[{"data":1,"prerenderedAt":1175},["ShallowReactive",2],{"layer:angles:understand":3},{"layer":4,"contentHash":1154,"dependencyHashes":1155,"approval":1169,"releaseId":1174},{"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":1149,"reviewStatus":1150,"authoring":1151},1,"angles","en","understand","Naming, sorting and pairing angles","Precise definitions, the seven types, and the angle pairs that let you find what you cannot measure","Define an angle as two rays with a common vertex, name it with ∠ABC, and use degrees and landmark angles. Pin down the seven types, clock and compass angles, then adjacent, complementary, supplementary, linear-pair, vertically opposite and around-a-point angles.",[13,14,15,16,17],"Define an angle, its vertex, arms, interior and exterior, and name angles correctly with three letters.","Classify any angle into the seven types using exact boundaries, and estimate sizes using landmark angles.","Find angles between clock hands at hours and half hours, and turns between the eight compass directions.","Recognise adjacent, complementary, supplementary, linear-pair and vertically opposite angles, and explain why vertically opposite angles are equal.","Find missing angles in several steps, giving a reason for each step.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Angles as turns; 90°, 180°, 360° (Discover)",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Angle lab, angle pairs ×3, sort game, match game",{"label":38,"value":39},"Key facts","90° · 180° · 360° · V.O. angles equal",[41,45,51,57,60,63,88,93,98,101,112,117,143,148,151,191,219,235,240,284,287,292,296,320,325,328,341,354,368,373,376,402,411,432,437,440,468,472,477,482,485,490,519,523,533,541,554,636,647,698,711,716,719,729,732,742,750,761,766,769,801,810,822,826,852,864,876,912,923,983,1102,1117,1123,1128,1132,1137],{"id":42,"type":43,"markdown":44},"intro","prose","In Discover you met angles as **turns**: doors, clock hands, scissors and compass directions. Now it is time to be precise, the way a mathematician is.\n\nThis layer answers four questions:\n\n1. **Exactly what is an angle**, and how do we name one so that everyone knows which angle we mean?\n2. **How big is it?** Degrees, landmarks, estimation and the seven types, with exact boundaries.\n3. **How do angles team up?** Adjacent angles, complementary and supplementary pairs, linear pairs, vertically opposite angles and angles around a point.\n4. **How do we use all this** to find an angle we cannot measure, and give a reason for every step?\n\nBy the end you will be able to look at a diagram of crossing lines, know one angle, and work out all the others, with reasons, like a detective.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-read","callout","observation","How to use this layer","Chapters 1 to 5 are about single angles; chapters 6 to 10 are about pairs and groups of angles. Worked examples show every step with a **reason** in brackets. Copy that habit: in geometry, a correct number without a reason is only half an answer.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","What exactly is an angle?","Chapter 01","1 Definition",{"id":58,"type":43,"markdown":59},"def-prose","Here is the precise definition used in NCERT textbooks and around the world:\n\n> An **angle** is the figure formed by **two rays with a common end point**.\n\n- The common end point is the **vertex** of the angle.\n- The two rays are the **arms** (or sides) of the angle.\n- The **measure** (size) of the angle is the **amount of rotation** needed to turn one arm onto the other, about the vertex.\n\nBoth ideas from Discover are inside this definition. The *two rays* describe what an angle **looks like**; the *amount of rotation* describes **how big** it is. Because rays go on forever, the arms have no length, so an angle's size can only depend on the turn.",{"id":61,"type":43,"markdown":62},"interior","Every angle divides the plane (the flat surface it is drawn on) into three parts:\n\n- the **interior** of the angle: all the points *between* the arms (on the side of the smaller opening);\n- the **exterior**: all the points outside the arms;\n- the **angle itself**: the points on the two arms, including the vertex.\n\nA point on an arm is **on** the angle, neither inside nor outside. This matters later, because when we say two angles are *adjacent*, we will need their interiors not to overlap.",{"id":64,"type":65,"caption":66,"columns":67,"rows":71},"table-point-where","table","Where is the point? For ∠PQR = 60° with vertex Q",[68,69,70],"Point","Where it is","Why",[72,76,80,84],[73,74,75],"Q","On the angle (the vertex)","It is the common end point of both arms.",[77,78,79],"A point on ray QP","On the angle","It lies on an arm.",[81,82,83],"A point between the arms, near Q","In the interior","It lies inside the 60° opening.",[85,86,87],"A point on the far side of Q, opposite the opening","In the exterior","It is outside both arms, in the reflex region.",{"id":89,"type":47,"variant":90,"title":91,"markdown":92},"misc-arms","misconception","“This angle is bigger because it is drawn bigger”","Textbook pictures are drawn at different sizes, and arms are often drawn longer just to fit a label. **An angle drawn with longer arms is not bigger.** If you trace ∠A on thin paper and it fits exactly on ∠B, turning and sliding as needed, the angles are equal, whatever the lengths of the drawn arms.\n\nA second trap: an angle drawn \"the other way up\" or tilted is still the same angle. A 40° angle pointing left, right or upwards is always 40°.",{"id":94,"type":53,"title":95,"eyebrow":96,"navLabel":97},"ch02","Naming angles","Chapter 02","2 Naming angles",{"id":99,"type":43,"markdown":100},"naming","There are three ways to name an angle. Suppose the vertex is **B**, one arm passes through point **A** and the other arm passes through point **C**.\n\n1. **Three letters, vertex in the middle:** ∠ABC or ∠CBA. The sign **∠** means *angle*. The middle letter must always be the vertex. ∠ABC and ∠CBA are the same angle (you can read the arms in either order).\n2. **Just the vertex:** ∠B. This is fine when only one angle has its vertex at B.\n3. **A number or small letter** written inside the angle, near the vertex: ∠1, ∠2, or ∠x. Handy in diagrams with many angles.\n\nWe write the **measure** of the angle as **∠ABC = 50°** (some books write m∠ABC = 50°).",{"id":102,"type":103,"title":104,"problem":105,"steps":106},"we-naming","worked_example","Name every angle at a crowded vertex","Three rays start from point O: ray OA, ray OB and ray OC, with ray OB lying between the other two. ∠AOB = 25° and ∠BOC = 40°. Name all the angles at O and give their sizes. Why can we not call any of them ∠O?",[107,108,109,110,111],"The rays in order are OA, OB, OC. Any two rays make an angle, so there are **three** angles at O.","∠AOB (or ∠BOA) = **25°** (given).","∠BOC (or ∠COB) = **40°** (given).","∠AOC is the whole opening from OA to OC. Because OB lies between them, ∠AOC = ∠AOB + ∠BOC = 25° + 40° = **65°**.","**∠O** would not tell us which of the three angles we mean, so we must use three letters with O in the middle.",{"id":113,"type":47,"variant":114,"title":115,"markdown":116},"careful-middle","careful","The vertex goes in the middle, always","∠ABC and ∠BAC are **different angles**. ∠ABC has its vertex at B; ∠BAC has its vertex at A. In a triangle ABC, these are two different corners. When you write an angle name, say to yourself: \"middle letter = corner\".",{"id":118,"type":119,"itemId":120,"prompt":121,"check":122,"hints":138,"feedback":140},"pr-naming","practice","angles.und-naming","In triangle PQR, which name means the angle at **corner Q**?",{"kind":123,"options":124,"correct":137},"choice",[125,128,131,134],{"id":126,"label":127},"a","∠QPR",{"id":129,"label":130},"b","∠PQR",{"id":132,"label":133},"c","∠PRQ",{"id":135,"label":136},"d","∠QRP",[129],[139],"The vertex letter goes in the middle.",{"correct":141,"incorrect":142},"Yes: ∠PQR has Q in the middle, so its vertex is Q. (∠RQP would also be correct.)","Look at the middle letter. Only ∠PQR has Q in the middle. ∠QPR is at P, and both ∠PRQ and ∠QRP are at R.",{"id":144,"type":53,"title":145,"eyebrow":146,"navLabel":147},"ch03","Degrees and landmark angles","Chapter 03","3 Degrees",{"id":149,"type":43,"markdown":150},"degrees","A **degree** is one 360th of a full turn: **1° = 1⁄360 of a full turn**. For finer work, each degree is split into **60 minutes** (60′) and each minute into **60 seconds** (60″). Map-makers and astronomers still use these, which is why a place's position is written like 28° 36′ N for Delhi.\n\nYou will learn to measure angles exactly with a protractor in the [measuring and constructing angles topic](\u002Ftopics\u002Fconstructing-angles). For now the goal is a good **eye**: to be able to glance at an angle and say \"about 60°\" or \"a bit more than a right angle\". Estimating first is also the best way to catch protractor mistakes, such as reading the wrong scale.",{"id":152,"type":65,"caption":153,"columns":154,"rows":158},"table-landmarks","Landmark angles to carry in your head",[155,156,157],"Angle","How to picture it","Fraction of a full turn",[159,163,167,171,175,179,183,187],[160,161,162],"30°","One hour-gap on a clock (12 to 1)","1⁄12",[164,165,166],"45°","Half a right angle; a square folded along its diagonal","1⁄8",[168,169,170],"60°","Each corner of an equilateral triangle; 12 to 2 on a clock","1⁄6",[172,173,174],"90°","A square corner; 12 to 3 on a clock","¼",[176,177,178],"120°","Between blades of a 3-blade fan; 12 to 4 on a clock","⅓",[180,181,182],"135°","A right angle plus half a right angle","⅜",[184,185,186],"180°","A straight line","½",[188,189,190],"270°","Three right angles","¾",{"id":192,"type":193,"title":194,"items":195},"steps-estimate","steps","How to estimate an angle",[196,199,203,207,211,215],{"title":197,"text":198},"Find the vertex and arms","Put your finger on the corner. Trace the two arms.",{"title":200,"tag":201,"text":202},"Compare with 90°","right angle?","Is the opening smaller than a square corner, about the same, or bigger?",{"title":204,"tag":205,"text":206},"Compare with 180°","straight line?","If bigger than 90°, is it smaller than a straight line? If not, it is reflex.",{"title":208,"tag":209,"text":210},"Halve the gap","45° or 135°","Is it closer to 0° or 90° (acute), or closer to 90° or 180° (obtuse)? Use 45° and 135° as halfway marks.",{"title":212,"tag":213,"text":214},"Use 30° steps","30, 60, 120, 150","Picture clock hour-gaps to refine: 30°, 60°, 120° and 150° are easy to see.",{"title":216,"tag":217,"text":218},"Say your estimate","about …°","Give a round number. Being within 10° is a very good eye.",{"id":220,"type":221,"component":222,"componentVersion":5,"config":223,"objective":229,"textAlternative":230,"help":231},"lab-estimate","interactive","angle-lab",{"modes":224,"allowReflex":227,"rounds":228},[225,226],"classify","estimate",true,8,"Name randomly tilted angles by type, then estimate their size in degrees; your score depends on how close you are.","Two games. In **classify**, the lab shows an angle tilted at random and you choose its type from zero, acute, right, obtuse, straight, reflex and complete. In **estimate**, the lab shows an angle with no numbers; you type your estimate, the lab reveals the true size, draws your guess as a dashed arm marked “you”, and gives more points the closer you were. Points, streaks and a best score are kept.\n\nGood strategy, as in the steps above: compare with a right angle (90°) and a straight line (180°) first, then use 45°, 135° and the 30° clock steps to refine. An angle a little wider than a square corner, about a third of the way to a straight line, is about 90° + 30° = 120°. For a reflex angle, estimate the small opening and subtract from 360°: a small opening of about 50° means a reflex angle of about 310°.",{"hints":232},[233,234],"First decide the type; that narrows the range.","For reflex angles, estimate the small opening and subtract from 360°.",{"id":236,"type":53,"title":237,"eyebrow":238,"navLabel":239},"ch04","The seven types, with exact boundaries","Chapter 04","4 Seven types",{"id":241,"type":65,"caption":242,"columns":243,"rows":248},"table-types","The seven types of angle. “Between” here means strictly between: 90° itself is right, not acute or obtuse",[244,245,246,247],"Type","Measure","Turn","Test",[249,254,259,264,269,274,279],[250,251,252,253],"Zero","0°","No turn","Arms lie on top of each other, with no rotation",[255,256,257,258],"Acute","more than 0°, less than 90°","Less than a quarter turn","Fits inside a right-angle checker",[260,261,262,263],"Right","= 90°","Quarter turn","Fits a right-angle checker exactly",[265,266,267,268],"Obtuse","more than 90°, less than 180°","More than a quarter, less than a half turn","Wider than a checker, arms not in a line",[270,271,272,273],"Straight","= 180°","Half turn","Arms point in opposite directions and form a line",[275,276,277,278],"Reflex","more than 180°, less than 360°","More than a half, less than a full turn","The “long way round” opening",[280,281,282,283],"Complete","= 360°","Full turn","Rotated all the way round back onto itself",{"id":285,"type":43,"markdown":286},"perpendicular","Two special words go with right angles:\n\n- Two lines that meet at a right angle are called **perpendicular**, written with the sign **⊥**. If line AB is perpendicular to line CD, we write AB ⊥ CD.\n- The four angles where two perpendicular lines cross are **all** right angles.\n\nAnd two special facts about types:\n\n- A **straight angle is two right angles**: 90° + 90° = 180°.\n- A **complete angle is four right angles** or two straight angles: 4 × 90° = 2 × 180° = 360°.",{"id":288,"type":47,"variant":289,"title":290,"markdown":291},"nuance-zero-complete","nuance","Zero and complete look the same, but are not","A zero angle and a complete angle both have one arm lying exactly on the other. The difference is the **turn**: zero means no rotation, complete means one full rotation. In a still picture you cannot tell them apart; you need to know the movement. A clock's minute hand at 1:00 and 2:00 is in the same place, but between those times it turned through a complete angle.",{"id":293,"type":47,"variant":90,"title":294,"markdown":295},"misc-reflex-drawn","“There is only one angle between two arms”","Two rays from a vertex always make **two** angles: the smaller one and the reflex one going the other way. They add to 360°. When a question says \"the angle between\", it usually means the smaller one, but when you are asked for a **reflex** angle, give 360° minus the smaller one. For clock hands at 8:00, the smaller angle is 120° and the reflex angle is 240°.",{"id":297,"type":119,"itemId":298,"prompt":299,"check":300,"hints":315,"feedback":317},"pr-type-reflex","angles.und-type-range","Which of these angles are **obtuse**? Choose all that apply: 89°, 90°, 91°, 179°, 180°, 181°.",{"kind":123,"options":301,"correct":314},[302,304,305,307,309,311],{"id":126,"label":303},"89°",{"id":129,"label":172},{"id":132,"label":306},"91°",{"id":135,"label":308},"179°",{"id":310,"label":184},"e",{"id":312,"label":313},"f","181°",[132,135],[316],"Obtuse means strictly more than 90° and strictly less than 180°.",{"correct":318,"incorrect":319},"Right: only 91° and 179° are obtuse. 89° is acute, 90° is right, 180° is straight and 181° is reflex.","Obtuse angles are strictly between 90° and 180°. So 91° and 179° qualify. The boundaries 90° (right) and 180° (straight) have their own names.",{"id":321,"type":53,"title":322,"eyebrow":323,"navLabel":324},"ch05","Clock angles, including half past","Chapter 05","5 Clock angles",{"id":326,"type":43,"markdown":327},"clock-hands","Two numbers let you find the angle between clock hands at almost any time.\n\n- The **minute hand** goes round once (360°) in 60 minutes, so it turns **360 ÷ 60 = 6° every minute**.\n- The **hour hand** goes from one number to the next (30°) in 60 minutes, so it turns **30 ÷ 60 = 0.5° every minute**, or half a degree.\n\nOn the hour, the minute hand is at 12 and the hour hand is exactly on a number, so the angle is simply gaps × 30°. But at half past, the hour hand has moved **half-way** to the next number, 15° further on. That is why the hands at 3:30 do **not** make a right angle, even though they point \"at 3 and 6\".",{"id":329,"type":103,"title":330,"problem":331,"steps":332,"help":338},"we-330","The angle at 3:30","Find the smaller angle between the hands of a clock at **3:30**.",[333,334,335,336,337],"Measure every position clockwise from 12.","Minute hand: 30 minutes × 6° = **180°** (pointing at 6).","Hour hand: at 3:00 it is at 3 × 30° = 90°. In 30 minutes it moves a further 30 × 0.5° = 15°. So it is at 90° + 15° = **105°**.","Angle between the hands = 180° − 105° = **75°**, an acute angle.","Sense check: the hour hand is half-way between 3 and 4, and the minute hand is at 6, which is two and a half gaps away: 2.5 × 30° = 75°. ✓",{"simplerExplanation":339,"anotherExample":340},"At half past, the hour hand has gone half-way to the next number. So count gaps carefully, including the half gap.","At 9:30, the hour hand is half-way from 9 to 10 and the minute hand is at 6: that is 3.5 gaps, 3.5 × 30° = 105°.",{"id":342,"type":343,"prompt":344,"options":345,"explanation":353},"predict-1215","prediction","What is the smaller angle between the clock hands at **12:15**?",[346,348,350,352],{"id":126,"label":347},"Exactly 90°",{"id":129,"label":349},"A little less than 90°",{"id":132,"label":351},"A little more than 90°",{"id":135,"label":251},"**A little less than 90°.** The minute hand is at 3, which is 15 × 6° = 90° from 12. But the hour hand has also crept forward from 12 by 15 × 0.5° = 7.5°. So the gap is 90° − 7.5° = **82.5°**. The hour hand never stands still!",{"id":355,"type":119,"itemId":356,"prompt":357,"check":358,"hints":363,"feedback":365},"pr-clock-630","angles.und-clock-630","Find the smaller angle between the hands of a clock at **6:30**.",{"kind":359,"answer":360,"tolerance":361,"unit":362},"number",15,0,"°",[364],"Where is the minute hand at half past? Where is the hour hand: exactly on 6, or half-way to 7?",{"correct":366,"incorrect":367},"Right: the minute hand is at 6 (180°) and the hour hand is half-way from 6 to 7 (195°), so they are 15° apart.","The minute hand points at 6, which is 180° from 12. The hour hand has moved half a gap past 6: 180° + 15° = 195°. The difference is 195° − 180° = 15°.",{"id":369,"type":53,"title":370,"eyebrow":371,"navLabel":372},"ch06","Directions and turns","Chapter 06","6 Directions",{"id":374,"type":43,"markdown":375},"compass","A compass rose has four **cardinal directions** 90° apart: **N, E, S, W**. Half-way between each pair are the four **intercardinal** (in-between) directions: **NE, SE, SW, NW**. Each is 45° from its neighbours, because 360° ÷ 8 = 45°.\n\nTurns have a size **and** a direction: **clockwise** (the way clock hands move: N → E → S → W) or **anticlockwise** (N → W → S → E). A clockwise turn of 90° from north takes you to east; an anticlockwise turn of 90° from north takes you to west.\n\nHere is a handy fact: turning **x° clockwise** ends in the same place as turning **(360 − x)° anticlockwise**. For example, 90° clockwise and 270° anticlockwise both take you from north to east.",{"id":377,"type":65,"caption":378,"columns":379,"rows":383},"table-compass","The eight compass points, measured clockwise from north",[380,381,382],"Direction","Clockwise from N","Anticlockwise from N",[384,386,389,391,394,396,398,400],[385,251,251],"N",[387,164,388],"NE","315°",[390,172,188],"E",[392,180,393],"SE","225°",[395,184,184],"S",[397,393,180],"SW",[399,188,172],"W",[401,388,164],"NW",{"id":403,"type":103,"title":404,"problem":405,"steps":406},"we-compass","Smallest turn between two directions","A traffic police officer faces **north-east (NE)** and must turn to face **south (S)**. What is the smallest turn, and in which direction?",[407,408,409,410],"Positions clockwise from north: NE is at 45°, S is at 180°.","Clockwise from NE to S: 180° − 45° = **135°**.","Anticlockwise from NE to S would be 360° − 135° = 225°.","So the smallest turn is **135° clockwise**. Check: NE → E (45°) → SE (45°) → S (45°) = 135°. ✓",{"id":412,"type":119,"itemId":413,"prompt":414,"check":415,"hints":426,"feedback":429},"pr-compass","angles.und-compass","Kabir faces **south-west (SW)** and turns **90° anticlockwise**. Which direction does he face now?",{"kind":123,"options":416,"correct":425},[417,419,421,423],{"id":126,"label":418},"North-west",{"id":129,"label":420},"South-east",{"id":132,"label":422},"North-east",{"id":135,"label":424},"West",[129],[427,428],"Anticlockwise goes N → NW → W → SW → S → SE → E …","90° is two 45° steps.",{"correct":430,"incorrect":431},"Yes: from SW, two 45° steps anticlockwise are S then SE.","Anticlockwise from SW: 45° brings you to S, another 45° to SE. So 90° anticlockwise from SW is south-east.",{"id":433,"type":53,"title":434,"eyebrow":435,"navLabel":436},"ch07","Adjacent angles","Chapter 07","7 Adjacent angles",{"id":438,"type":43,"markdown":439},"adjacent","Now we start working with **pairs** of angles. The first idea is about where two angles sit.\n\nTwo angles are **adjacent** (neighbours) when **all three** of these are true:\n\n1. They have a **common vertex**.\n2. They have a **common arm**.\n3. Their **interiors do not overlap**: the common arm lies between the other two arms.\n\nThe door and the frame, the two parts of a pizza cut by one extra slice, the angles on either side of a clock's hour hand between the other two hands: these are adjacent angles.",{"id":441,"type":65,"caption":442,"columns":443,"rows":449},"table-adjacent","Adjacent or not? Rays from O: OA, OB and OC, with OB between OA and OC",[444,445,446,447,448],"Pair","Common vertex?","Common arm?","Interiors separate?","Adjacent?",[450,456,461,465],[451,452,453,454,455],"∠AOB and ∠BOC","Yes (O)","Yes (OB)","Yes","**Yes**",[457,452,458,459,460],"∠AOB and ∠AOC","Yes (OA)","No: ∠AOB is inside ∠AOC","No",[462,463,464,454,460],"∠ABC and ∠BCD (corners of a square)","No (B and C)","Share segment BC",[466,460,460,467,460],"Two angles of 30° in different drawings","—",{"id":469,"type":47,"variant":90,"title":470,"markdown":471},"misc-adjacent-sum","“Adjacent angles add up to 90° or 180°”","Adjacent is only about **position**: sharing a vertex and an arm, side by side. Adjacent angles can add up to anything: 20° + 35° = 55°, or 100° + 150° = 250°. The sum is 180° only in the special case where the outer arms form a straight line (a **linear pair**), and 90° only when they fill a right angle.",{"id":473,"type":47,"variant":474,"title":475,"markdown":476},"aha-adjacent-add","aha","The useful fact about adjacent angles","When two angles are adjacent, **their sizes add**: ∠AOB + ∠BOC = ∠AOC. This sounds obvious, but it is the engine behind every missing-angle problem. If you know the whole angle and one part, subtract to find the other part.",{"id":478,"type":53,"title":479,"eyebrow":480,"navLabel":481},"ch08","Complementary and supplementary angles","Chapter 08","8 Comp. and supp.",{"id":483,"type":43,"markdown":484},"comp-supp","Now two ideas about the **sum** of two angles, whether or not they are adjacent.\n\n- Two angles are **complementary** if their sum is **90°**. Each is the **complement** of the other. The complement of an angle x is **90° − x**.\n- Two angles are **supplementary** if their sum is **180°**. Each is the **supplement** of the other. The supplement of an angle x is **180° − x**.\n\nFor example, 30° and 60° are complementary (30 + 60 = 90). 110° and 70° are supplementary (110 + 70 = 180). They do **not** have to touch: one could be at a corner of your book and the other at a corner of the blackboard.",{"id":486,"type":47,"variant":487,"title":488,"markdown":489},"memory-cs","example","Remembering which is which","**C** comes before **S** in the alphabet, and **90** comes before **180**. So **C**omplementary = 90°, **S**upplementary = 180°.\n\nAnother trick: a **C**orner is 90°; a **S**traight line is 180°.",{"id":491,"type":65,"caption":492,"columns":493,"rows":497},"table-comp-supp","Complements and supplements, computed as 90° − x and 180° − x",[494,495,496],"Angle x","Complement (90° − x)","Supplement (180° − x)",[498,502,506,507,508,512,514,516,517],[499,500,501],"10°","80°","170°",[503,504,505],"25°","65°","155°",[164,164,180],[168,160,176],[509,510,511],"72°","18°","108°",[303,513,306],"1°",[172,515,172],"none (x is not less than 90°)",[176,515,168],[518,515,160],"150°",{"id":520,"type":47,"variant":289,"title":521,"markdown":522},"nuance-no-complement","Not every angle has a partner","Only an **acute** angle has a complement, because the complement must be more than 0°. A right angle would need 0°, and 120° would need −30°, which is not an angle in this sense. Any angle smaller than 180° has a supplement. And two **acute** angles can never be supplementary, because their sum is less than 90° + 90° = 180°.",{"id":524,"type":103,"title":525,"problem":526,"steps":527},"we-comp","An angle and its complement","An angle is **twice** its complement. Find the angle.",[528,529,530,531,532],"Let the complement be c. Then the angle is 2c (it is twice its complement).","The angle and its complement add up to 90° (definition of complementary): c + 2c = 90°.","So 3c = 90°, and c = 90° ÷ 3 = 30°.","The angle = 2 × 30° = **60°**.","Check: 60° + 30° = 90° ✓ and 60° is twice 30° ✓.",{"id":534,"type":103,"title":535,"problem":536,"steps":537},"we-supp","Equal supplementary angles","Two supplementary angles are **equal**. What is each angle?",[538,539,540],"The two angles add up to 180° (definition of supplementary).","They are equal, so each is 180° ÷ 2 = **90°**.","So equal supplementary angles are both right angles. This fact is used to prove that two lines are perpendicular.",{"id":542,"type":221,"component":543,"componentVersion":5,"config":544,"objective":548,"textAlternative":549,"help":550},"lab-complementary","angle-pairs",{"scene":545,"initialAngle":546,"challenges":547},"complementary",35,5,"Split a right angle into two parts with a movable ray, watch them always total 90°, and find five missing complements.","A right angle (marked with a small square) with a movable ray inside it, splitting it into two adjacent angles ∠a and ∠b. It starts at 35° and 55°. A live table shows both sizes, and a button highlights the complementary pair. The ray can move between 5° and 85°.\n\nDrag the ray: the two parts always add up to **90°**, for example 20° and 70°, 45° and 45°, 80° and 10°. They are complementary angles that also happen to be adjacent.\n\nIn the five challenges the lab gives one part and asks for the other; type the number of degrees. If ∠a is 25°, then ∠b = 90° − 25° = 65°. The lab explains each answer.",{"simplerExplanation":551,"hints":552},"The two parts fill a square corner, so they add to 90°.",[553],"Complement = 90 − the angle you know.",{"id":555,"type":221,"component":556,"componentVersion":5,"config":557,"objective":634,"textAlternative":635},"lab-sort-cs","sort-game",{"prompt":558,"bins":559,"items":569,"seconds":361},"Is each pair of angles complementary, supplementary or neither?",[560,563,566],{"id":561,"label":562},"comp","Complementary (sum 90°)",{"id":564,"label":565},"supp","Supplementary (sum 180°)",{"id":567,"label":568},"neither","Neither",[570,574,578,582,586,590,594,598,602,606,610,614,618,622,626,630],{"id":571,"label":572,"bin":561,"why":573},"p0","30° and 60°","30° + 60° = 90°, a right angle.",{"id":575,"label":576,"bin":561,"why":577},"p1","45° and 45°","45° + 45° = 90°, a right angle.",{"id":579,"label":580,"bin":564,"why":581},"p2","120° and 60°","120° + 60° = 180°, a straight angle.",{"id":583,"label":584,"bin":564,"why":585},"p3","100° and 80°","100° + 80° = 180°, a straight angle.",{"id":587,"label":588,"bin":567,"why":589},"p4","40° and 40°","40° + 40° = 80°, which is neither 90° nor 180°.",{"id":591,"label":592,"bin":561,"why":593},"p5","15° and 75°","15° + 75° = 90°, a right angle.",{"id":595,"label":596,"bin":564,"why":597},"p6","135° and 45°","135° + 45° = 180°, a straight angle.",{"id":599,"label":600,"bin":564,"why":601},"p7","90° and 90°","90° + 90° = 180°, a straight angle.",{"id":603,"label":604,"bin":564,"why":605},"p8","70° and 110°","70° + 110° = 180°, a straight angle.",{"id":607,"label":608,"bin":567,"why":609},"p9","25° and 55°","25° + 55° = 80°, which is neither 90° nor 180°.",{"id":611,"label":612,"bin":561,"why":613},"p10","89° and 1°","89° + 1° = 90°, a right angle.",{"id":615,"label":616,"bin":567,"why":617},"p11","50° and 140°","50° + 140° = 190°, which is neither 90° nor 180°.",{"id":619,"label":620,"bin":561,"why":621},"p12","65° and 25°","65° + 25° = 90°, a right angle.",{"id":623,"label":624,"bin":564,"why":625},"p13","160° and 20°","160° + 20° = 180°, a straight angle.",{"id":627,"label":628,"bin":567,"why":629},"p14","60° and 60°","60° + 60° = 120°, which is neither 90° nor 180°.",{"id":631,"label":632,"bin":561,"why":633},"p15","37° and 53°","37° + 53° = 90°, a right angle.","Sort 16 angle pairs by their sum: 90°, 180° or something else.","A sorting game with three bins: complementary (sum 90°), supplementary (sum 180°) and neither.\n\nComplementary: 30° and 60°, 45° and 45°, 15° and 75°, 89° and 1°, 65° and 25°, 37° and 53°.\nSupplementary: 120° and 60°, 100° and 80°, 135° and 45°, 90° and 90°, 70° and 110°, 160° and 20°.\nNeither: 40° and 40° (80°), 25° and 55° (80°), 50° and 140° (190°), 60° and 60° (120°).\n\nThe only method needed is to add the pair and compare with 90° and 180°. Notice that 60° appears in all three bins, depending on its partner: being complementary or supplementary is about a **pair**, never one angle on its own.",{"id":637,"type":119,"itemId":638,"prompt":639,"check":640,"hints":642,"feedback":644},"pr-supp","angles.und-supp","Find the supplement of **47°**.",{"kind":359,"answer":641,"tolerance":361,"unit":362},133,[643],"Supplement = 180° − angle.",{"correct":645,"incorrect":646},"Correct: 180° − 47° = 133°.","Supplementary angles add to 180°, so the supplement is 180° − 47° = 133°.",{"id":648,"type":649,"title":650,"prompt":651,"options":652},"explorer-pairs-city","explorer","Angle pairs around town","Pick a scene to see which angle pairs it contains.",[653,665,676,687],{"id":654,"label":655,"chain":656,"badge":661,"note":664},"crossing","Crossroads",[657,658,659,660],"Two straight roads cross","Four angles at the centre","Opposite angles equal","Neighbours add to 180°",{"text":662,"tone":663},"Vertically opposite + linear pairs","yes","Where two straight roads cross, the four corners come as two equal pairs. If one corner is 70°, the corner diagonally across is also 70°, and the other two are 110° each. Traffic engineers prefer crossings close to 90°, where all four corners are equal and drivers can see clearly.",{"id":666,"label":667,"chain":668,"badge":673,"note":675},"level","Railway level crossing",[669,670,671,672],"Road meets the track","Track is a straight line","Road makes a linear pair","Sum 180°",{"text":674,"tone":663},"Linear pair","A road crossing a straight railway line makes two angles on each side of the road. On each side they form a linear pair adding to 180°. The two rails are parallel, so the road meets both at the same angle (you will see why in Deepen).",{"id":677,"label":678,"chain":679,"badge":684,"note":686},"clock","Clock at 2:00",[680,681,682,683],"Hands at 12 and 2","Angle 60°","Add the 30° to 3:00","60° + 30° = 90°",{"text":685,"tone":663},"Complementary","At 2:00 the hands make 60°. The angle from the hour hand on to the 3 is another 30°. These two adjacent angles fill the right angle from 12 to 3, so they are complementary.",{"id":688,"label":689,"chain":690,"badge":695,"note":697},"kite","Kite string and pole",[691,692,693,694],"Pole stands straight up","Ground is a line","String leans at an angle","Angles at the base add up",{"text":696,"tone":663},"Around a point \u002F on a line","A flagpole stands at 90° to flat ground. If a kite string tied at the base makes 35° with the ground, it makes 90° − 35° = 55° with the pole (complementary), and 180° − 35° = 145° with the ground on the other side (linear pair).",{"id":699,"type":343,"prompt":700,"options":701,"explanation":710},"predict-complement-big","An angle gets **bigger**. What happens to its complement and its supplement?",[702,704,706,708],{"id":126,"label":703},"Both get bigger",{"id":129,"label":705},"Both get smaller, by the same amount",{"id":132,"label":707},"The complement gets smaller, the supplement gets bigger",{"id":135,"label":709},"Nothing changes","**Both get smaller by the same amount.** If the angle grows by 10°, then 90° − x and 180° − x each shrink by 10°, because the totals (90° and 180°) are fixed. For example: 30° has complement 60° and supplement 150°; 40° has complement 50° and supplement 140°.",{"id":712,"type":53,"title":713,"eyebrow":714,"navLabel":715},"ch09","Linear pairs and vertically opposite angles","Chapter 09","9 Linear pairs, V.O.",{"id":717,"type":43,"markdown":718},"linear-pair","Stand a ray on a straight line, like a lane meeting a main road. It makes two adjacent angles whose outer arms point in **opposite directions**, forming a straight line. Such a pair is called a **linear pair**.\n\n**The angles of a linear pair are supplementary: they add up to 180°.** Reason: together they make the straight angle, which is 180°.\n\nSo a linear pair is *adjacent* **and** *supplementary* at the same time. Be careful with the reverse: two supplementary angles are not a linear pair unless they are also adjacent with outer arms on a line. 120° at one corner of your desk and 60° on the blackboard are supplementary but not a linear pair.",{"id":720,"type":221,"component":543,"componentVersion":5,"config":721,"objective":723,"textAlternative":724,"help":725},"lab-linear",{"scene":717,"initialAngle":722,"challenges":547},65,"Tilt the ray on a straight line, check that a linear pair always adds to 180°, and solve five missing-angle challenges.","A straight line with a ray standing on it, making the linear pair ∠a and ∠b, starting at 65° and 115°. A live table shows both sizes as you drag, and a button highlights the linear pair.\n\nOne angle grows exactly as fast as the other shrinks, and the sum stays **180°**. When the ray is upright the pair is 90° and 90°, and the ray is perpendicular to the line.\n\nIn the five challenges one angle is given and the other is asked for: type the number of degrees. If ∠a = 40°, then ∠b = 180° − 40° = 140°. The lab explains each answer with the reason “they sit side by side on a straight line”.",{"hints":726},[727,728],"The two angles fill a straight line, 180°.","Check your answer: the two numbers should add up to 180.",{"id":730,"type":43,"markdown":731},"vert-opp","Now cross two straight lines, like the two blades of open scissors or an X. Four angles form around the crossing point. The angles **directly across** from each other, which share only the vertex, are called **vertically opposite angles**. (*Vertical* here means *at the vertex*, not *up and down*.)\n\n**Vertically opposite angles are always equal.** Here is why. Call the four angles ∠1, ∠2, ∠3 and ∠4, going round, so ∠1 and ∠3 are opposite each other, and ∠2 and ∠4 are opposite each other.\n\n- ∠1 and ∠2 form a linear pair on one line, so ∠1 + ∠2 = 180°.\n- ∠2 and ∠3 form a linear pair on the other line, so ∠2 + ∠3 = 180°.\n- Both ∠1 and ∠3 are \"180° minus ∠2\". So **∠1 = ∠3**.\n\nThe same argument with ∠1 as the shared partner shows **∠2 = ∠4**. This is why the angle between the handles of scissors always matches the angle between the blades.",{"id":733,"type":103,"title":734,"problem":735,"steps":736},"we-vo","All four angles from one","Two lines cross at O. One of the four angles is **38°**. Find the other three.",[737,738,739,740,741],"Call the angles going round ∠1 = 38°, ∠2, ∠3, ∠4.","∠2 = 180° − 38° = **142°** (∠1 and ∠2 are a linear pair).","∠3 = ∠1 = **38°** (vertically opposite angles are equal).","∠4 = ∠2 = **142°** (vertically opposite angles are equal).","Check: 38 + 142 + 38 + 142 = 360°, a full turn around the point. ✓",{"id":743,"type":221,"component":543,"componentVersion":5,"config":744,"objective":748,"textAlternative":749},"lab-intersecting",{"scene":745,"initialAngle":746,"challenges":747},"intersecting",50,6,"Rotate one of two crossing lines, watch opposite angles stay equal, and find missing angles at the crossing.","Two straight lines crossing at O make four angles, ∠a, ∠b, ∠c and ∠d going round, starting at 50°, 130°, 50° and 130°. A live table shows all four sizes as you rotate one line, and buttons highlight the vertically opposite pairs (∠a with ∠c, ∠b with ∠d) or the linear pairs (neighbours).\n\nWhatever you do, opposite angles stay equal and neighbours add to 180°. When the lines are perpendicular, all four are 90°.\n\nIn the six challenges the lab gives one angle and asks for another: the one opposite (equal) or a neighbour (180° minus). If ∠a = 75°, then ∠c = 75° and ∠b = 105°. Type the number; the lab explains which fact it used.",{"id":751,"type":119,"itemId":752,"prompt":753,"check":754,"hints":756,"feedback":758},"pr-vo","angles.und-vo","Two roads cross. One angle at the crossing is **115°**. What is the angle **vertically opposite** to it?",{"kind":359,"answer":755,"tolerance":361,"unit":362},115,[757],"Vertically opposite angles are equal.",{"correct":759,"incorrect":760},"Right: vertically opposite angles are equal, so it is 115°.","The angle directly across the crossing is vertically opposite, and those angles are always equal: 115°. (The angles next to it would be 180° − 115° = 65°.)",{"id":762,"type":53,"title":763,"eyebrow":764,"navLabel":765},"ch10","Angles around a point, and finding missing angles","Chapter 10","10 Missing angles",{"id":767,"type":43,"markdown":768},"around-point","If several angles fill all the space around one point, with no gaps and no overlaps, they make a full turn. So:\n\n**Angles around a point add up to 360°.**\n\nTogether with the facts from earlier chapters, you now have a toolkit. The skill in geometry is choosing the right tool and **saying which one you used**.",{"id":770,"type":65,"caption":771,"columns":772,"rows":776},"table-toolkit","The angle toolkit: facts you can quote as reasons",[773,774,775],"Fact","In symbols","Short reason to write",[777,781,785,789,793,797],[778,779,780],"Adjacent angles add","∠AOB + ∠BOC = ∠AOC","adjacent angles",[782,783,784],"Angles in a right angle","sum = 90°","complementary angles",[786,787,788],"Angles on a straight line","sum = 180°","linear pair \u002F angles on a line",[790,791,792],"Vertically opposite angles","equal","vert. opp. angles",[794,795,796],"Angles around a point","sum = 360°","angles at a point",[798,799,800],"Right angle symbol","□ in the corner means 90°","given right angle",{"id":802,"type":103,"title":803,"problem":804,"steps":805},"we-point","Three angles around a point","Three angles around a point are **x**, **2x** and **150°**. Find x.",[806,807,808,809],"The three angles fill the space around the point, so x + 2x + 150° = 360° (angles at a point).","3x + 150° = 360°, so 3x = 360° − 150° = 210°.","x = 210° ÷ 3 = **70°**. The angles are 70°, 140° and 150°.","Check: 70 + 140 + 150 = 360°. ✓",{"id":811,"type":103,"title":812,"problem":813,"steps":814,"help":820},"we-chain","A chain of reasons","Lines AB and CD cross at O. Ray OE is perpendicular to AB, on the same side as C. ∠COE = 28°, and ray OC lies between OE and OA. Find ∠AOC, ∠BOD and ∠AOD.",[815,816,817,818,819],"∠AOE = 90° (OE ⊥ AB).","∠AOC = ∠AOE − ∠COE = 90° − 28° = **62°** (adjacent angles: OC lies inside ∠AOE).","∠BOD = ∠AOC = **62°** (vertically opposite angles).","∠AOD = 180° − ∠AOC = 180° − 62° = **118°** (linear pair on line CD).","Check around O: 62 + 118 + 62 + 118 = 360°. ✓",{"simplerExplanation":821},"Use the right angle first, then the equal opposite angle, then the straight line.",{"id":823,"type":47,"variant":114,"title":824,"markdown":825},"careful-reasons","Never assume from a picture","Diagrams in questions are often **not drawn to scale**. An angle that *looks* like 90° is only 90° if it is marked with a small square or the question says so. Two lines that *look* straight are straight only if they are named as lines. Use given facts and toolkit reasons, not your eyes.",{"id":827,"type":221,"component":828,"componentVersion":5,"config":829,"objective":850,"textAlternative":851},"lab-match-props","match-pairs",{"prompt":830,"mode":831,"pairs":832},"Match each angle pair to its property.","connect",[833,836,839,841,843,845,847],{"a":834,"b":835},"Complementary angles","Add up to 90°",{"a":837,"b":838},"Supplementary angles","Add up to 180°",{"a":674,"b":840},"Adjacent, and add up to 180°",{"a":790,"b":842},"Always equal",{"a":794,"b":844},"Add up to 360°",{"a":434,"b":846},"Common vertex and arm, no overlap",{"a":848,"b":849},"Perpendicular lines","Meet at 90°","Connect each type of angle pair or group to the property that defines it.","A matching game with seven pairs: complementary angles ↔ add up to 90°; supplementary angles ↔ add up to 180°; linear pair ↔ adjacent, and add up to 180°; vertically opposite angles ↔ always equal; angles around a point ↔ add up to 360°; adjacent angles ↔ common vertex and arm, no overlap; perpendicular lines ↔ meet at 90°.",{"id":853,"type":119,"itemId":854,"prompt":855,"check":856,"hints":858,"feedback":861},"pr-point","angles.und-point","Four angles around a point are **80°, 95°, 110°** and **x**. Find **x**.",{"kind":359,"answer":857,"tolerance":361,"unit":362},75,[859,860],"Angles around a point add to 360°.","Add the three you know, then subtract from 360.",{"correct":862,"incorrect":863},"Correct: 80 + 95 + 110 = 285, and 360 − 285 = 75°.","The four angles make a full turn: x = 360° − (80° + 95° + 110°) = 360° − 285° = 75°.",{"id":865,"type":119,"itemId":866,"prompt":867,"check":868,"hints":870,"feedback":873},"pr-linear-ratio","angles.und-linear-ratio","Two angles form a linear pair. One is **20° more** than the other. Find the **larger** angle.",{"kind":359,"answer":869,"tolerance":361,"unit":362},100,[871,872],"The two add to 180°.","If the smaller is s, the larger is s + 20°.",{"correct":874,"incorrect":875},"Yes: s + (s + 20) = 180 gives 2s = 160, s = 80°, so the larger is 100°.","Let the smaller be s. Then s + s + 20 = 180, so 2s = 160 and s = 80°. The larger angle is 80° + 20° = 100°.",{"id":877,"type":65,"caption":878,"columns":879,"rows":883},"table-mixups","Common mix-ups and how to fix them",[880,881,882],"Mix-up","Why it happens","Fix",[884,888,892,896,900,904,908],[885,886,887],"Writing ∠BAC for the angle at B","Forgetting the vertex rule","Middle letter = vertex. Say it aloud.",[889,890,891],"Complementary = 180°","Both words start the same way","C before S, 90 before 180: Corner and Straight.",[893,894,895],"Adjacent angles must add to 180°","Linear pairs are adjacent","Adjacent is about position only; a linear pair is a special case.",[897,898,899],"Assuming a 90° angle because it looks square","Diagrams are not to scale","Only use the right-angle mark or given facts.",[901,902,903],"Giving 60° when asked for a reflex angle","Only seeing the small opening","Reflex angle = 360° − small angle.",[905,906,907],"Clock at 3:30 is 90°","Forgetting the hour hand moves","The hour hand moves 0.5° per minute: 3:30 gives 75°.",[909,910,911],"Vertically opposite means up-and-down","The everyday meaning of vertical","It means sharing a vertex, directly across a crossing.",{"id":913,"type":119,"itemId":914,"prompt":915,"check":916,"hints":918,"feedback":920},"pr-adjacent","angles.und-adjacent","∠POQ = 35° and ∠QOR = 50°, and they are adjacent. What is ∠POR?",{"kind":359,"answer":917,"tolerance":361,"unit":362},85,[919],"Adjacent angles add up.",{"correct":921,"incorrect":922},"Right: ∠POR = 35° + 50° = 85°.","Because the angles are adjacent (common arm OQ, no overlap), ∠POR is the whole angle: 35° + 50° = 85°.",{"id":924,"type":925,"title":926,"terms":927},"glossary-understand","glossary","Words for angle pairs",[928,932,936,940,944,946,949,953,957,961,965,969,972,976,980],{"term":929,"meaning":930,"example":931},"∠ (angle sign)","The symbol for angle. ∠ABC is the angle with vertex B and arms through A and C.","∠ABC = 40°",{"term":933,"meaning":934,"example":935},"measure of an angle","The size of an angle in degrees: the amount of rotation from one arm to the other.","The measure of a right angle is 90°.",{"term":937,"meaning":938,"example":939},"minute of arc (′)","One sixtieth of a degree.","28° 36′",{"term":941,"meaning":942,"example":943},"perpendicular (⊥)","Meeting at a right angle.","The sides of a square are perpendicular.",{"term":780,"meaning":945,"example":451},"Two angles with a common vertex and a common arm, whose interiors do not overlap.",{"term":784,"meaning":947,"example":948},"Two angles whose measures add up to 90°.","25° and 65°",{"term":950,"meaning":951,"example":952},"complement","The angle that makes a given angle up to 90°: 90° − x.","The complement of 20° is 70°.",{"term":954,"meaning":955,"example":956},"supplementary angles","Two angles whose measures add up to 180°.","130° and 50°",{"term":958,"meaning":959,"example":960},"supplement","The angle that makes a given angle up to 180°: 180° − x.","The supplement of 20° is 160°.",{"term":962,"meaning":963,"example":964},"linear pair","Two adjacent angles whose outer arms form a straight line. They are supplementary.","A lane meeting a straight road makes a linear pair.",{"term":966,"meaning":967,"example":968},"vertically opposite angles","The pairs of opposite angles formed when two lines intersect. They are equal.","Scissor blades and handles.",{"term":796,"meaning":970,"example":971},"Angles that together fill all the space around a point. They add up to 360°.","The slices of a pizza at its centre.",{"term":973,"meaning":974,"example":975},"intersecting lines","Lines that cross at one point.","The two lines of an X.",{"term":977,"meaning":978,"example":979},"cardinal directions","The four main compass directions: north, east, south and west, 90° apart.","N, E, S, W",{"term":981,"meaning":982,"example":422},"intercardinal directions","The directions half-way between the cardinal ones, 45° from each: NE, SE, SW, NW.",{"id":984,"type":985,"title":986,"questions":987},"quiz-understand","quiz","Pairs, names and reasons",[988,1001,1014,1025,1038,1051,1064,1073,1084,1093],{"itemId":989,"prompt":990,"options":991,"correct":129,"why":1000},"angles.und-q-name","In ∠XYZ, which point is the vertex?",[992,994,996,998],{"id":126,"label":993},"X",{"id":129,"label":995},"Y",{"id":132,"label":997},"Z",{"id":135,"label":999},"It depends on the picture","The vertex is always the middle letter: Y.",{"itemId":1002,"prompt":1003,"options":1004,"correct":126,"why":1013},"angles.und-q-comp","What is the complement of 64°?",[1005,1007,1009,1011],{"id":126,"label":1006},"26°",{"id":129,"label":1008},"36°",{"id":132,"label":1010},"116°",{"id":135,"label":1012},"296°","90° − 64° = 26°.",{"itemId":1015,"prompt":1016,"options":1017,"correct":132,"why":1024},"angles.und-q-supp","What is the supplement of 64°?",[1018,1019,1021,1022],{"id":126,"label":1006},{"id":129,"label":1020},"106°",{"id":132,"label":1010},{"id":135,"label":1023},"126°","180° − 64° = 116°.",{"itemId":1026,"prompt":1027,"options":1028,"correct":132,"why":1037},"angles.und-q-lp","Which statement is always true about a linear pair?",[1029,1031,1033,1035],{"id":126,"label":1030},"Both angles are acute",{"id":129,"label":1032},"The angles are equal",{"id":132,"label":1034},"The angles add up to 180°",{"id":135,"label":1036},"The angles add up to 90°","A linear pair makes a straight line, so its angles are supplementary: sum 180°.",{"itemId":1039,"prompt":1040,"options":1041,"correct":129,"why":1050},"angles.und-q-vo","Two lines cross; one angle is 72°. What are the other three angles, going round?",[1042,1044,1046,1048],{"id":126,"label":1043},"72°, 72°, 72°",{"id":129,"label":1045},"108°, 72°, 108°",{"id":132,"label":1047},"18°, 72°, 18°",{"id":135,"label":1049},"108°, 108°, 72°","The neighbours are 180° − 72° = 108° (linear pairs), and the opposite angle is equal: 72°. Going round: 108°, 72°, 108°.",{"itemId":1052,"prompt":1053,"options":1054,"correct":126,"why":1063},"angles.und-q-adj","∠AOB = 40° and ∠AOC = 40°, with B and C on opposite sides of OA. Are ∠AOB and ∠AOC adjacent?",[1055,1057,1059,1061],{"id":126,"label":1056},"Yes: common vertex O, common arm OA, no overlap",{"id":129,"label":1058},"No: equal angles cannot be adjacent",{"id":132,"label":1060},"No: they do not add to 180°",{"id":135,"label":1062},"Only if they add to 90°","Adjacent only needs a common vertex, a common arm and no overlap. Sizes do not matter.",{"itemId":1065,"prompt":1066,"options":1067,"correct":129,"why":1072},"angles.und-q-clock","What is the smaller angle between the hands at 9:00?",[1068,1069,1070,1071],{"id":126,"label":160},{"id":129,"label":172},{"id":132,"label":184},{"id":135,"label":188},"From 9 to 12 is 3 gaps: 3 × 30° = 90°. (The reflex angle is 270°.)",{"itemId":1074,"prompt":1075,"options":1076,"correct":132,"why":1083},"angles.und-q-two-acute","Can two acute angles be supplementary?",[1077,1079,1081],{"id":126,"label":1078},"Yes, always",{"id":129,"label":1080},"Yes, sometimes",{"id":132,"label":1082},"No, never","Each acute angle is less than 90°, so two of them add to less than 180°.",{"itemId":1085,"prompt":1086,"options":1087,"correct":132,"why":1092},"angles.und-q-point","Five equal angles fill the space around a point. How big is each?",[1088,1089,1090,1091],{"id":126,"label":1008},{"id":129,"label":168},{"id":132,"label":509},{"id":135,"label":172},"360° ÷ 5 = 72°.",{"itemId":1094,"prompt":1095,"options":1096,"correct":129,"why":1101},"angles.und-q-compass","What is the smaller angle between north-west and east?",[1097,1098,1099,1100],{"id":126,"label":172},{"id":129,"label":180},{"id":132,"label":184},{"id":135,"label":393},"Clockwise from NW (315°) to E (90°) is 45° + 90° = 135°. The other way round is 225°, so the smaller angle is 135°.",{"id":1103,"type":1104,"title":1105,"points":1106},"cheat-sheet","summary","Cheat sheet",[1107,1108,1109,1110,1111,1112,1113,1114,1115,1116],"**Angle:** two rays with a common end point (vertex). Its **measure** is the rotation from one arm to the other. Name it ∠ABC with the **vertex in the middle**, or ∠B if there is only one angle at B.","**1° = 1⁄360 of a full turn.** Landmarks: 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, 360°.","**Types:** zero 0°, acute 0°–90°, right 90°, obtuse 90°–180°, straight 180°, reflex 180°–360°, complete 360° (boundaries belong to right, straight and complete).","**Clock:** minute hand 6° per minute, hour hand 0.5° per minute. At 3:30 the angle is 75°, not 90°.","**Compass:** N, E, S, W are 90° apart; NE, SE, SW, NW sit half-way (45°). x° clockwise = (360 − x)° anticlockwise.","**Adjacent:** common vertex, common arm, no overlap. Their sizes add.","**Complementary:** sum 90°, complement = 90° − x. **Supplementary:** sum 180°, supplement = 180° − x.","**Linear pair:** adjacent + outer arms on a line → sum 180°.","**Vertically opposite angles are equal** (both are 180° minus the same neighbour).","**Angles around a point:** sum 360°. Always give a **reason** for every step; never trust how a diagram looks.",{"id":1118,"type":1119,"conceptId":1120,"relation":1121,"explanation":1122},"conn-lines","connection","lines","helps_understand","Intersecting and perpendicular lines from the lines topic create the angle pairs in this layer.",{"id":1124,"type":1119,"conceptId":1125,"relation":1126,"explanation":1127},"conn-construct","constructing-angles","applied_in","Measuring with a protractor and constructing 60°, 90° or 45° with a compass are in the measuring and constructing angles topic.",{"id":1129,"type":1119,"conceptId":1130,"relation":1121,"explanation":1131},"conn-ops","four-operations","Every missing-angle problem is an addition or subtraction from 90°, 180° or 360°, and sometimes a division into equal parts.",{"id":1133,"type":1119,"conceptId":1134,"relation":1135,"explanation":1136},"conn-shape","shape-and-space","related_to","The corners of polygons are angles; a square's sides are perpendicular and a rectangle has four right angles.",{"id":1138,"type":1139,"sourceIds":1140},"sources-understand","sources",[1141,1142,1143,1144,1145,1146,1147,1148],"angles-ncert-class7-lines-angles","angles-mathsisfun-angles","angles-khan-angle-basics","angles-euclid-i15","angles-wiki-degree","angles-wiki-clock-angle","angles-ncert-class6-lines-angles","angles-mathsisfun-complementary",[1141,1142,1143,1144,1145,1146,1147,1148],"needs_review",{"generatedBy":1152,"notes":1153},"claude-code","Draft generated with Python generator scripts; every angle computed and asserted. Pending owner review.","cd6af75e4621119b559608cfab6476532dd3f22f1e59590ef14ec3cd3a60de0f",{"logic:practice":1156,"component:angle-lab@1":1157,"component:angle-pairs@1":1158,"component:sort-game@1":1159,"component:match-pairs@1":1160,"source:angles-euclid-i15":1161,"source:angles-khan-angle-basics":1162,"source:angles-mathsisfun-angles":1163,"source:angles-mathsisfun-complementary":1164,"source:angles-ncert-class6-lines-angles":1165,"source:angles-ncert-class7-lines-angles":1166,"source:angles-wiki-clock-angle":1167,"source:angles-wiki-degree":1168},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","cbb509e9a575a1b2ef133804e3450487de14952c1df934747fe56ff5d10d847e","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","d6d698be825d6f17fd34f22e885f8d2a9c0bee7634e87a36474309b3ab7e8c82","68741673c0dc6c96760fde888327818c508788257b5306bad48ab148615541a4","2dab8c7c8736bbad61c4a7223c297c3b250b20399d5b5544c02434cab39a0c9c","7656284145acec3d6ead6ded762ed0eef2bb3137446c527350b7b999562580df","36960f0a34d1e3f73475a665e4b12c1140dfbbeac1c1b2772bf71a8d3b9522a5","548a49fd09ac1b7cec7c74b038e1a87c5d3add389d6f27376ac054da17d20972","141f5516f8832cc0eea67b34a5726cd12ed7309036b178da74f753ca6e4ae706","79b5ead855ebdf6f15ad061e87052809bfff58b83c8dfb9f7a1f592debdf62e9",{"state":1170,"reviewer":1171,"selfReview":227,"reviewedAt":1172,"method":1173},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597207]