[{"data":1,"prerenderedAt":1161},["ShallowReactive",2],{"layer:lines:deepen":3},{"layer":4,"contentHash":1139,"dependencyHashes":1140,"approval":1154,"releaseId":1160},{"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":1134,"reviewStatus":1135,"authoring":1136},1,"lines","en","deepen","Why it must be so: reasoning about lines","Euclid's rules, proofs, counting arguments and the puzzle of parallels","Build geometry from Euclid's postulates, prove key facts about intersecting, parallel and perpendicular lines, count with pairs, and follow the 2,000-year story of the parallel postulate from Alexandria to curved space.",[13,14,15,16,17],"State Euclid's five postulates in plain words and Playfair's form of the fifth.","Write short proofs by contradiction about intersecting and parallel lines.","Prove and use pair-counting formulas, including cases with collinear points and polygon diagonals.","Explain why the perpendicular is the shortest distance and why the perpendicular bisector is the set of equidistant points.","Describe how replacing the parallel postulate leads to spherical and hyperbolic geometry, and how the Sulba Sutras used cords to make lines.",50,{"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","≈ 50 minutes",{"label":29,"value":30},"Prior knowledge","Understand and Investigate layers",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Labs","Postulate match, plane-or-sphere sort, line spotter",{"label":38,"value":39},"Key idea","Proof from a few starting rules",[41,45,51,57,60,65,70,73,108,113,144,149,152,163,173,182,187,212,217,220,238,265,276,285,299,311,322,331,340,352,356,361,364,374,384,390,395,398,447,452,486,489,521,525,543,548,551,556,561,607,612,615,640,645,659,664,667,693,702,722,726,731,734,790,794,799,802,812,824,834,838,847,867,879,883,894,899,909,956,1091,1095,1111,1116,1121],{"id":42,"type":43,"markdown":44},"intro-deepen","prose","So far we have described lines, drawn them, counted them and measured them. This layer asks a harder question: **how do we know?** How can we be *certain* that two lines never meet twice, or that the three perpendicular bisectors of every triangle pass through one point, when nobody can check every triangle that could ever be drawn?\n\nThe answer, invented in ancient Greece and India and polished over two thousand years, is **proof**: start from a few statements everyone accepts, and reason step by step to new ones. Along the way you will meet the most famous controversy in the history of geometry, a rule about parallel lines that took more than 2,000 years to understand.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how-d","callout","observation","How to read this layer","The arguments here are short but dense. Read each one twice: once to see where it is going, once to check every step. If a step seems obvious, ask *why* it is true; that is the habit this layer trains.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Euclid's Elements: geometry from a few rules","Chapter 01","1 Euclid",{"id":58,"type":43,"markdown":59},"euclid","Around 300 BCE, in the city of Alexandria in Egypt, a teacher named **Euclid** wrote the *Elements*, thirteen books that organised all the geometry known at the time. Its big idea was not any single fact but its **structure**: a short list of definitions and starting rules, followed by hundreds of results, each proved only from the rules and the results before it.\n\nThe *Elements* opens with definitions that should feel familiar from the Understand layer. In a well-known English translation: **\"A point is that which has no part.\"** **\"A line is breadthless length.\"** **\"The ends of a line are points.\"** Notice that Euclid's \"line\" is what we now call a segment or a curve, and his \"straight line\" is closer to our segment that can be extended. Words change over 2,300 years; ideas survive.",{"id":61,"type":47,"variant":62,"title":63,"markdown":64},"nuance-definitions","nuance","Why modern geometry leaves point and line undefined","Euclid's definitions describe but do not really define: \"has no part\" and \"breadthless\" need their own explanations. In 1899 the German mathematician David Hilbert rewrote the foundations and simply took point, line and plane as **undefined**, saying only how they relate (for example, \"two points lie on exactly one line\"). He joked that you could replace the words with \"tables, chairs and beer mugs\" and every proof would still work, because proofs use only the rules, never pictures.",{"id":66,"type":53,"title":67,"eyebrow":68,"navLabel":69},"ch02","The five postulates, in plain words","Chapter 02","2 Five postulates",{"id":71,"type":43,"markdown":72},"postulates-intro","A **postulate** (or **axiom**) is a starting statement accepted without proof. Euclid chose five about geometry, plus some \"common notions\" about quantities in general, such as *things equal to the same thing are equal to each other* and *the whole is greater than the part*.",{"id":74,"type":75,"caption":76,"columns":77,"rows":82},"table-postulates","table","Euclid's five postulates, simplified",[78,79,80,81],"No.","Euclid's postulate (simplified)","What it lets you do","Ruler and compass version",[83,88,93,98,103],[84,85,86,87],"1","A straight line can be drawn from any point to any point.","Join two points with a straight line (modern versions add: only one).","Use the straight edge.",[89,90,91,92],"2","A straight line segment can be extended as far as you like.","Turn a segment into a ray or a line.","Slide the ruler along.",[94,95,96,97],"3","A circle can be drawn with any centre and any radius.","Mark equal distances.","Use the compass.",[99,100,101,102],"4","All right angles are equal.","A right angle is the same size everywhere.","Your set square works on every page.",[104,105,106,107],"5","If a line crosses two lines and the two inside angles on one side add to less than two right angles, those two lines meet on that side.","Decide when lines meet; the root of the theory of parallels.","Tilted lines eventually meet.",{"id":109,"type":47,"variant":110,"title":111,"markdown":112},"def-playfair","definition","Playfair's axiom (the modern parallel postulate)","Through a point **not** on a given line, there is **exactly one** line parallel to the given line. This version, made popular by the Scottish mathematician John Playfair in 1795, says the same thing as Euclid's fifth postulate in school geometry, and is much easier to remember.",{"id":114,"type":115,"component":116,"componentVersion":5,"config":117,"objective":142,"textAlternative":143},"lab-match-postulates","interactive","match-pairs",{"prompt":118,"mode":119,"pairs":120},"Match each postulate to what it says in plain words.","connect",[121,124,127,130,133,136,139],{"a":122,"b":123},"Postulate 1","Any two points can be joined by a straight line",{"a":125,"b":126},"Postulate 2","A segment can be extended forever",{"a":128,"b":129},"Postulate 3","A circle can have any centre and radius",{"a":131,"b":132},"Postulate 4","All right angles are equal",{"a":134,"b":135},"Postulate 5","Lines tilted towards each other eventually meet",{"a":137,"b":138},"Playfair's axiom","Exactly one parallel through a point off a line",{"a":140,"b":141},"Common notion","Things equal to the same thing are equal","Connect each of Euclid's starting rules to its plain-words meaning.","Match each rule with its meaning.\n\n- **Postulate 1:** any two points can be joined by a straight line.\n- **Postulate 2:** a segment can be extended forever (into a line).\n- **Postulate 3:** a circle can be drawn with any centre and radius.\n- **Postulate 4:** all right angles are equal.\n- **Postulate 5:** if two lines are tilted towards each other (inside angles on one side add to less than 180°), they meet on that side.\n- **Playfair's axiom:** through a point not on a line there is exactly one parallel line.\n- **A common notion:** things equal to the same thing are equal to each other.",{"id":145,"type":53,"title":146,"eyebrow":147,"navLabel":148},"ch03","First proofs about lines","Chapter 03","3 First proofs",{"id":150,"type":43,"markdown":151},"proof-style","A proof is a chain of statements, each justified by a postulate, a definition or something already proved. One powerful style is **proof by contradiction**: assume the opposite of what you want, and show that it leads to something impossible.",{"id":153,"type":154,"title":155,"problem":156,"steps":157},"proof-meet-once","worked_example","Proof: two different lines meet in at most one point","Show that two different straight lines cannot have two points in common.",[158,159,160,161,162],"Suppose, for contradiction, that lines l and m are different but share two different points, P and Q.","Then l is a line through P and Q, and so is m.","But through two different points there is **exactly one** line (postulate 1, in its modern form).","So l and m must be the same line, which contradicts \"l and m are different\".","The assumption was false: two different lines share **at most one** point. ∎",{"id":164,"type":154,"title":165,"problem":166,"steps":167},"proof-transitive","Proof: parallel to the same line means parallel to each other","Lines l, m and n lie in one plane. l ∥ m and n ∥ m, and l and n are different lines. Show l ∥ n.",[168,169,170,171,172],"Suppose, for contradiction, that l and n are **not** parallel. Being in one plane, they must meet at some point P.","P is not on m (l is parallel to m, so l has no point on m, and P is on l).","Now l and n are two **different** lines through P, and both are parallel to m.","Playfair's axiom says there is **exactly one** line through P parallel to m. Two is too many: contradiction.","So l and n never meet, and l ∥ n. ∎",{"id":174,"type":154,"title":175,"problem":176,"steps":177},"proof-two-perps","Proof: two perpendiculars to one line are parallel","In one plane, lines a and b are both perpendicular to line l, at different points. Show a ∥ b.",[178,179,180,181],"Line l crosses a and b. On one side of l, the two inside angles are 90° and 90°, which add to exactly 180°: two right angles.","Postulate 5 says lines meet on a side where the inside angles add to **less** than 180°. Here they are not less on this side.","On the other side the inside angles are also 90° + 90° = 180°, again not less.","So a and b meet on neither side: a ∥ b. ∎ (A second proof: if they met at P, the triangle formed would have two right angles, and the angle sum of a triangle is only 180°.)",{"id":183,"type":47,"variant":184,"title":185,"markdown":186},"careful-circular","careful","Don't use what you are trying to prove","A common trap is **circular reasoning**: sneaking the conclusion into the argument. \"l ∥ n because they look parallel\" or \"because they will never meet\" just restates the claim. Every step must lean on something already accepted. In the proofs above, the key steps lean on postulate 1, postulate 5 or Playfair's axiom.",{"id":188,"type":189,"itemId":190,"prompt":191,"check":192,"hints":207,"feedback":209},"prac-which-rule","practice","lines.deepen-which-rule","Which starting rule is used in the proof that two different lines meet in at most one point?",{"kind":193,"options":194,"correct":206},"choice",[195,197,200,203],{"id":196,"label":132},"a",{"id":198,"label":199},"b","Exactly one line passes through two different points",{"id":201,"label":202},"c","A circle can be drawn with any centre",{"id":204,"label":205},"d","The whole is greater than the part",[198],[208],"What would it mean if two lines shared two points?",{"correct":210,"incorrect":211},"Yes: two shared points would give two lines through two points.","The contradiction comes from having two lines through the same two points, which breaks **\"exactly one line through two points\"**.",{"id":213,"type":53,"title":214,"eyebrow":215,"navLabel":216},"ch04","Counting arguments that always work","Chapter 04","4 Counting proofs",{"id":218,"type":43,"markdown":219},"handshake","In Investigate you found that n points on a line make n × (n − 1) ÷ 2 segments. Here is why it **must** be so, for every n.\n\n**The handshake argument.** Each segment is decided by a **pair** of points. Each of the n points can pair with the (n − 1) others, giving n × (n − 1) ordered pairs. But the pair (A, B) and the pair (B, A) are the same segment, so every segment has been counted exactly twice. Divide by 2. The same argument counts the handshakes when n people all shake hands once, which is where the name comes from.",{"id":221,"type":222,"items":223},"formulas-counting","formulas",[224,227,229,232,235],{"expression":225,"caption":226},"n × (n − 1) ÷ 2","Pairs from n things: segments on a line, lines through n points (no three collinear), handshakes.",{"expression":225,"caption":228},"Most crossing points of n lines: one per pair of lines (no two parallel, no three concurrent).",{"expression":230,"caption":231},"n(n−1)÷2 − k(k−1)÷2 + 1","Lines through n points when exactly k of them are collinear and no other three are.",{"expression":233,"caption":234},"n × (n − 3) ÷ 2","Diagonals of an n-sided polygon: all segments between corners minus the n sides.",{"expression":236,"caption":237},"2 × (n − 1)","Rays on a line with n marked points, each starting at one and passing through another.",{"id":239,"type":75,"caption":240,"columns":241,"rows":245},"table-pairs","The pair count n × (n − 1) ÷ 2, and polygon diagonals n × (n − 3) ÷ 2",[242,243,244],"n","Pairs (segments, lines, crossings)","Diagonals of an n-gon",[246,248,250,252,255,259,262],[94,94,247],"0",[99,249,89],"6",[104,251,104],"10",[249,253,254],"15","9",[256,257,258],"8","28","20",[251,260,261],"45","35",[33,263,264],"66","54",{"id":266,"type":154,"title":267,"problem":268,"steps":269},"we-collinear-k","Lines through 12 points with 5 collinear","Twelve points are marked in a plane. Exactly 5 of them lie on one line, and apart from these no three are collinear. How many different lines pass through at least two of the points? How many triangles have their corners among the points?",[270,271,272,273,274,275],"If no three were collinear: 12 × 11 ÷ 2 = 66 lines.","The 5 collinear points would have given 5 × 4 ÷ 2 = 10 lines, but in fact give only **1**.","Lines = 66 − 10 + 1 = **57**.","Triangles: any 3 points make a triangle unless they are collinear. Groups of 3 from 12: 12 × 11 × 10 ÷ 6 = 220.","Groups of 3 from the 5 collinear points: 5 × 4 × 3 ÷ 6 = 10. None of these makes a triangle.","Triangles = 220 − 10 = **210**.",{"id":277,"type":154,"title":278,"problem":279,"steps":280},"we-two-parallel","Points on two parallel lines","Five points are marked on line l and four points on line m, with l ∥ m. How many segments join a point of l to a point of m? How many different lines pass through two of the nine points? How many triangles can be formed?",[281,282,283,284],"Segments across: each of 5 points pairs with each of 4: 5 × 4 = **20**.","Lines: the 20 crossing lines are all different (no three of these points are collinear across the two lines), plus l and m themselves: 20 + 2 = **22**.","Triangles: all groups of 3 from 9 points: 84. Remove groups lying on one line: 10 on l and 4 on m.","Triangles = 84 − 10 − 4 = **70**.",{"id":286,"type":189,"itemId":287,"prompt":288,"check":289,"hints":293,"feedback":296},"prac-diagonals","lines.deepen-octagon-diagonals","How many diagonals does an octagon (8 sides) have?",{"kind":290,"answer":291,"tolerance":292},"number",20,0,[294,295],"Count all segments joining two of the 8 corners, then remove the sides.","8 × 7 ÷ 2 = 28 segments in total.",{"correct":297,"incorrect":298},"Correct: 28 − 8 = 20.","All segments between corners: 8 × 7 ÷ 2 = 28. Eight of them are sides. Diagonals = 28 − 8 = **20** (or 8 × 5 ÷ 2).",{"id":300,"type":189,"itemId":301,"prompt":302,"check":303,"hints":305,"feedback":308},"prac-handshake","lines.deepen-handshake","In a class of 30, every student shakes hands once with every other student. How many handshakes happen?",{"kind":290,"answer":304,"tolerance":292},435,[306,307],"Each handshake is a pair of students.","30 × 29 counts each handshake twice.",{"correct":309,"incorrect":310},"Yes: 30 × 29 ÷ 2 = 435.","Pairs from 30: 30 × 29 ÷ 2 = **435**. It is the same count as segments on a line with 30 marked points.",{"id":312,"type":154,"title":313,"problem":314,"steps":315},"proof-no-two","Proof: four lines never cross at exactly two points","In Investigate a computer check found that four lines in a plane can cross at 0, 1, 3, 4, 5 or 6 points, never exactly 2. Prove it.",[316,317,318,319,320,321],"Group the four lines by direction: lines in the same group are parallel. The possible group sizes are 4; 3 + 1; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1.","**4:** all parallel, 0 crossings. **3 + 1:** the single line cuts the 3 parallels at 3 different points: exactly 3.","**2 + 2:** each line of one pair cuts both lines of the other pair, giving 4 different points (like a hashtag #).","**2 + 1 + 1:** call the parallels a and b, the others c and d. c cuts a and b at 2 different points. d also cuts a and b at 2 points, and d can share at most one point with c. So at least 3 different points.","**1 + 1 + 1 + 1:** if all four pass through one point, 1 crossing. If not, some three are not concurrent, and three such lines already make 3 different points.","In no case is the count exactly 2. ∎",{"id":323,"type":154,"title":324,"problem":325,"steps":326},"we-k-parallel","Crossing points when some lines are parallel","Seven lines are drawn in a plane. Exactly three of them are parallel to each other, and otherwise no two are parallel and no three pass through one point. How many intersection points are there?",[327,328,329,330],"If no two were parallel: 7 × 6 ÷ 2 = 21 crossing points.","The 3 parallel lines make 3 × 2 ÷ 2 = 3 pairs that never meet.","Crossings = 21 − 3 = **18**.","Because no three lines are concurrent, no two of these crossings coincide, so none of the remaining points needs to be merged.",{"id":332,"type":154,"title":333,"problem":334,"steps":335},"we-k-concurrent","Crossing points when some lines are concurrent","Six lines are drawn. Exactly three of them pass through one point P; otherwise no two are parallel and no three meet at a point. How many intersection points are there?",[336,337,338,339],"In general position: 6 × 5 ÷ 2 = 15 points.","The 3 concurrent lines would have made 3 separate crossings, but they all happen at P: 3 points collapse into 1.","Points = 15 − 3 + 1 = **13**.","Compare the formula for lines through points with k collinear: the same \"subtract the pairs, add one back\" idea, with points and lines swapping roles.",{"id":341,"type":189,"itemId":342,"prompt":343,"check":344,"hints":346,"feedback":349},"prac-k-parallel","lines.deepen-k-parallel","Eight lines are drawn in a plane. Exactly four of them are parallel to one another; otherwise no two are parallel and no three are concurrent. How many intersection points are there?",{"kind":290,"answer":345,"tolerance":292},22,[347,348],"Start from 8 × 7 ÷ 2.","Remove the pairs among the 4 parallel lines.",{"correct":350,"incorrect":351},"Right: 28 − 6 = 22.","All pairs: 8 × 7 ÷ 2 = 28. Parallel pairs that never meet: 4 × 3 ÷ 2 = 6. Intersections = **22**.",{"id":353,"type":47,"variant":62,"title":354,"markdown":355},"nuance-duality","Points and lines swap places","Notice the mirror between two facts: **two points determine one line**, and **two (non-parallel) lines determine one point**. Many counting results come in matching pairs: lines through n points and crossing points of n lines both use n × (n − 1) ÷ 2, and \"k collinear points\" matches \"k concurrent lines\". In projective geometry, where parallel lines meet at infinity, this swap, called **duality**, becomes an exact rule: swap the words point and line in any theorem and you get another true theorem.",{"id":357,"type":53,"title":358,"eyebrow":359,"navLabel":360},"ch05","Why the perpendicular is the shortest","Chapter 05","5 Shortest distance",{"id":362,"type":43,"markdown":363},"shortest-intro","In Understand we measured the gap between parallel lines along a perpendicular. Why a perpendicular? Because the perpendicular segment from a point to a line is the **shortest** of all segments from that point to the line. Here is a proof using a fold, with no formulas at all.",{"id":365,"type":154,"title":366,"problem":367,"steps":368},"proof-shortest","Proof: the perpendicular is the shortest path to a line","P is a point not on line l. F is the point of l where the perpendicular from P meets it. Q is any other point of l. Show that PF is shorter than PQ.",[369,370,371,372,373],"Fold the page along l. P lands on a point P′ on the other side. Because the fold is along l, PF = P′F and PQ = P′Q.","PF meets l at 90°, so folding along l sends ray FP to the ray from F pointing straight out on the other side. So P, F and P′ are collinear and F is the midpoint of PP′.","Q is not on segment PP′ (the only point of l on that segment is F). So the path P → Q → P′ is a detour: PQ + QP′ > PP′.","Replace with equal lengths: PQ + PQ > PF + FP′ = PF + PF. So 2 × PQ > 2 × PF.","Therefore PQ > PF: the perpendicular is the shortest. ∎",{"id":375,"type":154,"title":376,"problem":377,"steps":378},"proof-perp-bisector","Proof: the perpendicular bisector is exactly the set of equidistant points","Show that (1) every point on the perpendicular bisector of AB is equally far from A and B, and (2) every point equally far from A and B lies on it.",[379,380,381,382,383],"(1) Let P lie on the perpendicular bisector, which meets AB at its midpoint M at 90°. Fold along the bisector: A lands on B (AM = MB and the angles at M match), and P stays put. So PA = PB.","(2) Now let Q be any point with QA = QB. Let M be the midpoint of AB and join QM.","Triangles QMA and QMB have QA = QB, MA = MB and share QM, so they are congruent (side–side–side).","So angle QMA = angle QMB. Together they make a straight angle of 180°, so each is 90°. QM is perpendicular to AB through its midpoint: Q is on the bisector. ∎","This is why the three perpendicular bisectors of a triangle meet at one point (the circumcentre): it is the point equally far from all three corners.",{"id":385,"type":386,"conceptId":387,"relation":388,"explanation":389},"conn-constructing-d","connection","constructing-angles","applied_in","The compass construction of a perpendicular bisector works because both arc crossings are equidistant from A and B.",{"id":391,"type":53,"title":392,"eyebrow":393,"navLabel":394},"ch06","The 2,000-year puzzle of parallels","Chapter 06","6 Parallel puzzle",{"id":396,"type":43,"markdown":397},"fifth-problem","The first four postulates are short and obviously true. The fifth is long, complicated and talks about lines meeting \"eventually\", far away where nobody can check. For centuries mathematicians felt it did not belong among the starting rules and tried to **prove** it from the other four.\n\nEvery attempt failed. Most accidentally assumed something equivalent to the fifth postulate along the way, such as \"the angles of a triangle add to 180°\", \"rectangles exist\" or \"parallel lines stay the same distance apart\". Each of those turns out to be just the fifth postulate in disguise.",{"id":399,"type":400,"title":401,"items":402},"timeline-parallel","timeline","The parallel postulate through history",[403,407,411,415,419,423,427,431,435,439,443],{"time":404,"title":405,"text":406},"c. 300 BCE","Euclid","Writes the *Elements*, and avoids using the fifth postulate for his first 28 results.",{"time":408,"title":409,"text":410},"5th c. CE","Proclus","Greek commentator argues the fifth postulate should be a theorem and offers a flawed proof.",{"time":412,"title":413,"text":414},"c. 1000","Ibn al-Haytham","In Cairo, argues by contradiction and brings motion into geometry; the quadrilateral with three right angles he uses, later named after Lambert, already assumes what he set out to prove.",{"time":416,"title":417,"text":418},"c. 1077","Omar Khayyam","The Persian poet-mathematician studies quadrilaterals with two right angles and equal sides.",{"time":420,"title":421,"text":422},"13th c.","Nasir al-Din al-Tusi","In Maragha, Persia, writes a critique of earlier attempts that later reached Europe.",{"time":424,"title":425,"text":426},"1733","Saccheri","Italian priest tries to disprove every alternative, and unknowingly proves results of a new geometry.",{"time":428,"title":429,"text":430},"1795","Playfair","Popularises the neat form: exactly one parallel through a point off a line.",{"time":432,"title":433,"text":434},"1820s","Gauss","Privately convinced a consistent geometry without the fifth postulate exists, but does not publish.",{"time":436,"title":437,"text":438},"1829–1832","Lobachevsky, Bolyai","Independently publish hyperbolic geometry: many parallels through one point.",{"time":440,"title":441,"text":442},"1854","Riemann","Describes geometries of curved space, including one with no parallels at all.",{"time":444,"title":445,"text":446},"1915","Einstein","General relativity describes gravity as curved space-time, using Riemann's geometry.",{"id":448,"type":47,"variant":449,"title":450,"markdown":451},"aha-independent","aha","The postulate cannot be proved, and that is the discovery","In the 1800s mathematicians realised why every attempt failed: you can **replace** the fifth postulate with a different rule and still get a perfectly consistent geometry. In **hyperbolic** geometry, many lines through a point miss a given line. In **spherical** (elliptic) geometry, there are no parallels at all. Euclid was right to make it a postulate: it is a genuine choice about what kind of space you are describing.",{"id":453,"type":75,"caption":454,"columns":455,"rows":460},"table-geometries","Three geometries compared",[456,457,458,459],"Feature","Euclidean (flat)","Spherical (on a globe)","Hyperbolic (saddle-like)",[461,466,471,476,481],[462,463,464,465],"Parallels through a point off a line","Exactly one","None","Infinitely many",[467,468,469,470],"Angles of a triangle add to","Exactly 180°","More than 180°","Less than 180°",[472,473,474,475],"\"Straight lines\" are","Ordinary lines","Great circles, like the equator","Curves that are shortest paths on the saddle",[477,478,479,480],"Two lines perpendicular to a third","Parallel","Meet (at the poles)","Never meet, and spread apart",[482,483,484,485],"Everyday picture","A page, a playground","The Earth's surface","A lettuce leaf, some coral",{"id":487,"type":43,"markdown":488},"disguises-intro","Here are some statements that turned out to be the parallel postulate in disguise. Each one can be proved from the other four postulates **plus** the parallel postulate, and each one, if assumed, is enough to prove the parallel postulate back.",{"id":490,"type":75,"caption":491,"columns":492,"rows":496},"table-disguises","Statements equivalent to the parallel postulate",[493,494,495],"Statement","Who used it","Why it seemed obvious",[497,501,505,509,513,517],[498,499,500],"Through a point off a line there is exactly one parallel","Playfair (1795), Proclus earlier","Try drawing a second one",[502,503,504],"The angles of every triangle add to 180°","Legendre (c. 1800) tried to prove it","It works for every triangle we measure on paper",[506,507,508],"Rectangles exist","Many builders, without thinking","Rooms, pages and courts are rectangles",[510,511,512],"Lines that are parallel stay the same distance apart","Clavius (1574), later Lambert","Railway rails do",[514,515,516],"Similar triangles of different sizes exist","John Wallis (1663)","Scale drawings and maps work",[518,519,520],"Pythagoras' theorem holds for every right triangle","Implicit in the Sulba Sutras and Euclid","It checks out for 3, 4, 5",{"id":522,"type":47,"variant":449,"title":523,"markdown":524},"aha-legendre","Why Legendre's proof never worked","Adrien-Marie Legendre spent decades trying to prove that triangle angles add to 180° from the first four postulates alone. He managed to prove that they add to **at most** 180°, but never exactly 180°. The reason is now clear: on a hyperbolic surface the first four postulates hold and triangles add to **less** than 180°. No proof from the first four can rule that world out.",{"id":526,"type":189,"itemId":527,"prompt":528,"check":529,"hints":538,"feedback":540},"prac-disguise","lines.deepen-disguise","Which statement is **not** equivalent to the parallel postulate?",{"kind":193,"options":530,"correct":537},[531,533,534,535],{"id":196,"label":532},"Triangle angles add to 180°",{"id":198,"label":506},{"id":201,"label":132},{"id":204,"label":536},"Parallel lines stay the same distance apart",[201],[539],"One of these is itself one of Euclid's other postulates.",{"correct":541,"incorrect":542},"Yes: \"all right angles are equal\" is postulate 4, which holds on spheres and saddles too.","\"All right angles are equal\" is Euclid's **fourth** postulate. It is true in spherical and hyperbolic geometry as well, so it cannot be equivalent to the fifth. The others all fail on a sphere or a saddle.",{"id":544,"type":53,"title":545,"eyebrow":546,"navLabel":547},"ch07","Lines on a ball: the Earth's surface","Chapter 07","7 Lines on a sphere",{"id":549,"type":43,"markdown":550},"sphere","On the surface of a ball, what is the \"straightest\" path between two points? Stretch a rubber band tightly between two points on a football: it settles along a **great circle**, a circle whose centre is the centre of the ball. The equator is a great circle; so is every line of longitude together with its partner on the far side.\n\nGreat circles behave like lines in some ways: they are the shortest paths, and they never bend left or right as you walk along them. But **any two great circles meet**, and in **two** points, exactly opposite each other. Lines of longitude are all perpendicular to the equator, yet they all meet at the North and South Poles. So on a sphere there are no parallel \"lines\" at all.",{"id":552,"type":47,"variant":553,"title":554,"markdown":555},"model-sphere","model_limit","Where the flat picture breaks","The flat-plane rules in this topic are a model. They work for anything you can draw on paper, lay out on a field or build as a house. Over hundreds of kilometres the Earth's curve matters: \"two points fix one line\" fails for opposite points such as the North and South Poles (every line of longitude passes through both), and triangles on the globe have angle sums bigger than 180°. Flights from Delhi to San Francisco follow great circles, which look curved on a flat map.",{"id":557,"type":47,"variant":558,"title":559,"markdown":560},"mis-latitude","misconception","“Lines of latitude are great circles too”","Only the equator is. The Tropic of Cancer, which passes through eight Indian states, is a smaller circle, and walking along it you would have to keep turning slightly towards the North Pole. Lines of latitude never meet each other, so they behave like parallels, but they are not the \"straight lines\" of the sphere.",{"id":562,"type":115,"component":563,"componentVersion":5,"config":564,"objective":605,"textAlternative":606},"lab-sort-geometry","sort-game",{"prompt":565,"bins":566,"items":573,"seconds":292},"Is each statement true on a flat plane only, or true on both a plane and a sphere (using great circles as lines)?",[567,570],{"id":568,"label":569},"plane","Flat plane only",{"id":571,"label":572},"both","Plane and sphere",[574,578,582,586,590,594,598,601],{"id":575,"label":576,"bin":568,"why":577},"parallel-exist","There are lines that never meet.","On a sphere every two great circles meet.",{"id":579,"label":580,"bin":568,"why":581},"one-line","Through any two points there is exactly one line.","Opposite points on a sphere have endlessly many great circles through them.",{"id":583,"label":584,"bin":568,"why":585},"meet-once","Two different lines meet at most once.","Great circles always meet twice, at opposite points.",{"id":587,"label":588,"bin":568,"why":589},"sum180","The angles of a triangle add to exactly 180°.","A globe triangle with corners at the North Pole and two points on the equator can have three right angles.",{"id":591,"label":592,"bin":571,"why":593},"shortest","A \"line\" is the shortest path between two nearby points.","Great circles are shortest paths on the sphere; straight lines are on the plane.",{"id":595,"label":596,"bin":571,"why":597},"circle","You can draw a circle with any centre and a small radius.","Postulate 3 works on a sphere for small circles.",{"id":599,"label":100,"bin":571,"why":600},"right-angles","A right angle is the same everywhere on both surfaces.",{"id":602,"label":603,"bin":568,"why":604},"perp-parallel","Two lines perpendicular to the same line never meet.","Lines of longitude are all perpendicular to the equator, and all meet at the poles.","Sort statements about lines by whether they hold only on a flat plane or also on a sphere.","Eight statements to sort, using great circles as the \"lines\" of a sphere.\n\n**True on a flat plane only:** lines that never meet exist; exactly one line through any two points (fails for opposite points on a sphere); two lines meet at most once (great circles meet twice); triangle angles add to exactly 180° (a globe triangle can have three right angles); two perpendiculars to the same line never meet (longitudes meet at the poles).\n\n**True on both:** lines are shortest paths between nearby points; circles can be drawn with any centre and a small radius; all right angles are equal.\n\nThe first four postulates mostly survive on a sphere; the parallel postulate does not.",{"id":608,"type":53,"title":609,"eyebrow":610,"navLabel":611},"ch08","Ropes and pegs: straight lines in ancient India","Chapter 08","8 Sulba Sutras",{"id":613,"type":43,"markdown":614},"sulba","Long before Euclid, priests in India needed exact geometry for building fire altars (vedi) of precise shapes and sizes. Their rules survive in the **Sulba Sutras**, texts of the first millennium BCE attached to the names of Baudhayana, Apastamba, Katyayana and Manava. Baudhayana's is the oldest, usually dated to roughly 800–500 BCE; Katyayana's is the latest. *Sulba* (or *shulba*) means **cord** or rope.\n\nThe tools were a stretched cord (rajju) and pegs (sanku). A cord pulled tight between two pegs gives a straight segment, exactly the idea of \"two points fix a line\". Marks on the cord allowed equal lengths to be copied, like a divider.\n\nOnly the latest of these texts, the **Katyayana Sulvasutra**, actually writes down a rule for finding the directions; the earlier ones start from an east–west line without saying how to get it.",{"id":616,"type":617,"title":618,"items":619},"steps-east-west","steps","Fixing the east–west line with a shadow stick (Katyayana's rule)",[620,624,628,632,636],{"title":621,"tag":622,"text":623},"Set a gnomon","a straight upright stick","Fix a stick vertically on level ground and draw a circle around its foot with a cord.",{"title":625,"tag":626,"text":627},"Morning mark","shadow touches the circle","In the morning, mark the point where the tip of the shadow touches the circle.",{"title":629,"tag":630,"text":631},"Afternoon mark","touches again","In the afternoon, mark where the shadow tip touches the circle again.",{"title":633,"tag":634,"text":635},"Join the marks","east–west line","The segment joining the two marks runs east–west.",{"title":637,"tag":638,"text":639},"Perpendicular","north–south","Its perpendicular bisector through the stick runs north–south, giving the altar's axis.",{"id":641,"type":47,"variant":642,"title":643,"markdown":644},"example-cord","example","A right angle from a knotted cord","Baudhayana's Sulba Sutra gives sets of cord lengths that make right angles, such as 3, 4 and 5 units, 5, 12 and 13, and 15, 36 and 39. Peg a cord of 12 equal parts into a triangle with sides 3, 4 and 5 and the angle between the 3 and the 4 is a right angle, so those two sides are perpendicular. Builders in India still sometimes square a foundation this way. The general rule behind it is what we now call Pythagoras' theorem.",{"id":646,"type":647,"prompt":648,"options":649,"explanation":658},"predict-3-4-5","prediction","A builder marks 60 cm along one wall from a corner and 80 cm along the other. For the walls to be perpendicular, how far apart should the two marks be?",[650,652,654,656],{"id":196,"label":651},"70 cm",{"id":198,"label":653},"100 cm",{"id":201,"label":655},"140 cm",{"id":204,"label":657},"120 cm","**100 cm.** 60, 80, 100 is the 3, 4, 5 pattern multiplied by 20. If the diagonal between the marks is exactly 100 cm, the corner is a right angle. If it is more, the corner is too wide; if less, too narrow.",{"id":660,"type":53,"title":661,"eyebrow":662,"navLabel":663},"ch09","Lines on graph paper","Chapter 09","9 Graph paper",{"id":665,"type":43,"markdown":666},"graph-intro","Draw a line on squared graph paper and walk along it from one crossing point of the grid to the next. You might go **2 squares across and 1 square up**, again and again. That repeated step describes the line's **steepness**, often called its **slope** or **gradient**: here 1 up for every 2 across, a slope of 1 ÷ 2.\n\nThis gives quick tests that work without a protractor. **Parallel lines** have the **same step** (the same slope). **Perpendicular lines** have steps that are the first one **turned through a quarter turn**: (2 across, 1 up) turned becomes (1 back, 2 up). And three points are **collinear** exactly when the steps between them are in the same ratio.",{"id":668,"type":75,"caption":669,"columns":670,"rows":675},"table-slopes","Steps and slopes on graph paper",[671,672,673,674],"Line's step","Slope (up ÷ across)","A parallel line's step","A perpendicular line's step",[676,681,685,689],[677,678,679,680],"2 across, 1 up","1 ÷ 2","4 across, 2 up (same ratio)","1 back, 2 up: slope −2",[682,84,683,684],"1 across, 1 up","3 across, 3 up","1 back, 1 up: slope −1",[686,247,687,688],"3 across, 0 up (horizontal)","Any horizontal line","Vertical lines",[690,94,691,692],"1 across, 3 up","2 across, 6 up","3 back, 1 up: slope −1 ÷ 3",{"id":694,"type":154,"title":695,"problem":696,"steps":697},"we-graph-collinear","Are the points collinear?","On graph paper, A is at (1, 1), B at (3, 2) and C at (7, 4), where the first number counts squares across and the second counts squares up. Are A, B and C collinear? What about P (0, 0), Q (2, 3) and R (4, 7)?",[698,699,700,701],"From A to B the step is 2 across, 1 up. From B to C it is 4 across, 2 up.","4 across, 2 up is just (2 across, 1 up) done twice: the same direction. So **A, B, C are collinear**.","From P to Q: 2 across, 3 up. From Q to R: 2 across, 4 up.","Same number across but a different number up, so the direction has changed: **P, Q, R are not collinear**.",{"id":703,"type":189,"itemId":704,"prompt":705,"check":706,"hints":716,"feedback":719},"prac-graph-perp","lines.deepen-graph-perp","Line l goes 3 squares across for every 1 square up. Which step gives a line **perpendicular** to l?",{"kind":193,"options":707,"correct":715},[708,710,711,713],{"id":196,"label":709},"3 across, 1 up",{"id":198,"label":690},{"id":201,"label":712},"1 back, 3 up",{"id":204,"label":714},"3 back, 1 up",[201],[717,718],"Turn the step (3 across, 1 up) through a quarter turn anticlockwise.","Across becomes up, and up becomes back.",{"correct":720,"incorrect":721},"Yes: a quarter turn sends (3 across, 1 up) to (1 back, 3 up).","A quarter turn anticlockwise turns 3 across into 3 up, and 1 up into 1 back, giving **(1 back, 3 up)**. Its slope is −3, and 1 ÷ 3 × (−3) = −1: slopes of perpendicular lines multiply to −1.",{"id":723,"type":47,"variant":62,"title":724,"markdown":725},"nuance-slopes","The −1 rule","In later classes you will write lines as equations such as y = 2x + 1, where 2 is the slope. Then: parallel lines have **equal** slopes, and perpendicular lines have slopes that **multiply to −1** (like 2 and −1 ÷ 2), except for horizontal and vertical lines, whose slopes are 0 and undefined. René Descartes' idea of describing points by numbers, published in 1637, is what turned geometry questions about lines into algebra.",{"id":727,"type":53,"title":728,"eyebrow":729,"navLabel":730},"ch10","Edge cases and conventions","Chapter 10","10 Edge cases",{"id":732,"type":43,"markdown":733},"edge-intro","Careful thinkers ask what happens at the edges of a definition. The answers are sometimes a matter of **convention**: an agreement about how to use a word, rather than a fact to be discovered.",{"id":735,"type":736,"title":737,"prompt":738,"options":739},"explorer-edge","explorer","Five questions at the edge of the definitions","Pick a question to see how mathematicians handle it.",[740,750,760,770,781],{"id":741,"label":742,"chain":743,"badge":746,"note":749},"self-parallel","Is a line parallel to itself?",[744,745],"Never meets? No, it shares every point","Same direction? Yes",{"text":747,"tone":748},"Depends on the book","no","School books in India define parallel lines as lines that never meet, so a line is not parallel to itself. Many university books define parallel as having the same direction, so every line is parallel to itself, which makes rules like \"parallel to the same line means parallel to each other\" work without exceptions. Both are fine; just be consistent.",{"id":751,"label":752,"chain":753,"badge":757,"note":759},"zero-seg","A segment of length 0?",[754,755,756],"Segment AB","Let B slide onto A","Only one point left",{"text":758,"tone":748},"Usually excluded","If A and B are the same point, \"segment AA\" is just a point. Most books require the end points of a segment to be different. Allowing it is called a degenerate case, like a triangle squashed flat.",{"id":761,"label":762,"chain":763,"badge":767,"note":769},"segments-cross","Do the segments meet?",[764,765,766],"Lines AB and CD cross at P","Is P on both segments?","Not necessarily",{"text":768,"tone":748},"No","Two segments on non-parallel lines may miss each other completely, because the crossing point of their lines lies beyond the end of one of them. Parallel is about lines; whether segments touch is a separate question.",{"id":771,"label":772,"chain":773,"badge":777,"note":780},"ray-endpoint","Does a ray include its start?",[774,775,776],"Ray AB","Starts at A","A is on the ray",{"text":778,"tone":779},"Yes","yes","The end point belongs to the ray. That is why two opposite rays QP and QR share exactly one point, Q, and together make the whole line.",{"id":782,"label":783,"chain":784,"badge":787,"note":789},"collinear-two","Are any two points collinear?",[785,786],"Take A and B","Line AB passes through both",{"text":788,"tone":779},"Always","Yes: two points always lie on a line. That is why \"collinear\" is only interesting for three or more points, and why three points in general position form a triangle.",{"id":791,"type":47,"variant":62,"title":792,"markdown":793},"nuance-segment-vs-line","Segments versus lines in questions","When a question asks whether two **segments** intersect, extend nothing; check whether they actually touch. When it asks about two **lines**, always imagine them extended. Many exam mistakes come from mixing the two.",{"id":795,"type":53,"title":796,"eyebrow":797,"navLabel":798},"ch11","More counting proofs and a hexagon puzzle","Chapter 11","11 Hexagon puzzle",{"id":800,"type":43,"markdown":801},"gauss-intro","In Investigate the segment count appeared as 1 + 2 + 3 + … + (n − 1). In this chapter's first proof it appeared as n × (n − 1) ÷ 2. Why are these equal? The answer uses a trick often told about the young Carl Friedrich Gauss.",{"id":803,"type":154,"title":804,"problem":805,"steps":806},"proof-gauss","Proof: 1 + 2 + … + (n − 1) = n × (n − 1) ÷ 2","Show that the sum of the whole numbers from 1 to n − 1 always equals n × (n − 1) ÷ 2.",[807,808,809,810,811],"Write the sum forwards: S = 1 + 2 + 3 + … + (n − 1).","Write it backwards underneath: S = (n − 1) + (n − 2) + … + 1.","Add the two rows column by column. Each column adds to n: 1 + (n − 1) = n, 2 + (n − 2) = n, and so on.","There are (n − 1) columns, so 2 × S = n × (n − 1).","Therefore S = n × (n − 1) ÷ 2. ∎ Two different ways of counting the same segments must give the same answer, and here they do.",{"id":813,"type":189,"itemId":814,"prompt":815,"check":816,"hints":818,"feedback":821},"prac-sum-99","lines.deepen-sum-99","Use the pairing trick: what is 1 + 2 + 3 + … + 99?",{"kind":290,"answer":817,"tolerance":292},4950,[819,820],"Here n − 1 = 99, so n = 100.","n × (n − 1) ÷ 2.",{"correct":822,"incorrect":823},"Yes: 100 × 99 ÷ 2 = 4,950.","With n = 100: 100 × 99 ÷ 2 = **4,950**. It is also the number of segments on a line with 100 marked points.",{"id":825,"type":154,"title":826,"problem":827,"steps":828},"we-hexagon","Parallel pairs in a regular hexagon","Draw a regular hexagon and all of its diagonals, so every pair of corners is joined. How many segments are there? How many pairs of them are parallel, and how many pairs are perpendicular?",[829,830,831,832,833],"Segments: 6 × 5 ÷ 2 = 15 (6 sides and 9 diagonals).","Sort them by direction. A computer check of all 15 finds 6 directions: three directions contain 3 segments each (a side, the opposite side and the long diagonal between them) and three contain 2 segments each (two short diagonals).","Parallel pairs: 3 groups × 3 pairs inside each + 3 groups × 1 pair = 9 + 3 = **12**.","Each 3-segment direction is at right angles to one 2-segment direction, giving 3 × 2 = 6 perpendicular pairs per matching, and there are 3 matchings: **18** perpendicular pairs.","So of the 105 pairs of segments, 12 are parallel, 18 perpendicular and the rest meet (or would meet) at other angles.",{"id":835,"type":47,"variant":558,"title":836,"markdown":837},"mis-drawing-proof","\\u201cI drew it and it worked, so it's proved\\u201d","A drawing shows **one** case. The perpendicular bisectors of *your* triangle may meet at one point, but that does not show they meet for *every* triangle. Students also often \\\"prove\\\" two lines are parallel because they look parallel. In this layer, a statement counts as proved only when every step is backed by a postulate, a definition or an earlier result.",{"id":839,"type":154,"title":840,"problem":841,"steps":842},"we-proof-perp-parallel","Proof: a line perpendicular to one of two parallels is perpendicular to the other","In a plane, l ∥ m, and line t is perpendicular to l at point A. Show that t is perpendicular to m.",[843,844,845,846],"t meets l, and l ∥ m. If t were parallel to m, then through A there would be two lines (l and t) parallel to m, which Playfair's axiom forbids. So t meets m at some point B.","Now t is a transversal of the parallel lines l and m. By the parallel postulate the two inside angles on one side must add to exactly 180° (if less, the lines would meet on that side; if more, they would meet on the other side).","The inside angle at A is 90°, so the inside angle at B is 180° − 90° = 90°.","So t ⊥ m. ∎ This is why the rungs of a ladder can be perpendicular to both side rails at once.",{"id":848,"type":189,"itemId":849,"prompt":850,"check":851,"hints":862,"feedback":864},"prac-rungs","lines.deepen-rungs","The two side rails of a ladder are parallel. A rung is perpendicular to the left rail. What must be true?",{"kind":193,"options":852,"correct":861},[853,855,857,859],{"id":196,"label":854},"The rung is perpendicular to the right rail too",{"id":198,"label":856},"The rung is parallel to the right rail",{"id":201,"label":858},"Nothing can be said",{"id":204,"label":860},"The rung and right rail are skew",[196],[863],"Use the result just proved.",{"correct":865,"incorrect":866},"Yes: a line perpendicular to one of two parallels is perpendicular to the other.","By the proof above, a line perpendicular to one of two parallel lines (in the same plane) is **perpendicular to the other** as well.",{"id":868,"type":189,"itemId":869,"prompt":870,"check":871,"hints":873,"feedback":876},"prac-concurrent-points","lines.deepen-concurrent-points","Nine lines are drawn. Exactly four of them pass through one point, and otherwise no two are parallel and no three are concurrent. How many intersection points are there?",{"kind":290,"answer":872,"tolerance":292},31,[874,875],"Start from 9 × 8 ÷ 2.","The 4 concurrent lines make 6 crossings that all collapse into 1.",{"correct":877,"incorrect":878},"Right: 36 − 6 + 1 = 31.","General position: 36. The 4 concurrent lines' 6 crossings become 1 point. Total = 36 − 6 + 1 = **31**.",{"id":880,"type":47,"variant":62,"title":881,"markdown":882},"nuance-gauss","About the Gauss story","The story says that the schoolboy Gauss, set the task of adding 1 to 100, answered 5,050 in moments by pairing the numbers. The details have grown with each retelling, and historians are not sure exactly what happened. The mathematics, however, is completely reliable: the pairing argument works for every n, which is what makes it a proof.",{"id":884,"type":189,"itemId":885,"prompt":886,"check":887,"hints":889,"feedback":891},"prac-hexagon-seg","lines.deepen-hexagon-diag","A regular hexagon has all its sides and diagonals drawn. Of the 15 segments, how many are **diagonals**?",{"kind":290,"answer":888,"tolerance":292},9,[890],"All segments between corners minus the sides.",{"correct":892,"incorrect":893},"Right: 15 − 6 = 9.","15 segments in total minus 6 sides = **9** diagonals, which matches n × (n − 3) ÷ 2 = 6 × 3 ÷ 2 = 9.",{"id":895,"type":53,"title":896,"eyebrow":897,"navLabel":898},"ch12","Summary and self-test","Chapter 12","12 Wrap-up",{"id":900,"type":115,"component":901,"componentVersion":5,"config":902,"objective":907,"textAlternative":908},"lab-spot-deep","line-spotter",{"modes":903,"rounds":906},[904,905],"kinds","pairs",16,"Play a long mixed round of line\u002Fray\u002Fsegment and parallel\u002Fperpendicular\u002Fintersecting spotting, aiming for a perfect streak.","A long mixed round of sixteen questions, for speed and accuracy.\n\n**Kinds:** read the ends. Two end points: segment. One end point and an arrow: ray. Two arrows: line.\n\n**Pairs:** parallel if the perpendicular gap is constant and they never meet; perpendicular if they meet at 90°; intersecting if they meet at any other angle.\n\nUse this layer's reasoning while you play: two lines meeting once can never meet again; if one angle at a crossing is 90°, all four are; and a ray is always named from its end point.",{"id":910,"type":911,"title":912,"terms":913},"glossary-deepen","glossary","Vocabulary for reasoning about lines",[914,918,922,926,929,932,936,940,944,948,952],{"term":915,"meaning":916,"example":917},"Postulate (axiom)","A starting statement accepted without proof.","Two points fix exactly one line",{"term":919,"meaning":920,"example":921},"Theorem","A statement proved from postulates and earlier theorems.","Two lines meet at most once",{"term":923,"meaning":924,"example":925},"Proof by contradiction","Assume the opposite of what you want, and show it leads to something impossible.","Assuming two lines share two points",{"term":137,"meaning":927,"example":928},"Through a point not on a line there is exactly one parallel to it.","The modern parallel postulate",{"term":930,"meaning":931,"example":532},"Euclidean geometry","The geometry of a flat plane, built on Euclid's five postulates.",{"term":933,"meaning":934,"example":935},"Non-Euclidean geometry","A consistent geometry where the parallel postulate is replaced.","Spherical and hyperbolic geometry",{"term":937,"meaning":938,"example":939},"Great circle","A circle on a sphere whose centre is the sphere's centre; the sphere's version of a line.","The equator",{"term":941,"meaning":942,"example":943},"Circular reasoning","An argument that assumes what it is trying to prove.","\"They are parallel because they never meet\"",{"term":945,"meaning":946,"example":947},"Degenerate case","A squashed or collapsed version of a figure.","A segment of length 0",{"term":949,"meaning":950,"example":951},"Diagonal","A segment joining two corners of a polygon that are not next to each other.","A hexagon has 9",{"term":953,"meaning":954,"example":955},"Sulba Sutras","Ancient Indian texts on geometry for altar building using cords and pegs.","Fixing an east–west line",{"id":957,"type":958,"title":959,"questions":960},"quiz-deepen","quiz","Reasoning about lines",[961,973,983,996,1008,1017,1030,1043,1056,1069,1078],{"itemId":962,"prompt":963,"options":964,"correct":198,"why":972},"lines.deepen-q-k-parallel","7 lines, exactly 3 parallel, otherwise general. How many intersection points?",[965,967,969,971],{"id":196,"label":966},"21",{"id":198,"label":968},"18",{"id":201,"label":970},"19",{"id":204,"label":253},"21 pairs minus the 3 parallel pairs that never meet.",{"itemId":974,"prompt":975,"options":976,"correct":196,"why":982},"lines.deepen-q-euclid","About when did Euclid write the Elements?",[977,978,980,981],{"id":196,"label":404},{"id":198,"label":979},"c. 1000 CE",{"id":201,"label":428},{"id":204,"label":440},"Euclid worked in Alexandria around 300 BCE.",{"itemId":984,"prompt":985,"options":986,"correct":198,"why":995},"lines.deepen-q-playfair","Playfair's axiom says that through a point not on a line there is…",[987,989,991,993],{"id":196,"label":988},"No parallel",{"id":198,"label":990},"Exactly one parallel",{"id":201,"label":992},"Two parallels",{"id":204,"label":994},"Infinitely many parallels","Exactly one parallel: the flat-plane version of the fifth postulate.",{"itemId":997,"prompt":998,"options":999,"correct":198,"why":1007},"lines.deepen-q-lines-12","12 points, exactly 5 collinear, no other three collinear. How many lines?",[1000,1002,1004,1005],{"id":196,"label":1001},"56",{"id":198,"label":1003},"57",{"id":201,"label":263},{"id":204,"label":1006},"62","66 − 10 + 1 = 57.",{"itemId":1009,"prompt":1010,"options":1011,"correct":198,"why":1016},"lines.deepen-q-hexagon","How many diagonals does a hexagon have?",[1012,1013,1014,1015],{"id":196,"label":249},{"id":198,"label":254},{"id":201,"label":33},{"id":204,"label":253},"6 × 3 ÷ 2 = 9, or 15 segments minus 6 sides.",{"itemId":1018,"prompt":1019,"options":1020,"correct":201,"why":1029},"lines.deepen-q-sphere","On a sphere, two different great circles meet at…",[1021,1023,1025,1027],{"id":196,"label":1022},"No points",{"id":198,"label":1024},"Exactly one point",{"id":201,"label":1026},"Exactly two opposite points",{"id":204,"label":1028},"Infinitely many points","Any two great circles cross at two points exactly opposite each other.",{"itemId":1031,"prompt":1032,"options":1033,"correct":198,"why":1042},"lines.deepen-q-shortest","The shortest segment from a point to a line is the one that…",[1034,1036,1038,1040],{"id":196,"label":1035},"Is horizontal",{"id":198,"label":1037},"Is perpendicular to the line",{"id":201,"label":1039},"Goes to the nearest labelled point",{"id":204,"label":1041},"Makes a 45° angle","The fold argument shows any other segment is longer.",{"itemId":1044,"prompt":1045,"options":1046,"correct":198,"why":1055},"lines.deepen-q-bisector","Point Q is 6 cm from A and 6 cm from B. Where must Q lie?",[1047,1049,1051,1053],{"id":196,"label":1048},"On segment AB",{"id":198,"label":1050},"On the perpendicular bisector of AB",{"id":201,"label":1052},"On line AB extended",{"id":204,"label":1054},"Anywhere","Points equidistant from A and B are exactly the points of the perpendicular bisector.",{"itemId":1057,"prompt":1058,"options":1059,"correct":198,"why":1068},"lines.deepen-q-noneuclid","Who independently published hyperbolic geometry around 1830?",[1060,1062,1064,1066],{"id":196,"label":1061},"Euclid and Playfair",{"id":198,"label":1063},"Lobachevsky and Bolyai",{"id":201,"label":1065},"Proclus and Saccheri",{"id":204,"label":1067},"Newton and Leibniz","Lobachevsky (1829) and Bolyai (1832).",{"itemId":1070,"prompt":1071,"options":1072,"correct":201,"why":1077},"lines.deepen-q-four-two","Four lines in a plane cannot have exactly how many crossing points?",[1073,1074,1075,1076],{"id":196,"label":247},{"id":198,"label":84},{"id":201,"label":89},{"id":204,"label":249},"The case-by-case proof shows 2 is impossible.",{"itemId":1079,"prompt":1080,"options":1081,"correct":196,"why":1090},"lines.deepen-q-sulba","What does the word sulba refer to?",[1082,1084,1086,1088],{"id":196,"label":1083},"A cord or rope",{"id":198,"label":1085},"A compass",{"id":201,"label":1087},"A fire",{"id":204,"label":1089},"A shadow","Sulba means cord: the Sulba Sutras are the rules of the cord.",{"id":1092,"type":1093,"prompt":1094},"reflect-deepen","reflection","Euclid chose five starting rules. If you were writing your own geometry from scratch, which of his rules would you keep, which would you change, and what kind of surface would your geometry describe?",{"id":1096,"type":1097,"title":1098,"points":1099},"cheat-deepen","summary","Cheat sheet",[1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110],"**Euclid's Elements (c. 300 BCE):** definitions, 5 postulates, common notions, then proofs.","**Modern view:** point, line, plane are undefined; only their relationships matter (Hilbert, 1899).","**Playfair's axiom:** through a point not on a line, exactly one parallel.","**Proved:** two lines meet at most once; parallel to the same line ⇒ parallel; two perpendiculars to one line ⇒ parallel.","**Counting pairs:** n(n − 1) ÷ 2 segments, lines (no three collinear), handshakes, crossings (general position).","**With k collinear points:** lines = n(n−1)÷2 − k(k−1)÷2 + 1. Diagonals of an n-gon: n(n − 3) ÷ 2.","**Four lines** cross at 0, 1, 3, 4, 5 or 6 points, never 2 (proof by cases).","**Perpendicular = shortest** path to a line (fold proof). Points equidistant from A and B = the perpendicular bisector.","**The fifth postulate** can't be proved from the others; replacing it gives spherical and hyperbolic geometry.","**On a sphere:** great circles are the lines; any two meet twice; there are no parallels.","**Sulba Sutras:** cord-and-peg geometry; Katyayana's shadow method for east–west; 3-4-5 and other cords for right angles.",{"id":1112,"type":386,"conceptId":1113,"relation":1114,"explanation":1115},"conn-angles-d","angles","helps_understand","The fifth postulate is about angles made by a transversal; angle facts about parallel lines depend on it.",{"id":1117,"type":386,"conceptId":1118,"relation":1119,"explanation":1120},"conn-shape-d","shape-and-space","related_to","Diagonals of polygons are counted with the same pair-counting argument as segments.",{"id":1122,"type":1123,"sourceIds":1124},"sources-deepen","sources",[1125,1126,1127,1128,1129,1130,1131,1132,1133],"lines-britannica-euclidean-geometry","lines-wiki-parallel-postulate","lines-wiki-shulba-sutras","lines-ncert-math-6","lines-ncert-math-7","lines-mathsisfun-parallel-perpendicular","lines-dani-katyayana-sulvasutra","lines-mactutor-non-euclidean","lines-straume-geometry-survey",[1125,1126,1127,1128,1129,1130,1131,1132,1133],"needs_review",{"generatedBy":1137,"notes":1138},"claude-code","Draft generated by a scripted generator; every count and length was computed in Python. Pending owner review.","7ab76f0b28c1b4cd637aa5d607a2c278101fefcbd9e2fb9c953c2355ed7d1ab1",{"component:match-pairs@1":1141,"logic:practice":1142,"component:sort-game@1":1143,"component:line-spotter@1":1144,"source:lines-britannica-euclidean-geometry":1145,"source:lines-dani-katyayana-sulvasutra":1146,"source:lines-mactutor-non-euclidean":1147,"source:lines-mathsisfun-parallel-perpendicular":1148,"source:lines-ncert-math-6":1149,"source:lines-ncert-math-7":1150,"source:lines-straume-geometry-survey":1151,"source:lines-wiki-parallel-postulate":1152,"source:lines-wiki-shulba-sutras":1153},"2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","1279a4e23634aba0fd1cb2bb76638a44d6b592131c326e194fdd1bf593a12f93","40d3ed4e883bb30133b96e69ed296da3721d5ef497178eea1c88023a8c2403c9","705e75261b4a1f9502f10c70908488953d028911f42c2f061fa572d8b5c76388","a1232b234e1f440e557a351815bc736f3469d23132b95c7662b6c0fcf21666a3","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","219993d3ebc010eeac3a16481eda537da1557ef707d475cbca1ba05a49dfff80","f380f754917bdc6d17093f871ff43201664a75572d28b860df518fdbf8f740be","fc9c2009ea75de9a47e553402b15524e2579e6e09d69e547d1dc50423219ad6d","197bbcf0b10e2a83a3b3bac20fa5140966e6e15fa8a10b0b238c50bc66a87d66","60acdd731cda7dbe3dd6e8e291ab83eb01ecdc72101b45c61a2ba14df47b9b0f",{"state":1155,"reviewer":1156,"selfReview":1157,"reviewedAt":1158,"method":1159},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598450]