[{"data":1,"prerenderedAt":1086},["ShallowReactive",2],{"questions:lines":3},{"bank":4,"contentHash":1075,"dependencyHashes":1076,"releaseId":1085},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1063,"reviewStatus":1071,"authoring":1072},1,"lines","Lines, rays and line segments: question bank","Practise everything about points, lines, rays and segments, from naming and measuring to counting crossings, parallel and perpendicular lines, skew edges and proofs. Questions are graded **foundation**, **core**, **stretch** and **challenge**. Every question has a worked solution: try it first, then read the solution even when you are right. Notation is written in words (line AB, ray AB, segment AB) because textbook bar symbols cannot be typed.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"points-lines","Points, lines and planes","What a point, a line and a plane are, how to name them, and how many lines pass through one or two points.",{"id":15,"title":16,"description":17},"rays","Rays","Rays, their end points and names, opposite rays, and counting rays on a line.",{"id":19,"title":20,"description":21},"segments-measuring","Line segments and measuring","Segments, lengths, midpoints, betweenness, measuring with a ruler and divider, and parallax error.",{"id":23,"title":24,"description":25},"collinear-counting","Collinear points and counting","Collinear and non-collinear points, counting segments, lines, triangles and diagonals.",{"id":27,"title":28,"description":29},"intersecting-concurrent","Intersecting and concurrent lines","Points of intersection, concurrent lines and how many crossings some lines can make.",{"id":31,"title":32,"description":33},"parallel","Parallel lines","Definition, the ∥ symbol, distance between parallels, transversals and the parallel postulate.",{"id":35,"title":36,"description":37},"perpendicular","Perpendicular lines","Right angles, the ⊥ symbol, checking with a set square and the perpendicular bisector.",{"id":39,"title":40,"description":41},"real-life-3d","Real life and three dimensions","Railways, cricket, maps, perspective and skew edges of solids.",{"id":43,"title":44,"description":45},"reasoning","Reasoning and puzzles","Always, sometimes or never; spotting mistakes; proofs; history; puzzles.",[47,72,83,101,111,130,148,158,176,193,211,223,234,252,264,273,282,292,309,319,330,340,358,376,385,399,409,419,431,439,448,465,474,483,492,500,509,518,527,535,551,562,577,584,592,600,613,630,647,664,676,694,711,724,734,742,751,768,775,793,804,814,831,841,858,872,881,899,907,923,932,941,950,968,985,1003,1021,1038,1046,1054],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":67,"solution":69,"skills":70},"lines.q001","foundation","Which of these has **no size at all**: no length, no width, no thickness?",{"kind":52,"options":53,"correct":66},"choice",[54,57,60,63],{"id":55,"label":56},"a","A dot made by a sharp pencil",{"id":58,"label":59},"b","A point",{"id":61,"label":62},"c","A grain of sand",{"id":64,"label":65},"d","A full stop",[58],[68],"Which one is an idea rather than a real object?","A **point** marks an exact position and has no size. A pencil dot, a grain of sand and a full stop are all *pictures* of points, and all of them have some size.",[71],"point",{"id":73,"section":11,"level":49,"prompt":74,"check":75,"hints":78,"solution":80,"skills":81},"lines.q002","How many end points does a **line** have?",{"kind":76,"answer":77,"tolerance":77},"number",0,[79],"Does a line stop anywhere?","A line goes on forever in **both** directions, so it never stops anywhere: it has **0** end points. (A segment has 2 and a ray has 1.)",[82],"line",{"id":84,"section":11,"level":49,"prompt":85,"check":86,"hints":97,"solution":98,"skills":99},"lines.q003","Points are usually named with…",{"kind":52,"options":87,"correct":96},[88,90,92,94],{"id":55,"label":89},"Small letters, like p",{"id":58,"label":91},"Capital letters, like P",{"id":61,"label":93},"Numbers, like 1",{"id":64,"label":95},"Greek letters only",[58],[],"By convention, **points get capital letters** (A, B, P). Lines are sometimes named with small letters (line l).",[100],"notation",{"id":102,"section":11,"level":103,"prompt":104,"check":105,"hints":106,"solution":108,"skills":109},"lines.q004","core","How many different straight lines can pass through **two** given points?",{"kind":76,"answer":5,"tolerance":77},[107],"Try to draw two different straight lines through the same two dots.","Through two different points there is **exactly one** line. Once a straight line passes through both, its direction is fixed.",[110],"two points fix a line",{"id":112,"section":11,"level":103,"prompt":113,"check":114,"hints":125,"solution":127,"skills":128},"lines.q005","How many straight lines can pass through **one** given point?",{"kind":52,"options":115,"correct":124},[116,118,120,122],{"id":55,"label":117},"1",{"id":58,"label":119},"2",{"id":61,"label":121},"4",{"id":64,"label":123},"Endlessly many",[64],[126],"Spin a ruler round a dot.","Every direction gives a different line through the point, and there are endlessly many directions. So **endlessly many** lines pass through one point.",[129],"lines through a point",{"id":131,"section":11,"level":103,"prompt":132,"check":133,"hints":144,"solution":145,"skills":146},"lines.q006","Which of these is **not** one of the three undefined starting ideas of school geometry?",{"kind":52,"options":134,"correct":143},[135,137,139,141],{"id":55,"label":136},"Point",{"id":58,"label":138},"Line",{"id":61,"label":140},"Plane",{"id":64,"label":142},"Ray",[64],[],"Point, line and plane are described but not defined. A **ray** is defined from them: part of a line starting at a point and going on forever one way.",[147],"undefined terms",{"id":149,"section":11,"level":103,"prompt":150,"check":151,"hints":153,"solution":155,"skills":156},"lines.q007","Three points P, Q and R lie on one line. How many different **names** of the form \"line XY\" can you write for this line, using two of the three letters in either order?",{"kind":76,"answer":152,"tolerance":77},6,[154],"Order matters for the name but not for the line.","Choose an ordered pair of different letters: 3 choices for the first, 2 for the second, 3 × 2 = **6** names (PQ, QP, PR, RP, QR, RQ). They all name the **same** line.",[100,157],"counting",{"id":159,"section":11,"level":160,"prompt":161,"check":162,"hints":171,"solution":173,"skills":174},"lines.q008","stretch","True or false: \"A line drawn 12 cm long in my notebook is 12 cm long.\"",{"kind":52,"options":163,"correct":170},[164,167],{"id":165,"label":166},"t","True",{"id":168,"label":169},"f","False",[168],[172],"What do the arrowheads on a line mean?","**False.** A line has no length: it goes on forever both ways. The 12 cm drawing is only a *picture* of part of the line, with arrows showing it continues. A **segment** could be 12 cm long.",[82,175],"misconception",{"id":177,"section":15,"level":49,"prompt":178,"check":179,"hints":188,"solution":190,"skills":191},"lines.q009","A drawing has a dot at one end and an arrowhead at the other. What is it?",{"kind":52,"options":180,"correct":187},[181,182,183,185],{"id":55,"label":138},{"id":58,"label":142},{"id":61,"label":184},"Line segment",{"id":64,"label":186},"Curve",[58],[189],"Count the end points.","One end point (the dot) and going on forever one way (the arrow): a **ray**.",[192],"ray",{"id":194,"section":15,"level":49,"prompt":195,"check":196,"hints":207,"solution":208,"skills":209},"lines.q010","Which everyday thing is the best picture of a ray?",{"kind":52,"options":197,"correct":206},[198,200,202,204],{"id":55,"label":199},"A cricket bat",{"id":58,"label":201},"A beam of light from a torch",{"id":61,"label":203},"A railway sleeper",{"id":64,"label":205},"A bangle",[58],[],"A torch beam starts at the bulb and travels on in one direction: a **ray**.",[192,210],"real life",{"id":212,"section":15,"level":103,"prompt":213,"check":214,"hints":219,"solution":221,"skills":222},"lines.q011","Ray AB starts at which point?",{"kind":215,"accept":216},"text",[217,218],"A","point A",[220],"Which letter comes first?","The **first** letter of a ray's name is always its end point, so ray AB starts at **A** and passes through B.",[100,192],{"id":224,"section":15,"level":103,"prompt":225,"check":226,"hints":231,"solution":232,"skills":233},"lines.q012","True or false: ray PQ and ray QP are the same ray.",{"kind":52,"options":227,"correct":230},[228,229],{"id":165,"label":166},{"id":168,"label":169},[168],[],"**False.** Ray PQ starts at P and goes through Q and beyond. Ray QP starts at Q and goes through P and beyond. Different starting points, opposite directions.",[192,100],{"id":235,"section":15,"level":103,"prompt":236,"check":237,"hints":248,"solution":250,"skills":251},"lines.q013","Points A, B and C lie on a line in that order. Which ray is the same as ray AB?",{"kind":52,"options":238,"correct":247},[239,241,243,245],{"id":55,"label":240},"Ray BA",{"id":58,"label":242},"Ray AC",{"id":61,"label":244},"Ray CA",{"id":64,"label":246},"Ray BC",[58],[249],"Same start, same direction?","Ray AB starts at A and heads towards B, and C lies further along in that direction. **Ray AC** has the same start and direction, so it is the same ray.",[192],{"id":253,"section":15,"level":160,"prompt":254,"check":255,"hints":260,"solution":262,"skills":263},"lines.q014","Q lies between P and R on a line. What do we call ray QP and ray QR?",{"kind":215,"accept":256},[257,258,259],"opposite rays","opposite ray","a pair of opposite rays",[261],"They share a start and point opposite ways.","They share the end point Q and point in exactly opposite directions: they are **opposite rays**. Together they make the whole line PR.",[257],{"id":265,"section":15,"level":160,"prompt":266,"check":267,"hints":269,"solution":271,"skills":272},"lines.q015","Points A, B, C, D, E lie on a line in that order. How many different rays start at one of these points and pass through at least one other of them?",{"kind":76,"answer":268,"tolerance":77},8,[270],"End points start one ray each; middle points start two.","The two end points A and E each start 1 such ray (pointing inward). Each of the 3 middle points starts 2 rays (one each way). Total = 1 + 1 + 3 × 2 = **8** = 2 × (5 − 1). Note ray AB = ray AC = ray AD = ray AE.",[157,192],{"id":274,"section":19,"level":49,"prompt":275,"check":276,"hints":278,"solution":279,"skills":280},"lines.q016","How many end points does a line segment have?",{"kind":76,"answer":277,"tolerance":77},2,[],"A segment starts at one point and stops at another: **2** end points. That is why it has a length.",[281],"segment",{"id":283,"section":19,"level":49,"prompt":284,"check":285,"hints":288,"solution":289,"skills":290},"lines.q017","A pencil lies along a ruler from the 0 cm mark to the 13.5 cm mark. How long is it?",{"kind":76,"answer":286,"tolerance":77,"unit":287},13.5,"cm",[],"It starts at 0, so its length is just the reading at the other end: **13.5 cm**.",[291],"measuring",{"id":293,"section":19,"level":49,"prompt":294,"check":295,"hints":306,"solution":307,"skills":308},"lines.q018","Why should you **not** start measuring from the very end of a ruler?",{"kind":52,"options":296,"correct":305},[297,299,301,303],{"id":55,"label":298},"The end is usually worn, and the 0 mark is a little way in",{"id":58,"label":300},"It is against the rules",{"id":61,"label":302},"Rulers are longer at the ends",{"id":64,"label":304},"The numbers are upside down there",[55],[],"On most rulers the 0 mark is a few millimetres in from the end, and ends get chipped. Measuring from the end gives lengths that are too big. **Put the 0 mark on the starting point.**",[291],{"id":310,"section":19,"level":103,"prompt":311,"check":312,"hints":315,"solution":317,"skills":318},"lines.q019","A segment starts at the 2.5 cm mark on a ruler and ends at the 9.1 cm mark. How long is it, in cm?",{"kind":76,"answer":313,"tolerance":314,"unit":287},6.6,0.001,[316],"Subtract the two readings.","Length = larger reading − smaller reading = 9.1 − 2.5 = **6.6 cm**.",[291],{"id":320,"section":19,"level":103,"prompt":321,"check":322,"hints":325,"solution":327,"skills":328},"lines.q020","Kabir's ruler has a broken end, so he starts a segment at the 1 cm mark. Its other end is at 8.4 cm. What is its length in millimetres?",{"kind":76,"answer":323,"tolerance":77,"unit":324},74,"mm",[326],"First find the length in cm.","Length = 8.4 − 1 = 7.4 cm. 1 cm = 10 mm, so 7.4 cm = **74 mm**.",[291,329],"units",{"id":331,"section":19,"level":103,"prompt":332,"check":333,"hints":335,"solution":337,"skills":338},"lines.q021","B lies on segment AC. AB = 3.8 cm and BC = 5.7 cm. Find AC in cm.",{"kind":76,"answer":334,"tolerance":314,"unit":287},9.5,[336],"The parts add up to the whole.","B is between A and C, so AC = AB + BC = 3.8 + 5.7 = **9.5 cm**.",[339],"betweenness",{"id":341,"section":19,"level":103,"prompt":342,"check":343,"hints":354,"solution":355,"skills":356},"lines.q022","What is **parallax error** when reading a ruler?",{"kind":52,"options":344,"correct":353},[345,347,349,351],{"id":55,"label":346},"Using a ruler that is too short",{"id":58,"label":348},"Reading the scale from the side instead of from directly above",{"id":61,"label":350},"Measuring in cm instead of mm",{"id":64,"label":352},"Starting at 1 cm instead of 0",[58],[],"The scale sits above the paper, so looking from the side lines up the wrong mark with the point. **Put your eye directly above the mark**, or use a divider.",[357],"parallax",{"id":359,"section":19,"level":103,"prompt":360,"check":361,"hints":372,"solution":373,"skills":374},"lines.q023","Which method of comparing two segments is the most reliable?",{"kind":52,"options":362,"correct":371},[363,365,367,369],{"id":55,"label":364},"Looking at them",{"id":58,"label":366},"Tracing one onto the other",{"id":61,"label":368},"Measuring with a divider and ruler",{"id":64,"label":370},"Guessing which looks longer",[61],[],"Observation is easily fooled (optical illusions), tracing is clumsy, but **measuring**, especially with a divider, gives an accurate number you can check.",[375],"comparing segments",{"id":377,"section":19,"level":160,"prompt":378,"check":379,"hints":381,"solution":383,"skills":384},"lines.q024","A, B and C lie on a line with B between A and C. AC = 12.4 cm and AB is 3 times as long as BC. Find BC in cm.",{"kind":76,"answer":380,"tolerance":314,"unit":287},3.1,[382],"Write AB as 3 × BC.","AB + BC = AC and AB = 3 × BC, so 3 × BC + BC = 4 × BC = 12.4. BC = 12.4 ÷ 4 = **3.1 cm** (and AB = 9.3 cm). Check: 9.3 + 3.1 = 12.4 ✓",[339,43],{"id":386,"section":19,"level":160,"prompt":387,"check":388,"hints":395,"solution":397,"skills":398},"lines.q025","PQ = 6 cm, QR = 4 cm and PR = 9 cm. Is Q on segment PR?",{"kind":52,"options":389,"correct":394},[390,392],{"id":55,"label":391},"Yes",{"id":58,"label":393},"No",[58],[396],"What would PQ + QR equal if Q were between P and R?","If Q were on segment PR we would need PQ + QR = PR. But 6 + 4 = 10 ≠ 9. So **Q is not on segment PR**; the three points form a triangle.",[339],{"id":400,"section":19,"level":160,"prompt":401,"check":402,"hints":404,"solution":406,"skills":407},"lines.q026","M is the midpoint of AB and N is the midpoint of AM. AB = 10 cm. Find NB in cm.",{"kind":76,"answer":403,"tolerance":314,"unit":287},7.5,[405],"Find AM first.","AM = MB = 5 cm. N is the midpoint of AM, so NM = 2.5 cm. NB = NM + MB = 2.5 + 5 = **7.5 cm**.",[408],"midpoint",{"id":410,"section":19,"level":411,"prompt":412,"check":413,"hints":415,"solution":417,"skills":418},"lines.q027","challenge","Points A, B, C, D lie on a line in that order. AC = 9 cm, BD = 11 cm and AD = 15 cm. Find BC in cm.",{"kind":76,"answer":414,"tolerance":77,"unit":287},5,[416],"Draw it. Which part is counted twice in AC + BD?","AC + BD covers the whole segment AD once, plus the overlap BC a second time. So AC + BD = AD + BC, giving 9 + 11 = 15 + BC, so BC = 20 − 15 = **5 cm**. Check: AB = 9 − 5 = 4 and CD = 11 − 5 = 6, and 4 + 5 + 6 = 15 ✓",[339,43],{"id":420,"section":23,"level":49,"prompt":421,"check":422,"hints":427,"solution":428,"skills":429},"lines.q028","Three beads are threaded on a tightly stretched string. Are they collinear?",{"kind":52,"options":423,"correct":426},[424,425],{"id":55,"label":391},{"id":58,"label":393},[55],[],"A tight string is straight, so all three beads lie on one line: they are **collinear**.",[430],"collinear",{"id":432,"section":23,"level":103,"prompt":433,"check":434,"hints":436,"solution":437,"skills":438},"lines.q029","Three non-collinear points are marked. How many different lines can be drawn through pairs of them?",{"kind":76,"answer":435,"tolerance":77},3,[],"No line passes through all three, so each pair (AB, BC, CA) gives its own line: **3** lines, like the sides of a triangle.",[430,157],{"id":440,"section":23,"level":103,"prompt":441,"check":442,"hints":443,"solution":445,"skills":446},"lines.q030","Four points are marked on a line. How many line segments have both end points among them?",{"kind":76,"answer":152,"tolerance":77},[444],"List them: AB, AC, …","Count pairs: 4 × 3 ÷ 2 = **6** (AB, AC, AD, BC, BD, CD).",[447],"counting segments",{"id":449,"section":23,"level":103,"prompt":450,"check":451,"hints":462,"solution":463,"skills":464},"lines.q031","Which set of points is **non-collinear**?",{"kind":52,"options":452,"correct":461},[453,455,457,459],{"id":55,"label":454},"The 1 cm, 4 cm and 7 cm marks on a ruler",{"id":58,"label":456},"The corners of a triangle",{"id":61,"label":458},"The two ends of a segment and its midpoint",{"id":64,"label":460},"The tops of three stumps in a row",[58],[],"If the corners of a triangle were on one line there would be no triangle. The others all lie along one straight line.",[430],{"id":466,"section":23,"level":160,"prompt":467,"check":468,"hints":470,"solution":472,"skills":473},"lines.q032","How many line segments are formed by **7** points marked on a line?",{"kind":76,"answer":469,"tolerance":77},21,[471],"Each point pairs with 6 others; each segment is counted twice.","Segments = n × (n − 1) ÷ 2 = 7 × 6 ÷ 2 = **21**.",[447],{"id":475,"section":23,"level":160,"prompt":476,"check":477,"hints":479,"solution":480,"skills":481},"lines.q033","Six points are marked in a plane, no three collinear. How many different lines pass through pairs of them?",{"kind":76,"answer":478,"tolerance":77},15,[],"Each pair gives its own line: 6 × 5 ÷ 2 = **15**.",[482],"counting lines",{"id":484,"section":23,"level":160,"prompt":485,"check":486,"hints":488,"solution":490,"skills":491},"lines.q034","Four points are marked with **exactly three** of them collinear. How many different lines pass through at least two of them?",{"kind":76,"answer":487,"tolerance":77},4,[489],"Treat the collinear three first.","The three collinear points share **1** line. The fourth point joins each of the other three by its own line: **3** more. Total **4**.",[430,482],{"id":493,"section":23,"level":160,"prompt":494,"check":495,"hints":497,"solution":498,"skills":499},"lines.q035","A straight bus route has 9 stops. A fare is needed for every pair of stops (a trip either way costs the same). How many different fares?",{"kind":76,"answer":496,"tolerance":77},36,[],"Each fare is a pair of stops, like a segment on a line with 9 points: 9 × 8 ÷ 2 = **36**.",[447,210],{"id":501,"section":23,"level":411,"prompt":502,"check":503,"hints":505,"solution":507,"skills":508},"lines.q036","Ten points are marked in a plane. Exactly four 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?",{"kind":76,"answer":504,"tolerance":77},40,[506],"Start as if no three were collinear, then correct.","If no three were collinear: 10 × 9 ÷ 2 = 45 lines. The 4 collinear points would have given 4 × 3 ÷ 2 = 6 lines but actually give 1. Lines = 45 − 6 + 1 = **40**.",[482,430],{"id":510,"section":23,"level":411,"prompt":511,"check":512,"hints":514,"solution":516,"skills":517},"lines.q037","How many triangles have all three corners among 8 points in a plane, if exactly 3 of the points are collinear and no other three are?",{"kind":76,"answer":513,"tolerance":77},55,[515],"Count all groups of 3, then remove those that lie on one line.","Groups of 3 from 8: 8 × 7 × 6 ÷ 6 = 56. Only one group, the 3 collinear points, fails to make a triangle. Triangles = 56 − 1 = **55**.",[157,430],{"id":519,"section":23,"level":411,"prompt":520,"check":521,"hints":523,"solution":524,"skills":525},"lines.q038","How many diagonals does a decagon (10 sides) have?",{"kind":76,"answer":522,"tolerance":77},35,[],"Segments joining 2 of the 10 corners: 10 × 9 ÷ 2 = 45. Subtract the 10 sides: 45 − 10 = **35**. Formula: n(n − 3) ÷ 2 = 10 × 7 ÷ 2 = 35.",[447,526],"polygons",{"id":528,"section":27,"level":49,"prompt":529,"check":530,"hints":531,"solution":532,"skills":533},"lines.q039","Two straight lines cross each other. At how many points do they meet?",{"kind":76,"answer":5,"tolerance":77},[],"Two different straight lines can meet at **at most one** point. Crossing lines meet at exactly **1**, the point of intersection.",[534],"intersecting",{"id":536,"section":27,"level":49,"prompt":537,"check":538,"hints":548,"solution":549,"skills":550},"lines.q040","The blades of an open pair of scissors are a picture of…",{"kind":52,"options":539,"correct":547},[540,541,543,545],{"id":55,"label":32},{"id":58,"label":542},"Intersecting lines",{"id":61,"label":544},"Skew lines",{"id":64,"label":546},"Curves",[58],[],"The blades cross at the screw, the **point of intersection**.",[534,210],{"id":552,"section":27,"level":103,"prompt":553,"check":554,"hints":559,"solution":560,"skills":561},"lines.q041","Three or more lines that all pass through one point are called…",{"kind":215,"accept":555},[556,557,558],"concurrent lines","concurrent","concurrent line",[],"They are **concurrent** lines, and the shared point is the point of concurrence. Think of the spokes of a wheel meeting at the hub.",[557],{"id":563,"section":27,"level":103,"prompt":564,"check":565,"hints":573,"solution":575,"skills":576},"lines.q042","Three lines in a plane. Which number of intersection points is **impossible**?",{"kind":52,"options":566,"correct":572},[567,569,570,571],{"id":55,"label":568},"0",{"id":58,"label":117},{"id":61,"label":119},{"id":64,"label":121},[64],[574],"How many pairs of lines are there?","Three lines make only 3 pairs, and each pair meets at most once, so the maximum is 3. **4 is impossible.** (0: all parallel; 1: concurrent; 2: two parallel and one crossing both; 3: a triangle.)",[534,43],{"id":578,"section":27,"level":103,"prompt":579,"check":580,"hints":581,"solution":582,"skills":583},"lines.q043","Two parallel lines are crossed by a third line. How many intersection points are there?",{"kind":76,"answer":277,"tolerance":77},[],"The parallel lines never meet each other, and the third line crosses each of them once: **2** points.",[534,31],{"id":585,"section":27,"level":160,"prompt":586,"check":587,"hints":588,"solution":590,"skills":591},"lines.q044","What is the **greatest** number of intersection points that 6 lines in a plane can have?",{"kind":76,"answer":478,"tolerance":77},[589],"Count pairs of lines.","Each pair of lines meets at most once. Pairs of 6 lines: 6 × 5 ÷ 2 = **15**, achieved when no two are parallel and no three are concurrent.",[534,157],{"id":593,"section":27,"level":411,"prompt":594,"check":595,"hints":596,"solution":598,"skills":599},"lines.q045","Four lines are drawn in a plane. Exactly two of them are parallel, and no three pass through one point. How many intersection points are there?",{"kind":76,"answer":414,"tolerance":77},[597],"Which pair does not meet?","Out of 4 × 3 ÷ 2 = 6 pairs, the parallel pair never meets, leaving 5 pairs. Since no three lines are concurrent, those 5 meeting points are all different: **5**.",[534,43],{"id":601,"section":27,"level":411,"prompt":602,"check":603,"hints":608,"solution":610,"skills":611},"lines.q046","Can four lines in a plane have exactly **2** intersection points?",{"kind":52,"options":604,"correct":607},[605,606],{"id":55,"label":391},{"id":58,"label":393},[58],[609],"Try the cases by how many lines are parallel.","**No.** Group the lines by direction. All parallel: 0 points. 3 parallel + 1: 3 points. 2 + 2: 4 points. 2 parallel + 2 others: the others each cut both parallels and share at most one point with each other, so at least 3 points. No two parallel: 1 point if all concurrent, otherwise some three are not concurrent and already give 3 points. Two is never possible.",[43,612],"proof",{"id":614,"section":31,"level":49,"prompt":615,"check":616,"hints":627,"solution":628,"skills":629},"lines.q047","Which pair is the best example of parallel lines?",{"kind":52,"options":617,"correct":626},[618,620,622,624],{"id":55,"label":619},"The two rails of a railway track",{"id":58,"label":621},"The hands of a clock at 3:00",{"id":61,"label":623},"The strokes of the letter X",{"id":64,"label":625},"Two roads crossing at a chowk",[55],[],"Railway rails stay the same distance apart and never meet: **parallel**.",[31,210],{"id":631,"section":31,"level":49,"prompt":632,"check":633,"hints":644,"solution":645,"skills":646},"lines.q048","Which symbol means \"is parallel to\"?",{"kind":52,"options":634,"correct":643},[635,637,639,641],{"id":55,"label":636},"⊥",{"id":58,"label":638},"∥",{"id":61,"label":640},"=",{"id":64,"label":642},"≠",[58],[],"**∥** means parallel (two upright strokes side by side). ⊥ means perpendicular.",[100],{"id":648,"section":31,"level":103,"prompt":649,"check":650,"hints":661,"solution":662,"skills":663},"lines.q049","Which condition must parallel lines satisfy?",{"kind":52,"options":651,"correct":660},[652,654,656,658],{"id":55,"label":653},"They lie in the same plane and never meet",{"id":58,"label":655},"They are both horizontal",{"id":61,"label":657},"They are the same length",{"id":64,"label":659},"They are exactly 1 cm apart",[55],[],"Parallel lines lie in the **same plane** and **never meet** however far they are extended. They can point in any direction and be any distance apart.",[31],{"id":665,"section":31,"level":103,"prompt":666,"check":667,"hints":672,"solution":674,"skills":675},"lines.q050","Two lines on a page are 2.0 cm apart at the left edge and 1.7 cm apart at the right edge. Are they parallel?",{"kind":52,"options":668,"correct":671},[669,670],{"id":55,"label":391},{"id":58,"label":393},[58],[673],"What must be true of the gap?","Parallel lines keep the **same** perpendicular distance everywhere. Here the gap shrinks, so the lines are slowly closing in and will meet somewhere to the right: **not parallel**.",[31,291],{"id":677,"section":31,"level":103,"prompt":678,"check":679,"hints":690,"solution":691,"skills":692},"lines.q051","How is the distance between two parallel lines measured?",{"kind":52,"options":680,"correct":689},[681,683,685,687],{"id":55,"label":682},"Along any slanting segment joining them",{"id":58,"label":684},"Along a segment perpendicular to both",{"id":61,"label":686},"Along one of the lines",{"id":64,"label":688},"It cannot be measured",[58],[],"The distance is the length of a segment **perpendicular** to both lines. It is the shortest gap and is the same everywhere.",[31,693],"distance",{"id":695,"section":31,"level":160,"prompt":696,"check":697,"hints":708,"solution":709,"skills":710},"lines.q052","Lines l, m and n lie in one plane. l ∥ m and m ∥ n, and l and n are different lines. What can you say about l and n?",{"kind":52,"options":698,"correct":707},[699,701,703,705],{"id":55,"label":700},"l ∥ n",{"id":58,"label":702},"l ⊥ n",{"id":61,"label":704},"l and n meet at one point",{"id":64,"label":706},"Nothing can be said",[55],[],"If l and n met at a point P, there would be two different lines through P both parallel to m, but through a point off a line there is exactly one parallel (Playfair's axiom). So **l ∥ n**.",[31,43],{"id":712,"section":31,"level":160,"prompt":713,"check":714,"hints":719,"solution":721,"skills":722},"lines.q053","True or false: two lines that never meet must be parallel.",{"kind":52,"options":715,"correct":718},[716,717],{"id":165,"label":166},{"id":168,"label":169},[168],[720],"Think about edges of a room.","**False** in three dimensions. The edge where the front wall meets the floor and the edge where the left wall meets the ceiling never meet, but are not parallel: they are **skew**. It is true for two lines in one plane.",[31,723],"skew",{"id":725,"section":31,"level":160,"prompt":726,"check":727,"hints":731,"solution":732,"skills":733},"lines.q054","A line crosses two other lines at two different points. What is it called?",{"kind":215,"accept":728},[729,730],"transversal","a transversal",[],"A line that crosses two or more lines at different points is a **transversal**.",[729],{"id":735,"section":31,"level":411,"prompt":736,"check":737,"hints":738,"solution":739,"skills":740},"lines.q055","Through a point P not on line l, how many lines can be drawn parallel to l (in the same plane)?",{"kind":76,"answer":5,"tolerance":77},[],"Exactly **1**. This is Playfair's axiom, the modern form of Euclid's fifth postulate. (On a sphere there would be none; in hyperbolic geometry, infinitely many.)",[741],"parallel postulate",{"id":743,"section":35,"level":49,"prompt":744,"check":745,"hints":748,"solution":749,"skills":750},"lines.q056","Perpendicular lines meet at an angle of how many degrees?",{"kind":76,"answer":746,"tolerance":77,"unit":747},90,"°",[],"Perpendicular lines meet at a **right angle**, 90°.",[35],{"id":752,"section":35,"level":49,"prompt":753,"check":754,"hints":765,"solution":766,"skills":767},"lines.q057","Which letter is made of two perpendicular strokes?",{"kind":52,"options":755,"correct":764},[756,758,760,762],{"id":55,"label":757},"T",{"id":58,"label":759},"V",{"id":61,"label":761},"X",{"id":64,"label":763},"Z",[55],[],"In **T** the top bar meets the upright stem at a right angle. V and X meet at slants; Z's parallel strokes are joined by a slant.",[35,210],{"id":769,"section":35,"level":103,"prompt":770,"check":771,"hints":772,"solution":773,"skills":774},"lines.q058","If two lines are perpendicular, how many of the four angles at their crossing are right angles?",{"kind":76,"answer":487,"tolerance":77},[],"If one angle is 90°, the angle next to it along a straight line is 180° − 90° = 90°, and so on round: all **4** are right angles.",[35],{"id":776,"section":35,"level":103,"prompt":777,"check":778,"hints":789,"solution":790,"skills":791},"lines.q059","How can you check whether two lines are perpendicular without a protractor?",{"kind":52,"options":779,"correct":788},[780,782,784,786],{"id":55,"label":781},"Use a set square or a folded-paper corner",{"id":58,"label":783},"Check whether one is horizontal",{"id":61,"label":785},"Measure their lengths",{"id":64,"label":787},"Look at them from the side",[55],[],"Fit the square corner of a set square, or a paper folded twice (crease onto itself), into the corner. If both edges lie exactly along the lines, they are **perpendicular**.",[35,792],"tools",{"id":794,"section":35,"level":103,"prompt":795,"check":796,"hints":801,"solution":802,"skills":803},"lines.q060","True or false: perpendicular lines must be one horizontal and one vertical.",{"kind":52,"options":797,"correct":800},[798,799],{"id":165,"label":166},{"id":168,"label":169},[168],[],"**False.** Perpendicular only means they meet at 90°. A tilted square still has perpendicular sides, although none is horizontal.",[35,175],{"id":805,"section":35,"level":160,"prompt":806,"check":807,"hints":809,"solution":811,"skills":812},"lines.q061","Segment AB is 14 cm long. Its perpendicular bisector meets AB at M. Point P is on the bisector with PA = 9 cm. Find AM + PB, in cm.",{"kind":76,"answer":808,"tolerance":77,"unit":287},16,[810],"Points on the perpendicular bisector are equidistant from A and B.","The bisector passes through the midpoint, so AM = 14 ÷ 2 = 7 cm. Every point on the perpendicular bisector is equally far from A and B, so PB = PA = 9 cm. AM + PB = 7 + 9 = **16 cm**.",[813],"perpendicular bisector",{"id":815,"section":35,"level":160,"prompt":816,"check":817,"hints":828,"solution":829,"skills":830},"lines.q062","In one plane, lines a and b are both perpendicular to line l. What can you say about a and b?",{"kind":52,"options":818,"correct":827},[819,821,823,825],{"id":55,"label":820},"They are parallel",{"id":58,"label":822},"They are perpendicular to each other",{"id":61,"label":824},"They meet on l",{"id":64,"label":826},"They are skew",[55],[],"Both meet l at 90°, so they point in the same direction. The inside angles on one side add to 90° + 90° = 180°, so by Euclid's fifth postulate they meet on neither side: **a ∥ b**.",[35,31,43],{"id":832,"section":35,"level":411,"prompt":833,"check":834,"hints":836,"solution":838,"skills":839},"lines.q063","A builder marks 90 cm along one wall from a corner and 120 cm along the other wall. For the walls to be perpendicular, how far apart (in cm) should the two marks be?",{"kind":76,"answer":835,"tolerance":77,"unit":287},150,[837],"90 : 120 is the same as 3 : 4.","Use the 3-4-5 rule from the Sulba Sutras: 90 = 3 × 30 and 120 = 4 × 30, so the diagonal should be 5 × 30 = **150 cm**. Check: 90² + 120² = 8,100 + 14,400 = 22,500 = 150².",[35,840],"right angle check",{"id":842,"section":39,"level":49,"prompt":843,"check":844,"hints":855,"solution":856,"skills":857},"lines.q064","The printed lines in a ruled notebook are…",{"kind":52,"options":845,"correct":854},[846,848,850,852],{"id":55,"label":847},"Parallel",{"id":58,"label":849},"Perpendicular",{"id":61,"label":851},"Intersecting",{"id":64,"label":853},"Concurrent",[55],[],"They are printed the same distance apart and never meet: **parallel**.",[210],{"id":859,"section":39,"level":103,"prompt":860,"check":861,"hints":869,"solution":870,"skills":871},"lines.q065","On a cricket pitch, the popping crease is 1.22 m in front of the bowling crease. What is their relationship?",{"kind":52,"options":862,"correct":868},[863,864,865,866],{"id":55,"label":847},{"id":58,"label":849},{"id":61,"label":853},{"id":64,"label":867},"Skew",[55],[],"The Laws of Cricket place the popping crease **parallel** to the bowling crease, 1.22 m (4 ft) in front of it. The return creases are perpendicular to both.",[210,31],{"id":873,"section":39,"level":103,"prompt":874,"check":875,"hints":877,"solution":878,"skills":879},"lines.q066","How many edges does a cube have?",{"kind":76,"answer":876,"tolerance":77},12,[],"A cube has 4 edges on top, 4 on the bottom and 4 upright: **12** edges.",[880],"3D",{"id":882,"section":39,"level":103,"prompt":883,"check":884,"hints":895,"solution":896,"skills":897},"lines.q067","Lines of longitude on a globe all meet at…",{"kind":52,"options":885,"correct":894},[886,888,890,892],{"id":55,"label":887},"The equator",{"id":58,"label":889},"The North and South Poles",{"id":61,"label":891},"India",{"id":64,"label":893},"Nowhere; they are parallel",[58],[],"Longitudes run north–south and **all pass through both poles**, so they are not parallel even though some flat maps draw them that way.",[898],"maps",{"id":900,"section":39,"level":160,"prompt":901,"check":902,"hints":903,"solution":905,"skills":906},"lines.q068","Pick one edge of a cube. How many of the other 11 edges are **skew** to it?",{"kind":76,"answer":487,"tolerance":77},[904],"Count parallel and intersecting edges first.","3 edges are parallel to it; 4 share one of its two corners (2 at each end) and so intersect it. The rest are skew: 11 − 3 − 4 = **4**.",[880,723],{"id":908,"section":39,"level":160,"prompt":909,"check":910,"hints":919,"solution":920,"skills":921},"lines.q069","In a perspective drawing of a straight railway track, the two rails are drawn…",{"kind":52,"options":911,"correct":918},[912,913,915,916],{"id":55,"label":847},{"id":58,"label":914},"Meeting at a vanishing point on the horizon",{"id":61,"label":849},{"id":64,"label":917},"Curving apart",[58],[],"Distant things look smaller, so the rails are drawn getting closer and **meeting at the vanishing point** on the horizon line, although they are parallel on the ground.",[922],"perspective",{"id":924,"section":39,"level":160,"prompt":925,"check":926,"hints":929,"solution":930,"skills":931},"lines.q070","Indian Standard Time is set by the 82.5° E line of longitude. The Earth turns 15° each hour. How many hours ahead of Greenwich is IST? (Answer as a decimal.)",{"kind":76,"answer":927,"tolerance":314,"unit":928},5.5,"hours",[],"82.5 ÷ 15 = **5.5** hours, that is, 5 hours 30 minutes: IST = UTC + 5:30.",[898,210],{"id":933,"section":39,"level":411,"prompt":934,"check":935,"hints":937,"solution":939,"skills":940},"lines.q071","A cuboid has 12 edges. How many **pairs** of its edges are skew?",{"kind":76,"answer":936,"tolerance":77},24,[938],"Count per edge, then divide by 2.","Every edge has 4 skew partners (11 others − 3 parallel − 4 intersecting). 12 edges × 4 = 48 counts each skew pair twice, so there are 48 ÷ 2 = **24** skew pairs. Check: 18 parallel + 24 intersecting + 24 skew = 66 = 12 × 11 ÷ 2 ✓",[880,723,157],{"id":942,"section":39,"level":411,"prompt":943,"check":944,"hints":945,"solution":947,"skills":948},"lines.q072","What is the greatest number of pieces a round cake can be cut into (looking from above) with 5 straight cuts?",{"kind":76,"answer":808,"tolerance":77},[946],"2, 4, 7, 11, …: how much does each new cut add?","The k-th cut crosses at most k − 1 earlier cuts, so it adds at most k pieces. Maximum = 1 + (1 + 2 + 3 + 4 + 5) = 1 + 15 = **16**.",[949,534],"puzzle",{"id":951,"section":43,"level":49,"prompt":952,"check":953,"hints":964,"solution":965,"skills":966},"lines.q073","Which of these is **always** true?",{"kind":52,"options":954,"correct":963},[955,957,959,961],{"id":55,"label":956},"Three points are collinear",{"id":58,"label":958},"A line segment has two end points",{"id":61,"label":960},"Perpendicular lines are vertical",{"id":64,"label":962},"Two lines meet",[58],[],"A segment always has **two end points**. Three points are collinear only sometimes; perpendicular lines can be tilted; parallel lines never meet.",[967],"always sometimes never",{"id":969,"section":43,"level":103,"prompt":970,"check":971,"hints":982,"solution":983,"skills":984},"lines.q074","Sanya says: \"Two different straight lines can meet at two points if they are long enough.\" Why is she wrong?",{"kind":52,"options":972,"correct":981},[973,975,977,979],{"id":55,"label":974},"Through two points there is exactly one line, so two lines sharing two points would be the same line",{"id":58,"label":976},"Lines are too thin to meet twice",{"id":61,"label":978},"Lines only meet at right angles",{"id":64,"label":980},"She is right",[55],[],"If two lines shared two points, both would be lines through those two points. But **exactly one** line passes through two points, so they would be the same line. Different lines meet **at most once**.",[612,43],{"id":986,"section":43,"level":103,"prompt":987,"check":988,"hints":999,"solution":1000,"skills":1001},"lines.q075","Spot the mistake: \"Line AB = 7 cm.\"",{"kind":52,"options":989,"correct":998},[990,992,994,996],{"id":55,"label":991},"A line has no length; it should say segment AB, or AB = 7 cm",{"id":58,"label":993},"It should say 70 mm",{"id":61,"label":995},"It should say line BA",{"id":64,"label":997},"There is no mistake",[55],[],"Lines go on forever and have no length. Write **AB = 7 cm** (the length of segment AB) or \"segment AB is 7 cm long\".",[100,1002],"spot the mistake",{"id":1004,"section":43,"level":160,"prompt":1005,"check":1006,"hints":1017,"solution":1018,"skills":1019},"lines.q076","Which of Euclid's postulates is equivalent to \"through a point not on a line there is exactly one parallel line\"?",{"kind":52,"options":1007,"correct":1016},[1008,1010,1012,1014],{"id":55,"label":1009},"The first",{"id":58,"label":1011},"The third",{"id":61,"label":1013},"The fourth",{"id":64,"label":1015},"The fifth",[64],[],"This is **Playfair's axiom**, the modern form of Euclid's **fifth** (parallel) postulate.",[1020,741],"history",{"id":1022,"section":43,"level":160,"prompt":1023,"check":1024,"hints":1035,"solution":1036,"skills":1037},"lines.q077","Mathematicians tried for about 2,000 years to prove the parallel postulate from the other four. What did they finally discover?",{"kind":52,"options":1025,"correct":1034},[1026,1028,1030,1032],{"id":55,"label":1027},"A simple proof by Playfair",{"id":58,"label":1029},"It cannot be proved: geometries without it are consistent",{"id":61,"label":1031},"It is false on paper",{"id":64,"label":1033},"It follows from postulate 3",[58],[],"In the 1800s Lobachevsky, Bolyai and others showed that replacing it gives **consistent** non-Euclidean geometries, so it cannot be proved from the other four.",[1020],{"id":1039,"section":43,"level":160,"prompt":1040,"check":1041,"hints":1042,"solution":1043,"skills":1044},"lines.q078","On the surface of a globe, taking great circles as \"lines\", how many points do two different lines share?",{"kind":76,"answer":277,"tolerance":77},[],"Any two great circles cross at **2** points, exactly opposite each other (like two lines of longitude meeting at both poles). So the flat-plane rule \"at most one\" fails on a sphere.",[1045,43],"sphere",{"id":1047,"section":43,"level":411,"prompt":1048,"check":1049,"hints":1050,"solution":1052,"skills":1053},"lines.q079","What is the smallest number of straight lines in a plane needed to make exactly 3 intersection points?",{"kind":76,"answer":435,"tolerance":77},[1051],"How many can 2 lines make?","Two lines give at most 1 intersection point, so you need at least 3. Three lines with no two parallel and not all through one point give exactly 3 points (a triangle). Answer **3**.",[43,534],{"id":1055,"section":43,"level":411,"prompt":1056,"check":1057,"hints":1059,"solution":1061,"skills":1062},"lines.q080","Nine dots form a 3 by 3 grid. How many different straight lines pass through at least two of the dots?",{"kind":76,"answer":1058,"tolerance":77},20,[1060],"Start with all pairs, then correct for rows, columns and diagonals.","36 pairs of dots (9 × 8 ÷ 2). Eight lines contain 3 dots each (3 rows, 3 columns, 2 diagonals); each was counted 3 times instead of once, so subtract 2 × 8 = 16. 36 − 16 = **20**. A computer check of every pair agrees.",[482,949],[1064,1065,1066,1067,1068,1069,1070],"lines-ncert-math-6","lines-ncert-ganita-prakash-6","lines-ncert-math-7","lines-mathsisfun-line","lines-mathsisfun-parallel-perpendicular","lines-wiki-parallel-postulate","lines-mcc-laws-creases","needs_review",{"generatedBy":1073,"notes":1074},"claude-code","Draft generated by a scripted generator; numeric answers computed in Python and cross-checked by brute-force enumeration where counts are involved. Pending owner review.","74fd2bf8257820a3331257bb7ab6e7ffd743f6b6c7f04d01016fcbdd85bdccc4",{"logic:questions":1077,"source:lines-mathsisfun-line":1078,"source:lines-mathsisfun-parallel-perpendicular":1079,"source:lines-mcc-laws-creases":1080,"source:lines-ncert-ganita-prakash-6":1081,"source:lines-ncert-math-6":1082,"source:lines-ncert-math-7":1083,"source:lines-wiki-parallel-postulate":1084},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","982f31ecb1e1e17e74ba13872e9e63e218010de6c5653901ebd0db8a12413bc3","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","29053ec2c573f86831c323ce2ecab04945b506bf04bddd43c7582c9fbf98ce8b","b7ec4502e5c7912ecccfeb371ab30a19043c5598016c6d51acbc678290fc40a2","219993d3ebc010eeac3a16481eda537da1557ef707d475cbca1ba05a49dfff80","f380f754917bdc6d17093f871ff43201664a75572d28b860df518fdbf8f740be","197bbcf0b10e2a83a3b3bac20fa5140966e6e15fa8a10b0b238c50bc66a87d66","preview-7e1cbbcc4f",1789899599911]