[{"data":1,"prerenderedAt":1072},["ShallowReactive",2],{"layer:angles:deepen":3},{"layer":4,"contentHash":1048,"dependencyHashes":1049,"approval":1065,"releaseId":1071},{"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":1043,"reviewStatus":1044,"authoring":1045},1,"angles","en","deepen","Why angles behave: proofs, parallels and polygons","From Babylonian 360 to Euclid's proofs: transversals, triangle and polygon angle sums, and hard missing-angle problems","Why a full turn is 360°, how to write a proof with reasons, why vertically opposite angles are equal, the angles made by a transversal on parallel lines and their converses, the triangle and polygon angle sums, bends and zigzags between parallels, and where 180° fails.",[13,14,15,16,17],"Explain the history and usefulness of 360° and convert between degrees, minutes and seconds.","Write short proofs with a reason for every step, including vertically opposite angles and bisectors of a linear pair.","Use corresponding, alternate and co-interior angles to find angles and to test whether lines are parallel.","Prove the triangle angle sum and use it to find angles in triangles, quadrilaterals and polygons.","Solve multi-step and algebraic missing-angle problems using construction lines.",55,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 55 minutes",{"label":29,"value":30},"Prior knowledge","Angle pairs and missing angles (Understand)",{"label":32,"value":33},"Chapters","9",{"label":35,"value":36},"Labs","Angle pairs ×3, sort game, match game",{"label":38,"value":39},"Key results","V.O. equal · F, Z, C · triangle 180°",[41,45,51,57,60,100,105,121,126,129,156,161,164,178,183,186,222,226,239,242,255,307,317,329,339,344,349,352,356,364,386,401,413,423,435,440,443,468,477,488,493,504,509,512,562,570,582,587,590,623,632,644,655,667,676,696,703,710,715,752,757,762,765,769,773,803,866,993,1008,1014,1019,1024,1028],{"id":42,"type":43,"markdown":44},"intro","prose","In earlier layers you learned angle facts, tested them, and saw them work every time. This layer asks the deeper question: **why must they be true?** And it pushes further, into parallel lines, triangles and polygons, where those simple facts combine into powerful results.\n\nYou will learn to write a short **proof**: a chain of statements, each backed by a reason, starting from things everyone agrees on. This is how mathematics has worked for more than 2,000 years, since Greek geometers collected their proofs in Euclid's *Elements*, and it is still how mathematicians convince each other today.\n\nAlong the way: why a full turn is 360°, what makes railway tracks parallel, why every triangle's angles add to 180° (and a surprising place where they do not), and how to solve missing-angle problems that would stump most adults.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-deepen","callout","observation","How to read a proof","A proof is not a trick. Read it slowly, one line at a time, and for each line ask: **what reason allows this?** If you can say the reason out loud, you understand the line. If a line feels like magic, go back one step. Try covering the reasons column in a proof table and filling it in yourself.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Why is a full turn 360°?","Chapter 01","1 Why 360?",{"id":58,"type":43,"markdown":59},"why360","Nobody *proved* that a full turn is 360°. It is a **choice**, a unit agreed by people, like choosing 100 paise in a rupee. So why this odd number?\n\nThe most common explanation goes back to astronomers in **ancient Babylon** (in today's Iraq), around 4,000 to 2,000 years ago:\n\n- They counted in **base 60** (sexagesimal), not base 10. We still see it in our clocks: 60 seconds in a minute, 60 minutes in an hour. Each degree is also split into 60 minutes (′) and each minute into 60 seconds (″).\n- A year has about **365 days**, and the Sun appears to move once round the sky in a year. **360** is close to 365, so a degree is roughly the Sun's movement against the stars in one day.\n- **360 has a huge number of divisors**: 24 of them (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360). A full turn can be split into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths with whole-number answers. Compare 100, which has only 9 divisors.\n\nHistorians still debate exactly how the choice was made, but the usefulness of 360 is not in doubt.",{"id":61,"type":62,"caption":63,"columns":64,"rows":69},"table-units","table","Different ways to divide a full turn",[65,66,67,68],"Unit","Parts in a full turn","Right angle","Where it is used",[70,75,80,85,90,95],[71,72,73,74],"Degree (°)","360","90°","School, maps, engineering, everyday life",[76,77,78,79],"Gradian (gon)","400","100 gon","Some surveying, mainly in parts of Europe",[81,82,83,84],"Radian (rad)","2π ≈ 6.283","π⁄2 ≈ 1.571 rad","Higher mathematics and physics (see Extend)",[86,87,88,89],"Turn (rev)","1","¼ turn","Motors (rpm), wheels, fans",[91,92,93,94],"Hour angle","24 hours","6 hours","Astronomy: the sky turns once a day",[96,97,98,99],"Clock minutes","60","15 minutes","Reading a clock face",{"id":101,"type":47,"variant":102,"title":103,"markdown":104},"aha-divisors","aha","Why a fussy number was a smart choice","With 360°, a regular polygon's turning angle is a whole number of degrees for triangles (120°), squares (90°), pentagons (72°), hexagons (60°), octagons (45°), nonagons (40°), decagons (36°) and 12-gons (30°). If a full turn were 100 units, a triangle's turn would be 33.33… units and a hexagon's 16.66…. Babylonian astronomers, who divided and redivided circles all day, benefited hugely from those divisors.",{"id":106,"type":107,"itemId":108,"prompt":109,"check":110,"hints":115,"feedback":118},"pr-dms","practice","angles.deep-dms","How many **minutes of arc** (′) are there in a right angle?",{"kind":111,"answer":112,"tolerance":113,"unit":114},"number",5400,0,"′",[116,117],"1° = 60′.","A right angle is 90°.",{"correct":119,"incorrect":120},"Correct: 90 × 60 = 5,400′.","Each degree has 60 minutes of arc, so 90° = 90 × 60 = 5,400′.",{"id":122,"type":53,"title":123,"eyebrow":124,"navLabel":125},"ch02","Proof: vertically opposite angles are equal","Chapter 02","2 V.O. proof",{"id":127,"type":43,"markdown":128},"proof-intro","In Understand you saw an argument that vertically opposite angles are equal. Now we set it out as a formal **proof**, the way it appears in Euclid's *Elements*, Book I, Proposition 15. The ancient writer Eudemus credited the discovery to **Thales of Miletus**, around 600 BCE.\n\nEvery proof has three parts:\n\n- **Given:** what we are told.\n- **To prove:** what we want to show.\n- **Proof:** statements, each with a reason. Reasons can only be definitions, given facts, earlier proved results or basic agreed facts (axioms).",{"id":130,"type":62,"caption":131,"columns":132,"rows":136},"table-proof-vo","Given: lines AB and CD intersect at O. To prove: ∠AOC = ∠BOD and ∠AOD = ∠BOC",[133,134,135],"Step","Statement","Reason",[137,140,144,148,152],[87,138,139],"∠AOC + ∠AOD = 180°","Linear pair: ray OA stands on line CD",[141,142,143],"2","∠AOD + ∠BOD = 180°","Linear pair: ray OD stands on line AB",[145,146,147],"3","∠AOC + ∠AOD = ∠AOD + ∠BOD","Both equal 180° (steps 1 and 2)",[149,150,151],"4","∠AOC = ∠BOD","Subtract ∠AOD from both sides of step 3",[153,154,155],"5","∠AOD = ∠BOC","Same argument, using ∠AOC as the shared partner",{"id":157,"type":47,"variant":158,"title":159,"markdown":160},"nuance-axiom","nuance","What are we allowed to assume?","Step 1 uses “a linear pair adds to 180°”. Why is *that* true? Because a straight angle is 180°, and adjacent angles add. Those are **definitions and basic facts** we agree on at the start. Every proof must start somewhere: from axioms, the few simple facts we accept without proof because they are obviously true and everyone agrees on them. Euclid began his *Elements* with just five postulates and some common notions, such as “things equal to the same thing are equal to each other”, which is exactly what step 3 uses.",{"id":162,"type":43,"markdown":163},"more-proofs","Once one fact is proved, it becomes a tool for proving the next. Here are two more short proofs that use only linear pairs.\n\n**Angles around a point add to 360°.** Draw any line through the point. The angles on one side of the line make a straight angle (180°) and the angles on the other side make another straight angle (180°). Together: 180° + 180° = 360°. (If a ray does not lie along the line, it simply splits one of those straight angles into parts that still add to 180°.)\n\n**Supplements of equal angles are equal.** If ∠P = ∠Q, then 180° − ∠P = 180° − ∠Q. This small fact is used again and again, for example to show that if one angle at a crossing is 90°, all four are 90°.",{"id":165,"type":166,"title":167,"problem":168,"steps":169,"help":176},"we-proof-bisector","worked_example","Prove: bisectors of a linear pair are perpendicular","Ray OC stands on line AB, making the linear pair ∠AOC and ∠COB. OP bisects ∠AOC and OQ bisects ∠COB. Prove ∠POQ = 90°.",[170,171,172,173,174,175],"Let ∠AOC = 2a and ∠COB = 2b (each is split into two equal halves, so it helps to call it “twice something”).","2a + 2b = 180° (linear pair).","So a + b = 90° (divide both sides by 2).","∠POC = a (OP bisects ∠AOC) and ∠COQ = b (OQ bisects ∠COB).","∠POQ = ∠POC + ∠COQ = a + b (adjacent angles, since OC lies between OP and OQ).","So **∠POQ = 90°**, and the bisectors are perpendicular. ∎",{"simplerExplanation":177},"Two halves of a straight line add up to half of 180°, which is 90°.",{"id":179,"type":53,"title":180,"eyebrow":181,"navLabel":182},"ch03","Parallel lines and a transversal","Chapter 03","3 Transversals",{"id":184,"type":43,"markdown":185},"transversal","**Parallel lines** are lines in the same flat surface that never meet, however far they are extended, like the two rails of a railway track or the lines on a ruled notebook page. We write **l ∥ m**.\n\nA line that crosses two (or more) other lines is called a **transversal**. When a transversal cuts two lines, it makes **eight angles**: four at each crossing. Number them 1 to 4 at the top crossing and 5 to 8 at the bottom crossing, in the same positions (top-left, top-right, bottom-right, bottom-left).\n\n- Angles **between** the two lines are **interior** angles (3, 4, 5, 6 in the usual picture).\n- Angles **outside** the two lines are **exterior** angles (1, 2, 7, 8).\n\nThe pairs formed get special names, whether or not the lines are parallel. But the magic happens **when the lines are parallel**.",{"id":187,"type":62,"caption":188,"columns":189,"rows":194},"table-transversal","Angle pairs made by a transversal. The last column holds only when the two lines are parallel",[190,191,192,193],"Pair","Where they are","Letter shape","If lines are parallel",[195,200,204,208,213,218],[196,197,198,199],"Corresponding angles","Same position at each crossing (e.g. top-left and top-left)","F shape","**Equal**",[201,202,203,199],"Alternate interior angles","Between the lines, on opposite sides of the transversal","Z shape",[205,206,207,199],"Alternate exterior angles","Outside the lines, on opposite sides of the transversal","—",[209,210,211,212],"Co-interior angles (interior angles on the same side)","Between the lines, on the same side of the transversal","C or U shape","**Add up to 180°**",[214,215,216,217],"Vertically opposite angles","Across each crossing point","X shape","Equal (always, parallel or not)",[219,220,207,221],"Linear pairs","Next to each other at one crossing","Add to 180° (always)",{"id":223,"type":47,"variant":158,"title":224,"markdown":225},"nuance-corr-axiom","The one fact we accept","In school geometry (NCERT included) the starting point is the **corresponding angles axiom**: *if a transversal cuts two parallel lines, each pair of corresponding angles is equal.* It captures what \"parallel\" means: both lines point in exactly the same direction, so the transversal meets them at the same angle. Everything else about transversals can be **proved** from this one fact plus linear pairs and vertically opposite angles.",{"id":227,"type":62,"caption":228,"columns":229,"rows":230},"table-proof-alt","Given: l ∥ m, transversal t; ∠5 corresponds to ∠1, ∠3 is vertically opposite ∠1. To prove: alternate interior angles ∠3 = ∠5",[133,134,135],[231,234,236],[87,232,233],"∠1 = ∠5","Corresponding angles, l ∥ m",[141,235,214],"∠1 = ∠3",[145,237,238],"∠3 = ∠5","Both equal ∠1 (steps 1 and 2)",{"id":240,"type":43,"markdown":241},"co-interior-proof","**Co-interior angles add to 180°.** Take ∠4 and ∠5, which are both interior and on the same side of the transversal. ∠4 and ∠3 form a linear pair at the top crossing, so ∠4 + ∠3 = 180°. We just proved ∠3 = ∠5. Replace ∠3 with ∠5: **∠4 + ∠5 = 180°**. ∎\n\nSo when the lines are parallel, the eight angles come in only **two sizes**: four equal \"small\" angles and four equal \"large\" angles, and small + large = 180° (unless the transversal is perpendicular, when all eight are 90°).",{"id":243,"type":244,"component":245,"componentVersion":5,"config":246,"objective":249,"textAlternative":250,"help":251},"lab-transversal","interactive","angle-pairs",{"scene":184,"initialAngle":247,"challenges":248},70,6,"Tilt a transversal across two parallel lines, highlight each kind of pair, and find missing angles.","Two parallel lines cut by a transversal you can tilt (between 20° and 160°). Four angles ∠a–∠d are at the top crossing and four, ∠e–∠h, at the bottom crossing, in matching positions. It starts at 70°. A live table lists all eight sizes.\n\nButtons highlight each kind of pair: **corresponding** (same position, F shape, equal), **alternate interior** (Z shape, equal), **alternate exterior** (equal), **co-interior** (C shape, adding to 180°), plus vertically opposite angles and linear pairs at each crossing. At 70° you see only two sizes, 70° and 110°.\n\nAs you tilt, all eight angles change together but only ever take two values that add to 180°. At 90° all eight are right angles.\n\nThe six challenges give one angle and ask for another somewhere in the picture; type the number of degrees. If an angle is 125°, its corresponding and alternate partners are 125° and its co-interior partner is 55°. The lab names the pair and the reason after each answer.",{"hints":252},[253,254],"Only two sizes appear: a small one and a large one adding to 180°.","F = corresponding (equal), Z = alternate (equal), C = co-interior (sum 180°).",{"id":256,"type":244,"component":257,"componentVersion":5,"config":258,"objective":305,"textAlternative":306},"lab-sort-transversal","sort-game",{"prompt":259,"bins":260,"items":267,"seconds":113},"Two parallel lines are cut by a transversal. Is each pair equal, or does it add up to 180°?",[261,264],{"id":262,"label":263},"equal","Equal",{"id":265,"label":266},"sum180","Add up to 180°",[268,271,274,277,281,285,289,293,297,301],{"id":269,"label":196,"bin":262,"why":270},"t1","Corresponding angles axiom: same position at each crossing.",{"id":272,"label":201,"bin":262,"why":273},"t2","Each equals the same corresponding angle (via vertically opposite).",{"id":275,"label":205,"bin":262,"why":276},"t3","Each is vertically opposite to one of a pair of equal alternate interior angles.",{"id":278,"label":279,"bin":265,"why":280},"t4","Co-interior angles (same side, between the lines)","One of them forms a linear pair with the other's alternate angle.",{"id":282,"label":283,"bin":262,"why":284},"t5","Vertically opposite angles at one crossing","Always equal, parallel lines or not.",{"id":286,"label":287,"bin":265,"why":288},"t6","A linear pair at one crossing","Adjacent angles on a straight line.",{"id":290,"label":291,"bin":265,"why":292},"t7","Exterior angles on the same side of the transversal","Each is vertically opposite to one of a co-interior pair.",{"id":294,"label":295,"bin":262,"why":296},"t8","The F-shape pair","The F shape outlines corresponding angles.",{"id":298,"label":299,"bin":262,"why":300},"t9","The Z-shape pair","The Z shape outlines alternate angles.",{"id":302,"label":303,"bin":265,"why":304},"t10","The C-shape (or U-shape) pair","The C shape outlines co-interior angles.","Sort transversal angle pairs into “equal” and “add up to 180°”.","A sorting game with two bins, for two parallel lines cut by a transversal.\n\nEqual: corresponding angles (F shape), alternate interior angles (Z shape), alternate exterior angles, and vertically opposite angles at one crossing.\n\nAdd up to 180°: co-interior angles (C or U shape), a linear pair at one crossing, and exterior angles on the same side of the transversal.\n\nRule of thumb: when the lines are parallel there are only two angle sizes. Two angles of the same size are equal; a small one and a large one add to 180°.",{"id":308,"type":166,"title":309,"problem":310,"steps":311},"we-transversal","All eight from one","Lines l ∥ m are cut by transversal t. The angle at the top crossing, above l and to the right of t, is **62°**. Find all eight angles.",[312,313,314,315,316],"Top crossing: the angle above l, right of t = 62° (given). The angle below l, left of t = 62° (vertically opposite).","The other two at the top crossing = 180° − 62° = **118°** each (linear pairs).","Bottom crossing: the angle above m, right of t corresponds to the given angle, so it is **62°** (corresponding angles, l ∥ m).","So the bottom crossing repeats the top one: 62°, 118°, 62°, 118° going round.","Check a co-interior pair: the angle below l on the right (118°) and the angle above m on the right (62°) add to 180°. ✓",{"id":318,"type":107,"itemId":319,"prompt":320,"check":321,"hints":323,"feedback":326},"pr-cointerior","angles.deep-cointerior","Two parallel lines are cut by a transversal. One co-interior angle is **(2x + 10)°** and the other is **(3x − 30)°**. Find **x**.",{"kind":111,"answer":322,"tolerance":113},40,[324,325],"Co-interior angles add to 180°.","(2x + 10) + (3x − 30) = 180.",{"correct":327,"incorrect":328},"Right: 5x − 20 = 180, 5x = 200, x = 40. The angles are 90° and 90°: the transversal is perpendicular!","Co-interior angles are supplementary: 2x + 10 + 3x − 30 = 180, so 5x = 200 and x = 40.",{"id":330,"type":166,"title":331,"problem":332,"steps":333},"we-trans-alg","Transversals with algebra","Lines l ∥ m are cut by a transversal. (i) A pair of **corresponding** angles are **(3x + 5)°** and **(4x − 20)°**. (ii) On another pair of parallel lines, **co-interior** angles are **y°** and **(2y + 30)°**. Find each angle.",[334,335,336,337,338],"(i) Corresponding angles on parallel lines are equal: 3x + 5 = 4x − 20.","So x = 25, and each angle is 3 × 25 + 5 = **80°** (check: 4 × 25 − 20 = 80).","(ii) Co-interior angles on parallel lines add to 180°: y + 2y + 30 = 180.","So 3y = 150, y = 50, and the angles are **50°** and **130°**.","Key habit: decide **equal or add to 180°** first, from the shape of the pair (F or Z: equal; C: add to 180°), and only then write the equation.",{"id":340,"type":47,"variant":341,"title":342,"markdown":343},"misc-cointerior-equal","misconception","“Co-interior angles are equal too”","Corresponding and alternate angles are equal, so many students assume co-interior angles are equal as well. They are **not** (unless both are 90°). Picture the C shape: one co-interior angle opens wide and the other narrow, like the inside of a ladder leaning against a wall. On parallel lines they **add to 180°**. A quick check: if you ever find two co-interior angles of 70° and 70°, the lines are not parallel.",{"id":345,"type":53,"title":346,"eyebrow":347,"navLabel":348},"ch04","Testing whether lines are parallel","Chapter 04","4 Testing parallels",{"id":350,"type":43,"markdown":351},"converse","The facts about transversals also work **backwards**. These backwards statements are called **converses**:\n\n- If a transversal makes **equal corresponding angles** with two lines, the lines are **parallel**.\n- If it makes **equal alternate interior angles**, the lines are parallel.\n- If it makes **co-interior angles that add to 180°**, the lines are parallel.\n\nThis is how carpenters, engineers and surveyors check parallel lines in real life. A railway track inspector cannot walk to infinity to check that the rails never meet, but can check that a straight sleeper (a transversal) meets both rails at equal angles. A carpenter uses a set-square sliding along a straight edge to draw parallel lines, because every line drawn meets the edge at the same angle.",{"id":353,"type":47,"variant":341,"title":354,"markdown":355},"misc-converse","“Alternate angles are always equal”","Alternate angles are **only** equal when the lines are parallel. If a transversal cuts two lines that are **not** parallel, it still makes pairs of alternate angles, but they have different sizes. This is exactly what lets us **test** for parallel lines: unequal alternate angles prove the lines are not parallel, and they will meet somewhere. Always check for the ∥ sign or arrow marks before using these facts.",{"id":357,"type":166,"title":358,"problem":359,"steps":360},"we-test-parallel","Are the shelves parallel?","A carpenter's diagonal brace crosses two shelves. The co-interior angles it makes with the shelves on one side measure **97°** and **83°**. On another bookcase they measure **97°** and **85°**. Which bookcase has parallel shelves?",[361,362,363],"Bookcase 1: 97° + 83° = **180°**. Co-interior angles add to 180°, so the shelves **are parallel** (converse of the co-interior property).","Bookcase 2: 97° + 85° = **182°**, which is not 180°. The shelves are **not parallel**: they would slowly get closer on one side (the side where the co-interior angles add to more than 180° is where they spread apart).","So only the first bookcase has parallel shelves.",{"id":365,"type":107,"itemId":366,"prompt":367,"check":368,"hints":381,"feedback":383},"pr-parallel-test","angles.deep-parallel-test","A transversal cuts two lines. A pair of **alternate interior** angles measure **74°** and **74°**. Are the lines parallel?",{"kind":369,"options":370,"correct":380},"choice",[371,374,377],{"id":372,"label":373},"a","Yes, because the alternate interior angles are equal",{"id":375,"label":376},"b","No, alternate angles should add to 180°",{"id":378,"label":379},"c","You cannot tell without measuring more angles",[372],[382],"Use the converse: equal alternate interior angles → parallel lines.",{"correct":384,"incorrect":385},"Correct: equal alternate interior angles guarantee the lines are parallel.","It is co-interior angles that add to 180°. Equal alternate interior angles are exactly the test for parallel lines, so yes, they are parallel.",{"id":387,"type":388,"prompt":389,"options":390,"explanation":400},"predict-not-parallel","prediction","A transversal cuts two lines that are **not** parallel. Which of these is still true?",[391,393,395,397],{"id":372,"label":392},"Corresponding angles are equal",{"id":375,"label":394},"Alternate interior angles are equal",{"id":378,"label":396},"Vertically opposite angles at each crossing are equal",{"id":398,"label":399},"d","Co-interior angles add to 180°","**Only vertically opposite angles.** Vertically opposite angles and linear pairs depend on just one crossing, so they always work. Corresponding, alternate and co-interior facts compare **two different crossings**, and they only work when the lines are parallel. That is why a question must tell you the lines are parallel (arrows or the ∥ sign) before you use them.",{"id":402,"type":107,"itemId":403,"prompt":404,"check":405,"hints":407,"feedback":410},"pr-alt-ext","angles.deep-alt-ext","l ∥ m, cut by transversal t. An **alternate exterior** angle at l measures **(5y − 15)°** and its partner at m measures **(2y + 36)°**. Find the angle.",{"kind":111,"answer":247,"tolerance":113,"unit":406},"°",[408,409],"Alternate exterior angles are equal when the lines are parallel.","5y − 15 = 2y + 36.",{"correct":411,"incorrect":412},"Correct: 3y = 51, y = 17, and each angle is 5 × 17 − 15 = 70°.","Set them equal: 5y − 15 = 2y + 36, so 3y = 51 and y = 17. Then 5 × 17 − 15 = 70°, and 2 × 17 + 36 = 70° as a check.",{"id":414,"type":166,"title":415,"problem":416,"steps":417},"we-eratosthenes","Measuring the Earth with alternate angles","About 2,250 years ago, Eratosthenes was told that at noon on midsummer day the Sun shone straight down a well at Syene (now Aswan, in Egypt). At the same time in Alexandria, reported to be about **5,000 stadia** to the north, a vertical pole cast a shadow showing the Sun's rays were **7.2°** from vertical — one fiftieth of a full turn. How did he estimate the distance round the Earth?",[418,419,420,421,422],"The Sun is so far away that its rays arriving at Syene and Alexandria are, for all practical purposes, **parallel lines**.","Imagine lines from Syene and from Alexandria to the centre of the Earth. The line through Syene points straight at the Sun, along a sunray. The line through Alexandria is a **transversal** crossing both parallel sunrays.","The 7.2° angle between the pole and the sunray at Alexandria and the angle at the Earth's centre between the two lines are **alternate angles**, so the angle at the centre is also **7.2°**.","7.2° is 360 ÷ 7.2 = **50** times smaller than a full turn, so the Earth's circumference is about 50 × 5,000 = **250,000 stadia**.","Historians argue about how long a stadion was, and the story itself survives only through later writers, so we cannot say exactly how accurate he was; most estimates put his answer within about 10–20% of today's value of roughly 40,000 km. An angle fact and a shadow measured the whole planet.",{"id":424,"type":107,"itemId":425,"prompt":426,"check":427,"hints":429,"feedback":432},"pr-cointerior-ratio","angles.deep-cointerior-ratio","Two parallel lines are cut by a transversal. A pair of **co-interior** angles are in the ratio **7 : 11**. Find the **larger** angle.",{"kind":111,"answer":428,"tolerance":113,"unit":406},110,[430,431],"Co-interior angles on parallel lines add to 180°.","7 + 11 = 18 equal parts.",{"correct":433,"incorrect":434},"Correct: one part is 180° ÷ 18 = 10°, so the angles are 70° and 110°.","Co-interior angles are supplementary, so the 18 parts make 180°. One part is 10°, and the larger angle is 11 × 10° = 110°.",{"id":436,"type":53,"title":437,"eyebrow":438,"navLabel":439},"ch05","The angles of a triangle add to 180°","Chapter 05","5 Triangle sum",{"id":441,"type":43,"markdown":442},"triangle-proof","In Investigate, torn paper corners suggested that a triangle's three angles make a straight line. Here is the proof, essentially Euclid's Proposition I.32.\n\n**Given:** triangle ABC with angles ∠A, ∠B and ∠C at its corners. **To prove:** ∠A + ∠B + ∠C = 180°.\n\n1. Through vertex A, draw line PQ **parallel to BC**. (Through a point not on a line there is exactly one parallel line.)\n2. Along PQ at A there are three angles side by side: ∠PAB, ∠BAC and ∠CAQ. They make a straight line, so **∠PAB + ∠BAC + ∠CAQ = 180°** (angles on a straight line).\n3. ∠PAB = ∠B, because they are **alternate interior angles** (PQ ∥ BC, transversal AB).\n4. ∠CAQ = ∠C, because they are **alternate interior angles** (PQ ∥ BC, transversal AC).\n5. Substitute into step 2: **∠B + ∠A + ∠C = 180°**. ∎\n\nThis is exactly the tear-the-corners experiment in disguise: the parallel line at A is where the torn corners ∠B and ∠C slide to.",{"id":444,"type":445,"tone":446,"items":447},"spec-triangle-consequences","spec","copper",[448,452,456,460,464],{"label":449,"big":450,"value":451},"Equilateral triangle","60° each","Three equal angles: 180° ÷ 3 = 60°.",{"label":453,"big":454,"value":455},"Right triangle","other two: 90° total","The two acute angles are complementary: 180° − 90° = 90°.",{"label":457,"big":458,"value":459},"At most one","right or obtuse","Two angles of 90° or more would already reach 180°, leaving nothing for the third.",{"label":461,"big":462,"value":463},"Isosceles","equal base angles","If the top angle is 40°, each base angle is (180° − 40°) ÷ 2 = 70°.",{"label":465,"big":466,"value":467},"Third angle","180° − the other two","Know two angles, and the third is fixed.",{"id":469,"type":166,"title":470,"problem":471,"steps":472},"we-tri-ratio","Angles in a ratio","The angles of a triangle are in the ratio **2 : 3 : 4**. Find each angle.",[473,474,475,476],"2 + 3 + 4 = 9 equal parts.","The three angles add to 180° (angle sum of a triangle), so one part = 180° ÷ 9 = 20°.","Angles: 2 × 20° = **40°**, 3 × 20° = **60°**, 4 × 20° = **80°**.","Check: 40 + 60 + 80 = 180°. ✓ All acute, so it is an acute-angled triangle.",{"id":478,"type":388,"prompt":479,"options":480,"explanation":487},"predict-tri-obtuse","Could a triangle have angles of 100° and 85°?",[481,483,485],{"id":372,"label":482},"Yes, the third angle would be −5°",{"id":375,"label":484},"No, because 100° + 85° is already more than 180°",{"id":378,"label":486},"Yes, if it is a very big triangle","**No.** 100° + 85° = 185°, which is already more than the total of 180° allowed for all three angles. Size makes no difference: a huge triangle and a tiny triangle both have angles summing to exactly 180°. This is also why a triangle can have at most one obtuse angle.",{"id":489,"type":47,"variant":490,"title":491,"markdown":492},"model-limit-sphere","model_limit","Where 180° fails: triangles on a globe","The proof used a parallel line through A, and the fact that there is **exactly one** such line. That is true on a flat page, but not on a curved surface. On a globe, draw a triangle from the North Pole down to the equator, a quarter of the way along the equator, and back up to the pole. **All three angles are 90°**, so the sum is **270°**! Geometry on curved surfaces (non-Euclidean geometry) was developed in the 1800s and is used today for flight routes, GPS and Einstein's theory of gravity. On a flat page, or a school playground, 180° is exact.",{"id":494,"type":107,"itemId":495,"prompt":496,"check":497,"hints":499,"feedback":501},"pr-tri-isos","angles.deep-tri-isos","An isosceles triangle has a top (apex) angle of **36°**. What is each of its two equal base angles?",{"kind":111,"answer":498,"tolerance":113,"unit":406},72,[500],"The two base angles share what is left of 180°.",{"correct":502,"incorrect":503},"Yes: (180° − 36°) ÷ 2 = 144° ÷ 2 = 72°. (This 36°–72°–72° triangle appears in the five-pointed star.)","The angles add to 180°. Take away the apex: 180° − 36° = 144°, shared equally: 144° ÷ 2 = 72°.",{"id":505,"type":53,"title":506,"eyebrow":507,"navLabel":508},"ch06","Quadrilaterals and polygons","Chapter 06","6 Polygons",{"id":510,"type":43,"markdown":511},"quad","A **quadrilateral** (four-sided shape) can always be cut by a **diagonal** into **two triangles**. The angles of the two triangles together make up exactly the four corner angles of the quadrilateral. So:\n\n**Angle sum of a quadrilateral = 2 × 180° = 360°.**\n\nThis works for squares, rectangles, kites, trapeziums and lopsided shapes alike, as long as the shape is not crossed over itself. (For a shape with a “dent” in it, choose the diagonal from the dented corner, which lies inside the shape.)\n\nThe same trick works for any polygon. From one corner, draw diagonals to all the non-neighbouring corners. A polygon with **n** sides splits into **n − 2** triangles, so:\n\n**Angle sum of an n-sided polygon = (n − 2) × 180°.**",{"id":513,"type":62,"caption":514,"columns":515,"rows":521},"table-polygons","Angle sums of polygons, and each angle when the polygon is regular",[516,517,518,519,520],"Sides (n)","Name","Triangles (n − 2)","Angle sum","Each angle if regular",[522,526,529,533,538,543,548,552,557],[145,523,87,524,525],"Triangle","180°","60°",[149,527,141,528,73],"Quadrilateral","360°",[153,530,145,531,532],"Pentagon","540°","108°",[534,535,149,536,537],"6","Hexagon","720°","120°",[539,540,153,541,542],"7","Heptagon","900°","≈ 128.6°",[544,545,534,546,547],"8","Octagon","1080°","135°",[33,549,539,550,551],"Nonagon","1260°","140°",[553,554,544,555,556],"10","Decagon","1440°","144°",[558,559,553,560,561],"12","Dodecagon","1800°","150°",{"id":563,"type":166,"title":564,"problem":565,"steps":566},"we-quad","A missing corner of a quadrilateral field","A farmer's four-sided field has three corner angles of **75°**, **110°** and **95°**. Find the fourth.",[567,568,569],"The four angles of a quadrilateral add to 360° (a diagonal splits it into two triangles).","75° + 110° + 95° = 280°.","Fourth angle = 360° − 280° = **80°**.",{"id":571,"type":107,"itemId":572,"prompt":573,"check":574,"hints":576,"feedback":579},"pr-polygon","angles.deep-polygon","What is each inside angle of a **regular octagon** (8 equal sides and angles), like many stop signs?",{"kind":111,"answer":575,"tolerance":113,"unit":406},135,[577,578],"Angle sum = (n − 2) × 180°.","Then share it equally among 8 corners.",{"correct":580,"incorrect":581},"Right: (8 − 2) × 180° = 1,080°, and 1,080° ÷ 8 = 135°.","An octagon splits into 8 − 2 = 6 triangles, so its angle sum is 6 × 180° = 1,080°. Each of 8 equal angles is 1,080° ÷ 8 = 135°.",{"id":583,"type":53,"title":584,"eyebrow":585,"navLabel":586},"ch07","Harder problems: bends, zigzags and algebra","Chapter 07","7 Harder problems",{"id":588,"type":43,"markdown":589},"bend","A classic puzzle: two parallel lines, and a path that goes from one to the other with a **bend** in it, pointing between the lines (like an arrowhead **>**). How big is the angle at the bend?\n\n**Trick: draw a third parallel line through the bend.** It splits the bend angle into two parts. The top part is an alternate angle with the angle at the top line, and the bottom part is an alternate angle with the angle at the bottom line. So:\n\n**angle at the bend = (angle at top line) + (angle at bottom line).**\n\nThis trick, adding an extra line to create angles you know, is called a **construction** in a proof. It is the same idea we used for the triangle sum. Good problem-solvers are always asking: *what extra line would help?*",{"id":591,"type":592,"title":593,"items":594},"steps-chase","steps","An angle-chasing strategy for hard problems",[595,599,603,607,611,615,619],{"title":596,"tag":597,"text":598},"Redraw big","neat sketch","Copy the diagram large, and mark every given angle, equal side and parallel line.",{"title":600,"tag":601,"text":602},"Name unknowns","x, y, a, b","Give letters to the angles you need; use one letter for angles known to be equal.",{"title":604,"tag":605,"text":606},"Harvest easy facts","lines and points","Linear pairs, vertically opposite angles, angles at a point, triangle sums.",{"title":608,"tag":609,"text":610},"Look for parallels","F, Z, C","If lines are parallel, find corresponding, alternate and co-interior pairs.",{"title":612,"tag":613,"text":614},"Add a line","construction","No progress? Extend a side, draw a parallel through a corner, or join two points.",{"title":616,"tag":617,"text":618},"Write equations","solve","Turn each fact into an equation and solve.",{"title":620,"tag":621,"text":622},"Check","second route","Confirm with a different fact, or check that totals are 180° or 360°.",{"id":624,"type":166,"title":625,"problem":626,"steps":627},"we-bend","The arrowhead between parallel lines","Lines AB ∥ CD, with AB above CD. Point E lies between them. ∠BAE = 40° and ∠DCE = 35°, where both angles open towards E. Find ∠AEC.",[628,629,630,631],"Through E draw line EF parallel to AB (and so also parallel to CD), pointing the same way as B and D.","∠AEF = ∠BAE = 40° (alternate interior angles, AB ∥ EF).","∠FEC = ∠DCE = 35° (alternate interior angles, EF ∥ CD).","∠AEC = ∠AEF + ∠FEC = 40° + 35° = **75°** (adjacent angles).",{"id":633,"type":166,"title":634,"problem":635,"steps":636,"help":642},"we-zigzag","A zigzag with three bends","A zigzag path runs down from line AB to a parallel line CD, bending three times. Going down the path, the angles are: **30°** at AB, then bends of **x**, **50°** and **65°**, then **45°** at CD. The angles alternate: 30°, 50° and 45° open towards the right, while x and 65° open towards the left. Find x.",[637,638,639,640,641],"Draw a line parallel to AB through every bend. Each bend splits into two parts, and each part is an alternate angle with a part of the bend (or line angle) just above or below it.","Adding everything up gives the zigzag rule: **the angles opening one way add up to the angles opening the other way**. (The arrowhead puzzle is the simplest case: bend = top + bottom.)","Opening right: 30° + 50° + 45° = 125°.","Opening left: x + 65°.","x + 65° = 125°, so x = **60°**.",{"simplerExplanation":643},"Think of the zigzag as several arrowheads glued together. Each bend splits into two alternate angles, so everything balances.",{"id":645,"type":107,"itemId":646,"prompt":647,"check":648,"hints":650,"feedback":652},"pr-bend","angles.deep-bend","AB ∥ CD, and a bent path joins them with the bend at E pointing between the lines. The angle at the top line is **52°**, and the bend angle ∠AEC is **110°**. What is the angle at the bottom line?",{"kind":111,"answer":649,"tolerance":113,"unit":406},58,[651],"Bend angle = top angle + bottom angle.",{"correct":653,"incorrect":654},"Correct: 110° − 52° = 58°.","Draw a parallel line through E. The bend is split into 52° (alternate with the top) and the rest, which is alternate with the bottom angle: 110° − 52° = 58°.",{"id":656,"type":107,"itemId":657,"prompt":658,"check":659,"hints":661,"feedback":664},"pr-algebra-vo","angles.deep-algebra-vo","Two lines cross. A pair of vertically opposite angles measure **(4x − 20)°** and **(2x + 30)°**. Find the size of **each of the other two angles**.",{"kind":111,"answer":660,"tolerance":113,"unit":406},100,[662,663],"Vertically opposite angles are equal: 4x − 20 = 2x + 30.","Then use a linear pair.",{"correct":665,"incorrect":666},"Yes: 2x = 50, so x = 25 and each angle is 80°. The other two are 180° − 80° = 100° each.","Set them equal: 4x − 20 = 2x + 30, so 2x = 50 and x = 25. Each of that pair is 4 × 25 − 20 = 80°. The other two are its linear-pair partners: 180° − 80° = 100°.",{"id":668,"type":166,"title":669,"problem":670,"steps":671},"we-two-step","A two-step chase across a transversal","Lines l ∥ m are cut by transversal t at P (on l) and Q (on m). The angle at P above l, on the left of t, is **125°**. Find the angle at Q below m, on the right of t, and the angle at Q above m, on the right of t.",[672,673,674,675],"At P: the angle below l on the right of t is vertically opposite the given angle, so it is **125°**.","That angle (below l, right of t) and the angle at Q below m, right of t are **corresponding** angles, so the angle below m on the right is **125°** too. (In one step: the two angles are alternate exterior angles, which are equal.)","The angle above m on the right of t forms a linear pair with it: 180° − 125° = **55°**.","Check: the angle below l on the right (125°) and the angle above m on the right (55°) are co-interior, and 125 + 55 = 180. ✓",{"id":677,"type":107,"itemId":678,"prompt":679,"check":680,"hints":691,"feedback":693},"pr-trans-choose","angles.deep-trans-choose","Two parallel lines are cut by a transversal. You are told one angle and asked for its **alternate exterior** partner. What do you do?",{"kind":369,"options":681,"correct":690},[682,684,686,688],{"id":372,"label":683},"Copy the same number: alternate exterior angles are equal",{"id":375,"label":685},"Subtract it from 180°",{"id":378,"label":687},"Subtract it from 90°",{"id":398,"label":689},"Double it",[372],[692],"Alternate exterior angles are each vertically opposite to one of a pair of alternate interior angles.",{"correct":694,"incorrect":695},"Right: alternate exterior angles are equal, just like alternate interior angles.","Alternate exterior angles are equal when the lines are parallel. Only co-interior angles and linear pairs need 180° minus.",{"id":697,"type":244,"component":245,"componentVersion":5,"config":698,"objective":701,"textAlternative":702},"lab-transversal-hard",{"scene":184,"initialAngle":699,"challenges":700},115,10,"Ten more transversal challenges, starting from 115°: decide which pair links the given and asked angles, then answer.","The same parallel lines and transversal, starting at 115° (angles of 115° and 65°). Use the highlight buttons to check your thinking, then switch to the challenges.\n\nThe ten challenges each give one angle and ask for another, chosen at random from all the pairs in the scene: corresponding, alternate interior, alternate exterior, co-interior, vertically opposite and linear pairs. The skill is to recognise the pair first: equal pairs (F, Z, alternate exterior, vertically opposite) give the same number; the co-interior and linear pairs give 180° minus it. Type the number; the lab explains the relation. Two-step and algebra problems, such as corresponding angles (3x + 5)° and (4x − 20)°, are in the worked examples and practice around this lab.",{"id":704,"type":244,"component":245,"componentVersion":5,"config":705,"objective":708,"textAlternative":709},"lab-intersecting-alg",{"scene":706,"initialAngle":707,"challenges":700},"intersecting",80,"See the vertically-opposite proof in action: as one line turns, ∠a and ∠c stay equal because both are 180° − ∠b.","Two lines crossing at O with angles ∠a, ∠b, ∠c, ∠d going round, starting at 80°, 100°, 80°, 100°. The live table shows all four sizes.\n\nLink it to the proof in chapter 2: highlight the linear pair ∠a and ∠b, then ∠b and ∠c. Both pairs total 180° at every position, so ∠a and ∠c are both 180° − ∠b and must be equal. Rotate the line and watch the table: ∠a and ∠c always match, and so do ∠b and ∠d.\n\nThe ten challenges give one angle and ask for another at the crossing; type the number of degrees (opposite: equal; neighbour: 180° minus). For algebra versions, such as (4x − 20)° and (2x + 30)°, see the practice questions in chapter 7.",{"id":711,"type":53,"title":712,"eyebrow":713,"navLabel":714},"ch08","A short history of angles","Chapter 08","8 History",{"id":716,"type":717,"title":718,"items":719},"timeline-angles","timeline","From Babylonian stars to Euclid and beyond",[720,724,728,732,736,740,744,748],{"time":721,"title":722,"text":723},"c. 1800 BCE","Babylonian base 60","Babylonian mathematicians write numbers in base 60 on clay tablets. Their astronomy later helps give us 360 parts in a circle, and 60 minutes in an hour and in a degree.",{"time":725,"title":726,"text":727},"800–500 BCE","Sulba Sutras, India","The Sulba Sutras of Baudhayana, Manava, Apastamba and Katyayana give cord-and-peg methods for laying out right angles and squares for fire altars, using triples such as 3, 4, 5, and state a rule equivalent to the Pythagorean theorem.",{"time":729,"title":730,"text":731},"c. 600 BCE","Thales of Miletus","Later Greek writers — Proclus, quoting the lost history of Eudemus, some 1,000 years afterwards — credit Thales with proving that the angles between two intersecting lines are equal and that the base angles of an isosceles triangle are equal.",{"time":733,"title":734,"text":735},"c. 300 BCE","Euclid's Elements","Euclid collects Greek geometry into 13 books. Book I proves vertically opposite angles equal (I.15) and the triangle angle sum (I.32) from a few postulates, including the famous parallel postulate.",{"time":737,"title":738,"text":739},"c. 150 CE","Ptolemy's tables","Ptolemy of Alexandria uses degrees, minutes and seconds of arc in his astronomy, tabulating chords of a circle for every half degree.",{"time":741,"title":742,"text":743},"499 CE","Aryabhata","The Aryabhatiya includes a table of sines (jya), used to calculate with angles in astronomy, measured with the circle divided into 21,600 minutes of arc (360 × 60).",{"time":745,"title":746,"text":747},"1820s","Non-Euclidean geometry","Lobachevsky, Bolyai and Gauss show that geometries where the parallel postulate fails are consistent. Triangle angle sums need not be 180°.",{"time":749,"title":750,"text":751},"1873","The radian is named","The word radian first appears in print on 5 June 1873, in examination questions set by James Thomson at Queen's College, Belfast, for the angle whose arc equals the radius. It becomes the standard unit in higher mathematics.",{"id":753,"type":47,"variant":754,"title":755,"markdown":756},"careful-history","careful","History is messier than a timeline","Dates for ancient mathematics are often approximate, and \"who discovered what\" usually comes from writers who lived centuries later. For example, we know about Thales's proofs only from Proclus, writing around 450 CE and quoting a history by Eudemus that is itself lost. Many cultures, including those of India, China and Egypt, used right angles and angle measures in building and astronomy long before any formal proofs were written down.",{"id":758,"type":53,"title":759,"eyebrow":760,"navLabel":761},"ch09","Edge cases and careful thinking","Chapter 09","9 Edge cases",{"id":763,"type":43,"markdown":764},"edge","Deep understanding means knowing where the rules stop working. A few edge cases:\n\n- **Reflex angles and naming.** ∠ABC normally means the smaller angle. If the reflex angle is meant, say “reflex ∠ABC”.\n- **Zero-width “triangles”.** If three points lie on one line, there is no triangle: two angles are 0° and the third is 180°. The angle sum is still 180°, a hint that the rule is very robust.\n- **Parallel lines in a triangle problem.** The triangle proof needs a parallel line; on a sphere there are no parallel lines, which is exactly why the sum changes there.\n- **Transversal at 90°.** All eight angles are right angles. Corresponding, alternate and co-interior facts still hold, but they all look the same.\n- **More than 360°.** A turn can be 450° (a full turn plus a quarter turn). As a direction, it ends in the same place as 90°. Wheels, fans and spinning bowlers can turn through thousands of degrees.",{"id":766,"type":47,"variant":341,"title":767,"markdown":768},"misc-sum-triangle-exterior","“A bigger triangle has bigger angles”","Enlarge a triangle to twice its size (like a photocopy at 200%) and its **sides double** but its **angles stay exactly the same**. This is why a small map and the real land have the same angles, and why a photo of a building has the same angles as the building. Shapes with the same angles but different sizes are called **similar** shapes.",{"id":770,"type":771,"prompt":772},"reflect-proof","reflection","Why do mathematicians bother to **prove** that a triangle's angles add to 180°, when every triangle anyone has ever measured already shows it? Give two reasons, and use the idea of the triangle on a globe in one of them.",{"id":774,"type":244,"component":775,"componentVersion":5,"config":776,"objective":801,"textAlternative":802},"lab-match-reasons","match-pairs",{"prompt":777,"mode":778,"pairs":779},"Match each result with the key reason in its proof.","connect",[780,783,786,789,792,795,798],{"a":781,"b":782},"Vertically opposite angles equal","Two linear pairs share an angle",{"a":784,"b":785},"Alternate interior angles equal","Corresponding angles + vertically opposite",{"a":787,"b":788},"Co-interior angles sum to 180°","Alternate angles + a linear pair",{"a":790,"b":791},"Triangle angle sum 180°","Parallel line through a vertex",{"a":793,"b":794},"Quadrilateral angle sum 360°","A diagonal makes two triangles",{"a":796,"b":797},"Linear-pair bisectors perpendicular","Half of 180° is 90°",{"a":799,"b":800},"Bend between parallels = sum","Extra parallel line through the bend","Connect seven angle results with the key idea used to prove each one.","A matching game of results and reasons: vertically opposite angles are equal ↔ two linear pairs share an angle; alternate interior angles are equal ↔ corresponding angles plus vertically opposite angles; co-interior angles sum to 180° ↔ alternate angles plus a linear pair; triangle angle sum is 180° ↔ a parallel line drawn through one vertex; quadrilateral angle sum is 360° ↔ a diagonal makes two triangles; the bisectors of a linear pair are perpendicular ↔ half of 180° is 90°; the bend angle between parallel lines equals the sum of the two outer angles ↔ an extra parallel line drawn through the bend.",{"id":804,"type":805,"title":806,"terms":807},"glossary-deepen","glossary","Words for proofs and parallels",[808,812,816,820,823,827,830,834,838,842,846,850,854,858,862],{"term":809,"meaning":810,"example":811},"proof","A chain of statements, each justified by a definition, axiom, given fact or earlier result, that shows something must be true.","The proof that vertically opposite angles are equal.",{"term":813,"meaning":814,"example":815},"axiom (postulate)","A basic statement accepted without proof, used as a starting point.","Corresponding angles on parallel lines are equal.",{"term":817,"meaning":818,"example":819},"theorem","A statement that has been proved.","The angle sum of a triangle is 180°.",{"term":350,"meaning":821,"example":822},"The statement you get by swapping the “if” and “then” parts. It may or may not be true.","If corresponding angles are equal, the lines are parallel.",{"term":824,"meaning":825,"example":826},"parallel lines (∥)","Lines in the same plane that never meet.","The rails of a straight railway track.",{"term":184,"meaning":828,"example":829},"A line that crosses two or more other lines.","A sleeper across the rails.",{"term":831,"meaning":832,"example":833},"corresponding angles","Angles in the same position at each crossing of a transversal. Equal when the lines are parallel.","The F shape.",{"term":835,"meaning":836,"example":837},"alternate interior angles","Angles between two lines, on opposite sides of a transversal. Equal when the lines are parallel.","The Z shape.",{"term":839,"meaning":840,"example":841},"alternate exterior angles","Angles outside two lines, on opposite sides of a transversal. Equal when the lines are parallel.","Top-left at one crossing and bottom-right at the other.",{"term":843,"meaning":844,"example":845},"co-interior angles","Angles between two lines, on the same side of a transversal. They add to 180° when the lines are parallel. Also called interior angles on the same side.","The C or U shape.",{"term":847,"meaning":848,"example":849},"angle sum property","The angles of a triangle add up to 180°.","40° + 60° + 80° = 180°",{"term":851,"meaning":852,"example":853},"quadrilateral","A polygon with four sides; its angles add to 360°.","A kite, a trapezium.",{"term":855,"meaning":856,"example":857},"diagonal","A line segment joining two corners of a polygon that are not next to each other.","A diagonal splits a quadrilateral into two triangles.",{"term":859,"meaning":860,"example":861},"sexagesimal","Counting in base 60, as the Babylonians did.","60 minutes in a degree.",{"term":863,"meaning":864,"example":865},"non-Euclidean geometry","Geometry on curved surfaces, where the parallel postulate fails and triangle angle sums are not 180°.","A triangle on a globe with three 90° angles.",{"id":867,"type":868,"title":869,"questions":870},"quiz-deepen","quiz","Proofs, parallels and polygons",[871,884,896,909,922,935,947,957,970,983],{"itemId":872,"prompt":873,"options":874,"correct":398,"why":883},"angles.deep-q-360","Which is NOT a reason often given for choosing 360 parts in a full turn?",[875,877,879,881],{"id":372,"label":876},"Babylonian astronomers counted in base 60",{"id":375,"label":878},"360 is close to the number of days in a year",{"id":378,"label":880},"360 has many divisors",{"id":398,"label":882},"360 is a prime number","360 is far from prime: it has 24 divisors, which is exactly why it is useful.",{"itemId":885,"prompt":886,"options":887,"correct":375,"why":895},"angles.deep-q-vo-reason","In the proof that vertically opposite angles are equal, which fact is used twice?",[888,889,891,893],{"id":372,"label":392},{"id":375,"label":890},"A linear pair adds to 180°",{"id":378,"label":892},"A triangle's angles add to 180°",{"id":398,"label":894},"Angles at a point add to 360°","Each of the two opposite angles forms a linear pair with the same neighbour.",{"itemId":897,"prompt":898,"options":899,"correct":375,"why":908},"angles.deep-q-z","For parallel lines cut by a transversal, the Z-shape angles are…",[900,902,904,906],{"id":372,"label":901},"corresponding and equal",{"id":375,"label":903},"alternate and equal",{"id":378,"label":905},"co-interior and add to 180°",{"id":398,"label":907},"always 90°","The Z shape outlines alternate interior angles, which are equal when the lines are parallel.",{"itemId":910,"prompt":911,"options":912,"correct":375,"why":921},"angles.deep-q-cointerior","Co-interior angles on two lines cut by a transversal are 100° and 75°. The lines are…",[913,915,917,919],{"id":372,"label":914},"parallel",{"id":375,"label":916},"not parallel",{"id":378,"label":918},"perpendicular",{"id":398,"label":920},"impossible to judge","100° + 75° = 175°, not 180°, so the lines are not parallel.",{"itemId":923,"prompt":924,"options":925,"correct":375,"why":934},"angles.deep-q-tri","A triangle has angles 57° and 68°. The third angle is…",[926,928,930,932],{"id":372,"label":927},"45°",{"id":375,"label":929},"55°",{"id":378,"label":931},"65°",{"id":398,"label":933},"125°","180° − 57° − 68° = 55°.",{"itemId":936,"prompt":937,"options":938,"correct":375,"why":946},"angles.deep-q-right-tri","In a right-angled triangle, the two other angles are always…",[939,940,942,944],{"id":372,"label":262},{"id":375,"label":941},"complementary",{"id":378,"label":943},"supplementary",{"id":398,"label":945},"obtuse","They share 180° − 90° = 90°, so they add to 90°: complementary.",{"itemId":948,"prompt":949,"options":950,"correct":378,"why":956},"angles.deep-q-hex","What is the angle sum of a hexagon?",[951,952,953,954],{"id":372,"label":528},{"id":375,"label":531},{"id":378,"label":536},{"id":398,"label":955},"1,080°","(6 − 2) × 180° = 720°.",{"itemId":958,"prompt":959,"options":960,"correct":375,"why":969},"angles.deep-q-bend","AB ∥ CD with a bend E pointing between them. The outer angles are 25° and 48°. The bend angle is…",[961,963,965,967],{"id":372,"label":962},"23°",{"id":375,"label":964},"73°",{"id":378,"label":966},"107°",{"id":398,"label":968},"155°","Draw a parallel through E: the bend is the sum of two alternate angles, 25° + 48° = 73°.",{"itemId":971,"prompt":972,"options":973,"correct":375,"why":982},"angles.deep-q-globe","On a globe, a triangle can have three 90° angles. Why does this not break the flat-page proof?",[974,976,978,980],{"id":372,"label":975},"Globes are inaccurate",{"id":375,"label":977},"The proof needs exactly one parallel through a point, which fails on a sphere",{"id":378,"label":979},"90° angles are not real angles",{"id":398,"label":981},"It does break it: the proof is wrong","The flat proof relies on a unique parallel line. On a sphere there are no parallel lines, so the proof does not apply.",{"itemId":984,"prompt":985,"options":986,"correct":378,"why":992},"angles.deep-q-ratio","The angles of a triangle are in the ratio 1 : 2 : 3. The largest angle is…",[987,988,990,991],{"id":372,"label":525},{"id":375,"label":989},"80°",{"id":378,"label":73},{"id":398,"label":537},"6 parts make 180°, so one part is 30°, and the largest is 3 × 30° = 90°.",{"id":994,"type":995,"title":996,"points":997},"cheat-sheet","summary","Cheat sheet",[998,999,1000,1001,1002,1003,1004,1005,1006,1007],"**360°** is a human choice from Babylonian base-60 astronomy; 360 has 24 divisors. 1° = 60′, 1′ = 60″.","**Proof** = statements + reasons, starting from definitions and axioms. Mark the end with ∎.","**Vertically opposite angles are equal:** both are 180° minus the same neighbour (Euclid I.15).","**Transversal on parallel lines:** corresponding (F) equal, alternate (Z) equal, co-interior (C) add to 180°. Only two sizes appear.","**Converses test for parallels:** equal corresponding or alternate angles, or co-interior angles adding to 180°, mean the lines are parallel.","**Triangle angle sum = 180°**, proved with a parallel line through one vertex (Euclid I.32).","Consequences: equilateral 60° each; right triangle's acute angles complementary; at most one right or obtuse angle.","**Quadrilateral = 360°**; **n-sided polygon = (n − 2) × 180°**.","**Bend between parallels** = sum of the outer angles (draw an extra parallel). For zigzags, left-pointing angles total the right-pointing ones.","On curved surfaces (a globe), triangle sums can exceed 180°: non-Euclidean geometry.",{"id":1009,"type":1010,"conceptId":1011,"relation":1012,"explanation":1013},"conn-lines","connection","lines","helps_understand","Parallel and intersecting lines from the lines topic are the setting for transversal angle pairs.",{"id":1015,"type":1010,"conceptId":1016,"relation":1017,"explanation":1018},"conn-shape","shape-and-space","applied_in","Triangle and quadrilateral angle sums, and the angles of regular polygons, describe the shapes studied in shape and space.",{"id":1020,"type":1010,"conceptId":1021,"relation":1022,"explanation":1023},"conn-hcf","hcf-and-lcm","related_to","360 was chosen partly for its 24 factors; finding factors and common divisors is the heart of the HCF and LCM topic.",{"id":1025,"type":1010,"conceptId":1026,"relation":1017,"explanation":1027},"conn-construct","constructing-angles","Drawing parallel lines with a set-square or compass relies on equal corresponding or alternate angles.",{"id":1029,"type":1030,"sourceIds":1031},"sources-deepen","sources",[1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042],"angles-ncert-class7-lines-angles","angles-ncert-class7-triangle","angles-euclid-i15","angles-euclid-i32","angles-mathsisfun-parallel","angles-wiki-degree","angles-wiki-radian","angles-wiki-eratosthenes","angles-mactutor-thales","angles-wiki-shulba-sutras","angles-wiki-aryabhata-sine",[1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042],"needs_review",{"generatedBy":1046,"notes":1047},"claude-code","Draft generated with Python generator scripts; every angle computed and asserted. Pending owner review.","e9ad6f9e6d061d93144391bdc937af69c7cd78bab8f286aee3c18ce6bf9fc2ca",{"logic:practice":1050,"component:angle-pairs@1":1051,"component:sort-game@1":1052,"component:match-pairs@1":1053,"source:angles-euclid-i15":1054,"source:angles-euclid-i32":1055,"source:angles-mactutor-thales":1056,"source:angles-mathsisfun-parallel":1057,"source:angles-ncert-class7-lines-angles":1058,"source:angles-ncert-class7-triangle":1059,"source:angles-wiki-aryabhata-sine":1060,"source:angles-wiki-degree":1061,"source:angles-wiki-eratosthenes":1062,"source:angles-wiki-radian":1063,"source:angles-wiki-shulba-sutras":1064},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","cbb509e9a575a1b2ef133804e3450487de14952c1df934747fe56ff5d10d847e","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","d6d698be825d6f17fd34f22e885f8d2a9c0bee7634e87a36474309b3ab7e8c82","336834da6f32f1f0c454d6e5e6a714b7a3f52963b223288563bc48c1085bbc46","32bf74caeb49dd57134ef46f7ad420fba32afd065f9005e8383b329745bcacb1","3cc0370817500ac4458d13f159f32213300db7e939f725c2dc0f989a699c8609","548a49fd09ac1b7cec7c74b038e1a87c5d3add389d6f27376ac054da17d20972","286f44f488c463767261100625e631e8edb823f1f296976d251359a4bc9677ee","343b3e5097d5424a0cf405da2c24ed7908cfd62cbf599c03ecf532bf2aa8c2fe","79b5ead855ebdf6f15ad061e87052809bfff58b83c8dfb9f7a1f592debdf62e9","0808aac8a334060e03820abfdbeb54eede2550b7058c9b8576f91c92fec37279","d3ba163093746b28c02ebab9f522ab7f88f2bf57f4b01e73547910f83286d311","baa79f2e279b961dd19774fd7f44f19b036afb3c93d8459df1c91e789e449e53",{"state":1066,"reviewer":1067,"selfReview":1068,"reviewedAt":1069,"method":1070},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597339]