. The angle starts at 90°, **shrinks** to almost nothing near 3:16 (the hands overlap), then **grows** again, passes 90° between 3:30 and 3:35, reaches almost 180° near 3:49, and then shrinks towards 120° at 4:00.\n\nThe minute hand gains on the hour hand by **6° − 0.5° = 5.5° every minute**. That single number explains everything:\n\n- From 3:00 the gap of 90° closes at 5.5° per minute, so the hands overlap after 90 ÷ 5.5 = **16 4⁄11 minutes**, at about **3:16**.\n- The next right angle comes when the minute hand is 90° **ahead**: that needs a total gain of 180°, which takes 180 ÷ 5.5 = **32 8⁄11 minutes**, at about **3:32 and 44 seconds**.",{"id":578,"type":88,"prompt":579,"options":580,"explanation":589},"predict-right-per-day","How many times in a full day (24 hours) do the hands of a clock make a right angle?",[581,583,585,587],{"id":92,"label":582},"24 times",{"id":95,"label":584},"44 times",{"id":98,"label":586},"48 times",{"id":261,"label":588},"96 times","**44 times.** In 12 hours the minute hand goes round 12 times but the hour hand goes round once, so the minute hand **laps** the hour hand 11 times. In each lap the hands are at right angles twice (once with the minute hand 90° behind, once 90° ahead). That gives 11 × 2 = 22 right angles in 12 hours, and **44** in 24 hours. It is not 48 (two per hour), because the minute hand laps the hour hand only 11 times in 12 hours, not 12: the slow hour hand is also moving forward.",{"id":591,"type":248,"itemId":592,"prompt":593,"check":594,"hints":596,"feedback":598},"pr-clock-overlap","angles.inv-clock-overlap","How many times in **24 hours** do the hour and minute hands **overlap** (point in exactly the same direction)?",{"kind":348,"answer":595,"tolerance":243},22,[597],"How many times does the minute hand lap the hour hand in 12 hours?",{"correct":599,"incorrect":600},"Right: 11 laps in 12 hours, so 22 overlaps in 24 hours. They overlap every 12 ÷ 11 hours, about every 65 minutes.","In 12 hours the minute hand goes round 12 times and the hour hand once, so the minute hand passes the hour hand 12 − 1 = 11 times. In 24 hours that is 22 overlaps.",{"id":602,"type":248,"itemId":603,"prompt":604,"check":605,"hints":609,"feedback":612},"pr-clock-straight-3","angles.inv-clock-straight-3","Starting at 3:00, after how many minutes are the clock hands first in a **straight line** (pointing in opposite directions)? Give your answer to one decimal place.",{"kind":348,"answer":606,"tolerance":607,"unit":608},49.1,0.1,"min",[610,611],"At 3:00 the minute hand is 90° behind the hour hand.","To be 180° ahead, it must gain 90° + 180° = 270°, at 5.5° per minute.",{"correct":613,"incorrect":614},"Correct: 270 ÷ 5.5 = 49 1⁄11 ≈ 49.1 minutes, at about 3:49.","The minute hand starts 90° behind and must end 180° ahead, a total gain of 270°. It gains 5.5° per minute, so this takes 270 ÷ 5.5 = 49 1⁄11 ≈ 49.1 minutes.",{"id":616,"type":47,"variant":617,"title":618,"markdown":619},"try-clock","try_it","Check with a real clock","If you have a clock or watch with hands, set it to 3:00, then turn the minute hand slowly forward. Stop when you think the hands overlap. Is it a little after 3:16? Keep going: when do the hands form a straight line? (Answer: at about 3:49, when the minute hand has gained 270°: 270 ÷ 5.5 = 49 1⁄11 minutes.)",{"id":621,"type":82,"title":622,"eyebrow":623,"navLabel":624},"ch07","Walking around shapes","Chapter 07","7 Turning walks",{"id":626,"type":43,"markdown":627},"turtle","Imagine a tiny robot that can only do two things: walk forward, and turn on the spot. It walks around a square: forward, turn, forward, turn, forward, turn, forward, turn, and it is back where it started, **facing the same way**.\n\nHow much did it turn altogether? Facing the same way again means it made exactly **one full turn: 360°**. There were 4 equal turns, so each was 360° ÷ 4 = **90°**.\n\nWhat about walking around an equilateral triangle? Again it ends facing the way it started, so the **total turning is 360°**, and each of the 3 equal turns is 360° ÷ 3 = **120°**. Notice: the robot turns 120° at each corner, but the angle **inside** each corner of the triangle is 180° − 120° = **60°**. The turn and the inside angle form a linear pair.",{"id":629,"type":136,"caption":630,"columns":631,"rows":636},"table-turns","Robot walks around regular shapes: turn at each corner and inside angle",[632,633,634,635],"Shape","Corners","Turn at each corner","Inside angle (180° − turn)",[637,639,641,645,647,651,655],[638,448,539,532],"Equilateral triangle",[640,446,151,151],"Square",[642,453,643,644],"Regular pentagon","72°","108°",[646,449,532,539],"Regular hexagon",[648,451,649,650],"Regular octagon","45°","135°",[652,33,653,654],"Regular decagon","36°","144°",[656,455,527,543],"Regular 12-gon",{"id":658,"type":88,"prompt":659,"options":660,"explanation":666},"predict-hexagon","A robot walks around a regular hexagon (6 equal sides, 6 equal corners). How much does it turn at each corner?",[661,662,663,664],{"id":92,"label":532},{"id":95,"label":151},{"id":98,"label":539},{"id":261,"label":665},"720°","**60°.** Whatever the shape, the robot ends facing the way it started, so its total turn is 360°. Six equal turns: 360° ÷ 6 = 60°. The angle inside each corner of the hexagon is then 180° − 60° = **120°**. This is why a honeycomb's hexagons fit together perfectly: three 120° corners make exactly 360° around a point.",{"id":668,"type":47,"variant":669,"title":670,"markdown":671},"aha-honeycomb","aha","Why tiles and honeycombs work","Shapes can cover a floor with no gaps only if their corners fit together to make exactly **360° around every point**. Squares work (4 × 90°). Equilateral triangles work (6 × 60°). Regular hexagons work (3 × 120°). Regular pentagons do **not**: their inside angle is 108°, and 360 ÷ 108 is not a whole number, so there is always a gap or an overlap. Look at floor tiles in a railway station or temple courtyard and check which shapes are used.",{"id":673,"type":82,"title":674,"eyebrow":675,"navLabel":676},"ch08","Turn sequences and compass puzzles","Chapter 08","8 Turn puzzles",{"id":678,"type":43,"markdown":679},"turn-seq","Treat clockwise turns as **positive** and anticlockwise turns as **negative**. Then a whole sequence of turns can be added up like ordinary numbers, and the **net turn** tells you where you end up facing.\n\nFor example, +90° (right turn), +90°, −45°, +180° gives a net turn of 90 + 90 − 45 + 180 = **315°** clockwise. Turning 315° clockwise ends in the same direction as turning 45° anticlockwise, because 315° + 45° = 360°.\n\nInvestigation question: **does the order of the turns matter?** Try +90°, then −45°, then +180°, and compare with +180°, then +90°, then −45°. Predict first.",{"id":681,"type":88,"prompt":682,"options":683,"explanation":690},"predict-order","Priya faces north and makes three turns: 90° clockwise, 45° anticlockwise, 180° clockwise. Sam faces north and makes the same three turns in a different order: 180° clockwise, 90° clockwise, 45° anticlockwise. Do they end up facing the same way?",[684,686,688],{"id":92,"label":685},"Yes, always: only the total turn matters",{"id":95,"label":687},"No, the order changes the final direction",{"id":98,"label":689},"Only if the first turn is the same","**Yes.** Turning on the spot is like adding numbers: 90 − 45 + 180 = 180 + 90 − 45 = 225°. Both end up facing **south-west** (225° clockwise from north). For turns about one fixed point, the order does not matter. (Careful: if you **walk** between the turns, the order does change where you end up standing, even though you face the same way.)",{"id":692,"type":136,"caption":693,"columns":694,"rows":699},"table-turn-seq","Net turns from north (clockwise positive); the final direction depends only on the net turn modulo 360°",[695,696,697,698],"Turns","Net turn","Same as","Facing",[700,705,710,713,718,722],[701,702,703,704],"+90, +90, +90","+270°","90° anticlockwise","W",[706,707,708,709],"+180, −45","+135°","135° clockwise","SE",[711,712,708,709],"−90, −90, −45","−225°",[714,715,716,717],"+45 eight times","+360°","no turn","N",[719,720,147,721],"+270, +270","+540°","S",[723,724,716,717],"−135, +45, −270","−360°",{"id":726,"type":248,"itemId":727,"prompt":728,"check":729,"hints":731,"feedback":734},"pr-net-turn","angles.inv-net-turn","A drone faces **east**. It turns **135° clockwise**, then **270° anticlockwise**. What is the **smallest** single turn (in degrees) that would have taken it straight to its final direction?",{"kind":348,"answer":730,"tolerance":243,"unit":350},135,[732,733],"Net turn = +135 − 270.","A net turn of −135° is 135° anticlockwise.",{"correct":735,"incorrect":736},"Yes: 135 − 270 = −135°, so a single 135° anticlockwise turn does the same job. It ends facing north-west.","Add the turns with signs: +135 − 270 = −135°. That is 135° anticlockwise, which is already smaller than 180°, so the smallest single turn is 135°.",{"id":738,"type":47,"variant":617,"title":739,"markdown":740},"try-scissors","Scissors test for vertically opposite angles","Open a pair of scissors and lay them on paper. Trace along both blades and both handles with a pencil, so you get two crossing lines. Measure (or compare by tracing) the angle between the blades and the angle between the handles.\n\nOpen the scissors wider and repeat three times. You should find the two angles are equal every time, even though both change. Why must the handle angle grow exactly as fast as the blade angle? (Hint: each blade is one straight piece of metal with its handle.)",{"id":742,"type":82,"title":743,"eyebrow":744,"navLabel":745},"ch09","Tearing corners: an experiment","Chapter 09","9 Tear the corners",{"id":747,"type":47,"variant":617,"title":748,"markdown":749},"try-tear","The tear-the-corners experiment","1. Cut out any triangle from paper. Make it lopsided, not a neat one.\n2. Colour the three corners with three different colours.\n3. Tear off the three corners (tear, do not cut, so you can see which edge is which).\n4. Place the three corners side by side along a straight line, with their tips meeting at one point and no gaps.\n\nWhat do you notice? Try again with a very thin triangle and a very fat one. Then try a four-sided shape: tear off all four corners and fit them around a point.",{"id":751,"type":43,"markdown":752},"tear-result","Almost everyone finds the same thing: the three corners of **any** triangle fit together to make a **straight line**, 180°. The four corners of any four-sided shape fit together **all the way round a point**, 360°.\n\nAn experiment is powerful evidence, but it is **not a proof**. Paper tears roughly, and we could never test every triangle in the universe. Is it *always* true? In the Deepen layer you will prove it using parallel lines, so that nobody ever needs to tear another triangle.",{"id":754,"type":136,"caption":755,"columns":756,"rows":762},"table-tri-data","Class results for three measured angles of five triangles (to the nearest degree)",[757,758,759,760,761],"Triangle","Angle 1","Angle 2","Angle 3","Sum",[763,768,770,772,777],[764,765,313,766,767],"A (thin)","12°","149°","181°",[769,151,308,307,147],"B (right-angled)",[771,532,532,532,147],"C (equilateral)",[773,774,775,776,767],"D (lopsided)","47°","71°","63°",[778,293,779,780,781],"E (tall)","81°","18°","179°",{"id":783,"type":47,"variant":784,"title":785,"markdown":786},"model-limit-measure","model_limit","Why the measured sums are not all exactly 180°","In the class data above, two triangles sum to 181° and one to 179°. The true sums are all exactly 180°; the small differences come from **measuring errors**, because a protractor can usually only be read to the nearest degree, and three small errors can add up. Real data is always a little messy. A mathematician asks: is the pattern clear despite the mess, and can I find a reason? Here the answer to both is yes.",{"id":788,"type":248,"itemId":789,"prompt":790,"check":791,"hints":792,"feedback":794},"pr-tri-missing","angles.inv-tri-missing","Two torn corners of a triangle measure **48°** and **67°**. Using the tear-the-corners discovery, how big is the third corner?",{"kind":348,"answer":428,"tolerance":243,"unit":350},[793],"The three corners fit along a straight line.",{"correct":795,"incorrect":796},"Right: 180° − 48° − 67° = 65°.","The three corners together make a straight line (180°). So the third corner is 180° − 48° − 67° = 65°.",{"id":798,"type":82,"title":799,"eyebrow":800,"navLabel":801},"ch10","Missing-angle detective","Chapter 10","10 Detective cases",{"id":803,"type":43,"markdown":804},"detective","A good detective does not guess; they **reason from clues**. A good missing-angle solver does the same:\n\n1. **Mark** everything you are told on a sketch.\n2. **Look** for straight lines (180°), right angles (90°), crossing lines (equal opposite angles) and points with angles all round (360°).\n3. **Find** one new angle, write its reason, and mark it.\n4. **Repeat** until you reach the angle you want.\n5. **Check** with a different route or a total (360° round a point).",{"id":806,"type":118,"title":807,"problem":808,"steps":809},"we-case1","Case 1: the kite string","A straight kite string AB passes through point O. Rays OC and OD are on the same side of AB, in the order A, C, D, B. ∠AOC = 3x, ∠COD = 90° and ∠DOB = x. Find x and ∠AOC.",[810,811,812,813,814],"∠AOC, ∠COD and ∠DOB sit side by side along the straight string, so 3x + 90° + x = 180° (angles on a straight line).","4x + 90° = 180°, so 4x = 90°.","x = 90° ÷ 4 = **22.5°**.","∠AOC = 3x = 3 × 22.5° = **67.5°**.","Check: 67.5 + 90 + 22.5 = 180.0°. ✓",{"id":816,"type":118,"title":817,"problem":818,"steps":819},"we-case2","Case 2: the crossing and the lamp post","Lines PQ and RS cross at O. A lamp post OT stands perpendicular to PQ, between OS and OQ, so ∠POT = 90°. ∠SOT = 34°. Find ∠POR.",[820,821,822],"OS lies inside ∠POT (between OP and OT), so ∠POS = ∠POT − ∠SOT = 90° − 34° = **56°** (adjacent angles).","∠POR and ∠POS form a linear pair on line RS, so ∠POR = 180° − 56° = **124°**.","Check another way: ∠QOS = 180° − ∠POS = 124° (linear pair on PQ), and ∠POR is vertically opposite ∠QOS, so ∠POR = 124°. ✓ Two routes, same answer.",{"id":824,"type":118,"title":825,"problem":826,"steps":827},"we-case-scissor-gate","Case 3: the scissor gate","A folding scissor gate at a shop front is made of straight metal strips that cross at pins. At one pin, the angle between two strips at the top is **(2x + 16)°** and the angle at the side of the same pin is **(4x − 10)°**. The shopkeeper pulls the gate until the top angle becomes **100°**. Find x before the pull, and the side angle after it.",[828,829,830,831],"At a pin two straight strips cross, so the top angle and the side angle are **neighbours** at the crossing: a linear pair (sum 180°).","Before: (2x + 16) + (4x − 10) = 180, so 6x + 6 = 180, 6x = 174 and **x = 29**. The top angle was 74° and the side angle 106°.","After: the top angle is 100°, so the side angle is 180° − 100° = **80°** (linear pair).","The bottom angle always equals the top angle (vertically opposite), so it is also 100° after the pull. Every pin in the gate repeats this pattern, which is why the whole gate opens smoothly together.",{"id":833,"type":47,"variant":413,"title":834,"markdown":835},"misc-spot","Spot the mistake: “They look equal”","Asha writes: *\"∠1 and ∠2 are vertically opposite, so they are equal.\"* But in her diagram, ∠1 and ∠2 are at **different** crossing points of two different pairs of lines. Vertically opposite angles must share the **same vertex** and be formed by the **same two lines**. Angles that merely sit across from each other on a page are not vertically opposite.",{"id":837,"type":248,"itemId":838,"prompt":839,"check":840,"hints":841,"feedback":844},"pr-case3","angles.inv-case3","Around a point there are four angles: **x**, **x + 20°**, **2x** and **100°**. Find **x**.",{"kind":348,"answer":349,"tolerance":243,"unit":350},[842,843],"All four add to 360°.","x + (x + 20) + 2x + 100 = 360.",{"correct":845,"incorrect":846},"Correct: 4x + 120 = 360, so 4x = 240 and x = 60°. The angles are 60°, 80°, 120° and 100°.","Add them: x + x + 20 + 2x + 100 = 4x + 120 = 360, so 4x = 240 and x = 60°.",{"id":848,"type":103,"component":849,"componentVersion":5,"config":850,"objective":875,"textAlternative":876},"lab-match-evidence","match-pairs",{"prompt":851,"mode":852,"pairs":853},"Match each claim with the evidence or reason that settles it.","connect",[854,857,860,863,866,869,872],{"a":855,"b":856},"Two acute angles can make an obtuse angle","Example: 50° + 60° = 110°",{"a":858,"b":859},"Supplement − complement = 90°","(180 − x) − (90 − x) = 90",{"a":861,"b":862},"Linear-pair bisectors are perpendicular","Half of 180° is 90°",{"a":864,"b":865},"Right angles 44 times a day","11 laps × 2 × 2",{"a":867,"b":868},"Hexagons tile a floor","3 × 120° = 360°",{"a":870,"b":871},"Pentagons do not tile","360 ÷ 108 is not whole",{"a":873,"b":874},"Three lines through a point","a + b + c = 180°","Connect seven investigation findings with the example or reason behind each.","A matching game linking findings to evidence. Two acute angles can make an obtuse angle ↔ the example 50° + 60° = 110°. Supplement minus complement is 90° ↔ (180 − x) − (90 − x) = 90. The bisectors of a linear pair are perpendicular ↔ half of 180° is 90°. Clock hands make a right angle 44 times a day ↔ 11 laps × 2 right angles per lap × 2 halves of the day. Regular hexagons tile a floor ↔ 3 × 120° = 360°. Regular pentagons do not tile ↔ 360 ÷ 108 is not a whole number. Three lines through a point ↔ three neighbouring angles make a straight line, a + b + c = 180°.",{"id":878,"type":879,"prompt":880},"reflect-evidence","reflection","In this layer you met two kinds of evidence: **examples** (like the torn triangles and the class data) and **reasons** (like the algebra for supplement − complement). Which kind convinces you more that something is *always* true? Could an example ever prove an 'always' claim? Could it disprove one?",{"id":882,"type":883,"title":884,"terms":885},"glossary-investigate","glossary","Investigation words",[886,890,894,897,901,905,909,913],{"term":887,"meaning":888,"example":889},"conjecture","A statement you think is true because of the examples you have tried, but have not yet proved.","“The corners of a triangle always make 180°.”",{"term":891,"meaning":892,"example":893},"counterexample","One example that shows a claim is not always true.","50° + 60° = 110° breaks “acute + acute is acute”.",{"term":78,"meaning":895,"example":896},"Whether a claim holds for every case, some cases or no cases.","Vertically opposite angles are always equal.",{"term":898,"meaning":899,"example":900},"angle bisector","A ray that splits an angle into two equal angles.","The bisector of 70° makes two 35° angles.",{"term":902,"meaning":903,"example":904},"regular polygon","A shape with all sides equal and all angles equal.","A square; an equilateral triangle.",{"term":906,"meaning":907,"example":908},"tiling (tessellation)","Covering a flat surface with shapes, with no gaps and no overlaps.","Square floor tiles.",{"term":910,"meaning":911,"example":912},"turning angle","The angle you turn through at a corner when walking around a shape.","90° at each corner of a square.",{"term":914,"meaning":915,"example":916},"measurement error","The small difference between a measured value and the true value.","Measuring 181° for a triangle's angle sum.",{"id":918,"type":919,"title":920,"questions":921},"quiz-investigate","quiz","Investigator's check",[922,935,948,959,972,982,995,1004,1013],{"itemId":923,"prompt":924,"options":925,"correct":98,"why":934},"angles.inv-q-counter","Which is a counterexample to “the sum of two acute angles is always acute”?",[926,928,930,932],{"id":92,"label":927},"30° + 40°",{"id":95,"label":929},"10° + 20°",{"id":98,"label":931},"60° + 70°",{"id":261,"label":933},"5° + 5°","60° + 70° = 130°, which is obtuse, so the claim fails.",{"itemId":936,"prompt":937,"options":938,"correct":95,"why":947},"angles.inv-q-obtuse","The sum of two obtuse angles is…",[939,941,943,945],{"id":92,"label":940},"always obtuse",{"id":95,"label":942},"always reflex",{"id":98,"label":944},"sometimes straight",{"id":261,"label":946},"sometimes acute","Each is between 90° and 180°, so the sum is between 180° and 360°: always reflex.",{"itemId":949,"prompt":950,"options":951,"correct":98,"why":958},"angles.inv-q-sc","For an acute angle x, supplement − complement equals…",[952,954,956,957],{"id":92,"label":953},"x",{"id":95,"label":955},"2x",{"id":98,"label":151},{"id":261,"label":147},"(180 − x) − (90 − x) = 90, whatever x is.",{"itemId":960,"prompt":961,"options":962,"correct":98,"why":971},"angles.inv-q-lp","In a linear pair, one angle is acute. The other is…",[963,965,967,969],{"id":92,"label":964},"acute",{"id":95,"label":966},"right",{"id":98,"label":968},"obtuse",{"id":261,"label":970},"reflex","The pair adds to 180°. If one is less than 90°, the other is more than 90° but less than 180°: obtuse.",{"itemId":973,"prompt":974,"options":975,"correct":95,"why":981},"angles.inv-q-three","Three lines meet at a point. Two neighbouring angles are 45° and 85°. The third neighbouring angle is…",[976,977,978,979],{"id":92,"label":649},{"id":95,"label":303},{"id":98,"label":317},{"id":261,"label":980},"230°","Three neighbouring angles make a straight line: 180 − 45 − 85 = 50°.",{"itemId":983,"prompt":984,"options":985,"correct":95,"why":994},"angles.inv-q-overlap","Starting from 12:00, after about how long do the hands overlap again?",[986,988,990,992],{"id":92,"label":987},"60 minutes",{"id":95,"label":989},"about 65 minutes",{"id":98,"label":991},"about 72 minutes",{"id":261,"label":993},"90 minutes","The minute hand gains 5.5° per minute and must gain 360°: 360 ÷ 5.5 = 65 5⁄11 minutes, about 1:05.",{"itemId":996,"prompt":997,"options":998,"correct":95,"why":1003},"angles.inv-q-octagon","A robot walks round a regular octagon. How much does it turn at each corner?",[999,1000,1001,1002],{"id":92,"label":527},{"id":95,"label":649},{"id":98,"label":650},{"id":261,"label":166},"Total turn 360°, shared by 8 corners: 360 ÷ 8 = 45°.",{"itemId":1005,"prompt":1006,"options":1007,"correct":98,"why":1012},"angles.inv-q-tile","Which regular shape can NOT tile a floor on its own?",[1008,1009,1010,1011],{"id":92,"label":638},{"id":95,"label":640},{"id":98,"label":642},{"id":261,"label":646},"Its inside angle is 108°, and 360 is not a multiple of 108, so gaps appear.",{"itemId":1014,"prompt":1015,"options":1016,"correct":95,"why":1025},"angles.inv-q-evidence","Five measured triangles have angle sums 179°, 180°, 181°, 180°, 180°. What is the best conclusion?",[1017,1019,1021,1023],{"id":92,"label":1018},"Triangle sums vary between 179° and 181°",{"id":95,"label":1020},"The sum is 180°, with small measuring errors",{"id":98,"label":1022},"Only some triangles sum to 180°",{"id":261,"label":1024},"Protractors are useless","The values cluster tightly around 180°, and the differences are the size of normal measuring errors.",{"id":1027,"type":1028,"title":1029,"points":1030},"cheat-sheet","summary","Cheat sheet",[1031,1032,1033,1034,1035,1036,1037,1038,1039],"Test claims as **always \u002F sometimes \u002F never**. One **counterexample** disproves “always”; only a **reason** proves it.","Arm length never changes an angle. Estimate with 90°, 180°, 45°, 135° and 30° steps, and measure your own error.","Acute + acute: sometimes acute, right or obtuse. Acute + right: always obtuse. Obtuse + obtuse: always reflex.","**Supplement − complement = 90°** for every acute angle. The complement of the complement is the angle itself.","A linear pair has at least one angle of 90° or more. Its bisectors are always perpendicular.","Two crossing lines: one angle gives all four. Three lines through a point: a + b + c = 180°, and each angle repeats opposite.","Clock: the minute hand gains 5.5° per minute. Overlaps 22 times a day; right angles 44 times a day.","A walk round any shape turns 360° in total. Regular n-gon: turn 360° ÷ n, inside angle 180° − turn.","Tearing corners suggests triangle = 180°, four-sided shape = 360°. Deepen proves it.",{"id":1041,"type":1042,"conceptId":1043,"relation":1044,"explanation":1045},"conn-patterns","connection","patterns","related_to","Tables of turning angles for regular polygons, and times when clock hands meet, are number patterns you can predict.",{"id":1047,"type":1042,"conceptId":1048,"relation":1049,"explanation":1050},"conn-shape","shape-and-space","applied_in","Which regular shapes tile a floor depends on whether their inside angles fit exactly into 360°.",{"id":1052,"type":1042,"conceptId":1053,"relation":1044,"explanation":1054},"conn-data","data-handling","Measured angle sums (179°, 180°, 181°) show real data varying around a true value; averages summarise estimation errors.",{"id":1056,"type":1042,"conceptId":1057,"relation":1049,"explanation":1058},"conn-construct","constructing-angles","Constructing angle bisectors with a compass lets you check that the bisectors of a linear pair are perpendicular.",{"id":1060,"type":1061,"sourceIds":1062},"sources-investigate","sources",[1063,1064,1065,1066,1067,1068,1069],"angles-ncert-class7-lines-angles","angles-mathsisfun-angles","angles-khan-angle-basics","angles-wiki-clock-angle","angles-ncert-class7-triangle","angles-ncert-class6-lines-angles","angles-mathsisfun-complementary",[1063,1064,1065,1066,1067,1068,1069],"needs_review",{"generatedBy":1073,"notes":1074},"claude-code","Draft generated with Python generator scripts; every angle computed and asserted. Pending owner review.","3ace636a746d1d3845715b5bbc2a35fe4036ec797d689d1c6fedd6e48cf64e0a",{"component:angle-lab@1":1077,"component:sort-game@1":1078,"logic:practice":1079,"component:angle-pairs@1":1080,"component:match-pairs@1":1081,"source:angles-khan-angle-basics":1082,"source:angles-mathsisfun-angles":1083,"source:angles-mathsisfun-complementary":1084,"source:angles-ncert-class6-lines-angles":1085,"source:angles-ncert-class7-lines-angles":1086,"source:angles-ncert-class7-triangle":1087,"source:angles-wiki-clock-angle":1088},"1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","cbb509e9a575a1b2ef133804e3450487de14952c1df934747fe56ff5d10d847e","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","68741673c0dc6c96760fde888327818c508788257b5306bad48ab148615541a4","2dab8c7c8736bbad61c4a7223c297c3b250b20399d5b5544c02434cab39a0c9c","7656284145acec3d6ead6ded762ed0eef2bb3137446c527350b7b999562580df","36960f0a34d1e3f73475a665e4b12c1140dfbbeac1c1b2772bf71a8d3b9522a5","548a49fd09ac1b7cec7c74b038e1a87c5d3add389d6f27376ac054da17d20972","286f44f488c463767261100625e631e8edb823f1f296976d251359a4bc9677ee","141f5516f8832cc0eea67b34a5726cd12ed7309036b178da74f753ca6e4ae706",{"state":1090,"reviewer":1091,"selfReview":108,"reviewedAt":1092,"method":1093},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597396]