[{"data":1,"prerenderedAt":1172},["ShallowReactive",2],{"layer:angles:extend":3},{"layer":4,"contentHash":1147,"dependencyHashes":1148,"approval":1166,"releaseId":1171},{"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":1142,"reviewStatus":1143,"authoring":1144},1,"angles","en","extend","Angles at work and play","Clock formulas, exterior angles, bearings, radians, real-world angles, olympiad puzzles and projects","Use |30h − 5.5m| for any clock time, prove and use the exterior angle property, navigate with bearings and runway numbers, meet the radian, see angles in ramps, ladders, bowling and pie charts, and tackle olympiad-style angle chases, projects and open questions.",[13,14,15,16,17],"Derive and use the clock-angle formula, including finding when hands overlap or make right angles.","Prove the exterior angle property of a triangle and use exterior angles of polygons.","Use three-figure bearings and back bearings, and explain runway numbering.","Convert between degrees and radians for common angles, and explain what a radian is.","Solve multi-step angle-chasing puzzles and apply angles to real-life designs, data and careers.",55,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 55 minutes",{"label":29,"value":30},"Prior knowledge","Transversals, triangle sum (Deepen)",{"label":32,"value":33},"Chapters","9",{"label":35,"value":36},"Labs","Sort, match, angle pairs ×2, angle lab",{"label":38,"value":39},"Big formulas","|30h − 5.5m| · 180° = π rad",[41,45,51,57,60,79,127,138,153,167,172,187,192,204,209,212,221,224,236,247,257,268,273,276,317,328,333,338,346,357,362,402,411,485,527,536,547,552,564,569,572,608,634,639,650,664,669,672,683,691,701,713,724,736,748,757,765,774,786,798,805,814,819,871,876,907,911,916,921,925,978,1094,1107,1113,1117,1122,1126],{"id":42,"type":43,"markdown":44},"intro","prose","You now know what angles are, how they pair up, and why the rules are true. This layer takes angles **out of the textbook**: into clocks and compasses, airports and cricket grounds, ramps, ladders, roofs and pie charts, and into the puzzles that appear in mathematics olympiads.\n\nYou will meet a formula that finds the angle between clock hands at **any** time, the exterior angle property of triangles, the bearings that pilots and sailors use, a completely different unit for angles called the **radian**, and a set of puzzles that need every tool you have. The layer ends with projects to try, careers that use angles every day, and questions nobody has fully answered.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-extend","callout","observation","How to use this layer","Chapters stand on their own, so pick the ones that interest you. Puzzles are marked by difficulty. Try each one for at least five minutes before opening the solution: the struggle is where the learning happens.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","A formula for any clock time","Chapter 01","1 Clock formula",{"id":58,"type":43,"markdown":59},"clock-formula","At h hours and m minutes, measure both hands clockwise from 12.\n\n- The **minute hand** turns 6° per minute, so it is at **6m** degrees.\n- The **hour hand** turns 30° per hour plus 0.5° per minute, so it is at **30h + 0.5m** degrees.\n\nThe angle between them is the difference: (30h + 0.5m) − 6m = 30h − 5.5m. Since we only care about the size, we take the value without its sign:\n\n**angle = |30h − 5.5m|**\n\nThe bars | | mean “ignore any minus sign” (the absolute value). If the answer is more than 180°, subtract it from 360° to get the smaller angle. Use h from 0 to 11, writing 12 o'clock as h = 0.",{"id":61,"type":62,"items":63},"formulas-clock","formulas",[64,67,70,73,76],{"expression":65,"caption":66},"angle = |30h − 5.5m|","Angle between the hands at h:m. If it exceeds 180°, the smaller angle is 360° minus it.",{"expression":68,"caption":69},"minute hand: 6° per min","360° in 60 minutes.",{"expression":71,"caption":72},"hour hand: 0.5° per min","30° in 60 minutes.",{"expression":74,"caption":75},"gain: 5.5° per min","The minute hand gains 6 − 0.5 = 5.5° on the hour hand every minute.",{"expression":77,"caption":78},"overlap every 65 5⁄11 min","360 ÷ 5.5 = 720⁄11 minutes between overlaps: 11 overlaps in 12 hours.",{"id":80,"type":81,"caption":82,"columns":83,"rows":89},"table-clock-formula","table","The formula at work (smaller angle shown)",[84,85,86,87,88],"Time","30h","5.5m","|30h − 5.5m|","Smaller angle",[90,95,100,106,111,117,122],[91,92,93,94,94],"4:20","120","110","10°",[96,97,98,99,99],"7:45","210","247.5","37.5°",[101,102,103,104,105],"10:10","300","55","245°","115°",[107,108,109,110,110],"2:30","60","165","105°",[112,113,114,115,116],"9:15","270","82.5","187.5°","172.5°",[118,119,120,121,121],"5:24","150","132","18°",[123,124,125,126,126],"11:55","330","302.5","27.5°",{"id":128,"type":129,"title":130,"problem":131,"steps":132},"we-clock-when","worked_example","When do the hands first overlap after 4:00?","Find the exact time after 4:00 when the hour and minute hands first point the same way.",[133,134,135,136,137],"Overlap means angle = 0, so 30 × 4 − 5.5m = 0.","5.5m = 120, so m = 120 ÷ 5.5 = 1200 ÷ 55 = 240 ÷ 11.","240 ÷ 11 = **21 9⁄11 minutes** (because 11 × 21 = 231, remainder 9).","9⁄11 of a minute is 540⁄11 ≈ 49.1 seconds. So the hands overlap at about **4:21:49**.","Check: at 4:21 9⁄11 the minute hand is at 6 × 240⁄11 = 1440⁄11 ≈ 130.9°; the hour hand at 120 + 0.5 × 240⁄11 = 120 + 120⁄11 ≈ 130.9°. ✓",{"id":139,"type":140,"itemId":141,"prompt":142,"check":143,"hints":148,"feedback":150},"pr-clock-formula","practice","angles.ext-clock-formula","Use the formula to find the smaller angle between the hands at **8:20**.",{"kind":144,"answer":145,"tolerance":146,"unit":147},"number",130,0,"°",[149],"h = 8, m = 20: compute |240 − 110|.",{"correct":151,"incorrect":152},"Correct: |240 − 110| = 130°.","30 × 8 = 240 and 5.5 × 20 = 110, so the angle is |240 − 110| = 130°. That is less than 180°, so it is already the smaller angle.",{"id":154,"type":140,"itemId":155,"prompt":156,"check":157,"hints":161,"feedback":164},"pr-clock-right","angles.ext-clock-right","Between 2:00 and 3:00, at how many minutes past 2 are the hands first at **right angles**? Give your answer to one decimal place.",{"kind":144,"answer":158,"tolerance":159,"unit":160},27.3,0.1,"min",[162,163],"You need |60 − 5.5m| = 90.","At 2:00 the minute hand is behind; it must get 90° ahead: 5.5m − 60 = 90.",{"correct":165,"incorrect":166},"Yes: 5.5m = 150, so m = 150 ÷ 5.5 = 27 3⁄11 ≈ 27.3 minutes, about 2:27:16.","Solve 5.5m − 60 = 90 (the minute hand ends up 90° ahead). 5.5m = 150, m = 300⁄11 = 27 3⁄11 ≈ 27.3 minutes.",{"id":168,"type":47,"variant":169,"title":170,"markdown":171},"aha-clock-sym","aha","Mirror times","The angle at 4:20 is 10°, and the angle at 7:40 is also 10°. Times that are mirror images across the 12–6 line always give the same angle, because the whole clock face is just reflected. Check with the formula: |30 × 7 − 5.5 × 40| = |210 − 220| = 10°.",{"id":173,"type":174,"prompt":175,"options":176,"explanation":186},"predict-315","prediction","At 3:15 the minute hand points exactly at 3. Are the two hands exactly on top of each other?",[177,180,183],{"id":178,"label":179},"a","Yes, both point at 3: the angle is 0°",{"id":181,"label":182},"b","No, the hour hand has moved on: the angle is 7.5°",{"id":184,"label":185},"c","No, the angle is 15°","**No: the angle is 7.5°.** In 15 minutes the hour hand has crept 15 × 0.5° = 7.5° past the 3. The formula agrees: |30 × 3 − 5.5 × 15| = |90 − 82.5| = 7.5°. The hands do not overlap until about 3:16:22 (90 ÷ 5.5 = 16 4⁄11 minutes).",{"id":188,"type":47,"variant":189,"title":190,"markdown":191},"nuance-24h","nuance","Railway time and the 24-hour clock","Indian Railways timetables and airline tickets use the 24-hour clock: 15:40 means 3:40 p.m. The clock face only has 12 hours, so before using the formula, subtract 12 from any hour above 12: at 15:40, use h = 3 and m = 40, giving |90 − 220| = 130°. At 00:30 (just after midnight), use h = 0: |0 − 165| = 165°.",{"id":193,"type":140,"itemId":194,"prompt":195,"check":196,"hints":198,"feedback":201},"pr-clock-straight","angles.ext-clock-straight","How many times between 12 noon and 12 midnight do the hour and minute hands form a **straight line** (point in exactly opposite directions)?",{"kind":144,"answer":197,"tolerance":146},11,[199,200],"The minute hand must be 180° ahead of the hour hand, once per lap.","How many laps does the minute hand gain in 12 hours?",{"correct":202,"incorrect":203},"Right: the minute hand gains 11 laps in 12 hours, so it is exactly opposite the hour hand 11 times (6:00 is one of them).","The minute hand gains 5.5° per minute, and 360° per lap, so it gains 11 laps in 12 hours. It is exactly 180° ahead once in each lap: 11 times.",{"id":205,"type":53,"title":206,"eyebrow":207,"navLabel":208},"ch02","The exterior angle of a triangle","Chapter 02","2 Exterior angles",{"id":210,"type":43,"markdown":211},"exterior","Extend one side of a triangle beyond a corner. The angle between the extended side and the next side is called an **exterior angle** of the triangle. The two angles of the triangle at the *other* corners are the **interior opposite angles**.\n\n**Exterior angle property:** an exterior angle of a triangle equals the **sum of the two interior opposite angles**.\n\n**Proof.** Let the triangle be ABC, and extend BC beyond C to D. Call the exterior angle ∠ACD.\n\n1. ∠ACB + ∠ACD = 180° (linear pair on line BD).\n2. ∠A + ∠B + ∠ACB = 180° (angle sum of a triangle).\n3. Both expressions equal 180°, so ∠ACB + ∠ACD = ∠A + ∠B + ∠ACB.\n4. Take ∠ACB away from both sides: **∠ACD = ∠A + ∠B**. ∎\n\nA handy consequence: an exterior angle is always **bigger than each** interior opposite angle.",{"id":213,"type":129,"title":214,"problem":215,"steps":216},"we-exterior","A ladder of exterior angles","In triangle PQR, side QR is extended to S. ∠PRS = 128° and ∠P = 55°. Find ∠Q and ∠PRQ.",[217,218,219,220],"∠PRS is the exterior angle at R, and its interior opposite angles are ∠P and ∠Q.","∠PRS = ∠P + ∠Q (exterior angle property), so 128° = 55° + ∠Q, giving ∠Q = **73°**.","∠PRQ = 180° − 128° = **52°** (linear pair with the exterior angle).","Check: 55 + 73 + 52 = 180°. ✓",{"id":222,"type":43,"markdown":223},"exterior-polygon","Now take **every** exterior angle of a shape, one at each corner, all going the same way round. Remember the robot from Investigate, walking round a shape and turning at each corner? Its turns are exactly the exterior angles. It ends facing the way it started, so:\n\n**The exterior angles of any convex polygon add up to 360°.**\n\nFor a triangle, check: each exterior angle is 180° minus the interior angle, so the three exterior angles total 3 × 180° − 180° = 360°. ✓ For a regular polygon with n sides, each exterior angle is 360° ÷ n, which gives a fast way to find the interior angle: 180° − 360° ÷ n. A regular 20-gon has exterior angles of 18° and interior angles of 162°.",{"id":225,"type":140,"itemId":226,"prompt":227,"check":228,"hints":230,"feedback":233},"pr-sides","angles.ext-sides","Each interior angle of a regular polygon is **156°**. How many sides does it have?",{"kind":144,"answer":229,"tolerance":146},15,[231,232],"Find each exterior angle first: 180° − 156°.","The exterior angles add to 360°.",{"correct":234,"incorrect":235},"Right: each exterior angle is 24°, and 360 ÷ 24 = 15 sides.","Exterior angle = 180° − 156° = 24°. The exterior angles total 360°, so there are 360 ÷ 24 = 15 of them: 15 sides.",{"id":237,"type":174,"prompt":238,"options":239,"explanation":246},"predict-exterior-acute","Can an exterior angle of a triangle be **acute**?",[240,242,244],{"id":178,"label":241},"No, exterior angles are always obtuse",{"id":181,"label":243},"Yes, when the triangle's angle at that corner is obtuse",{"id":184,"label":245},"Only in a right-angled triangle","**Yes.** The exterior angle and the interior angle at the same corner form a linear pair. If the interior angle is obtuse, say 130°, the exterior angle is 180° − 130° = 50°, which is acute. And since the exterior angle equals the sum of the two other angles, 50° is also the sum of the triangle's two acute angles. A triangle can have at most one acute exterior angle, because it has at most one obtuse angle.",{"id":248,"type":129,"title":249,"problem":250,"steps":251},"we-exterior-alg","Exterior angle with algebra","The exterior angle at C of triangle ABC is (3x + 10)°. The interior opposite angles are ∠A = x° and ∠B = 50°. Find x and all three angles of the triangle.",[252,253,254,255,256],"Exterior angle = sum of interior opposite angles: 3x + 10 = x + 50.","2x = 40, so **x = 20**.","∠A = 20°, ∠B = 50°, exterior angle = 3 × 20 + 10 = 70°.","∠C = 180° − 70° = **110°** (linear pair with the exterior angle).","Check: 20 + 50 + 110 = 180°. ✓",{"id":258,"type":140,"itemId":259,"prompt":260,"check":261,"hints":262,"feedback":265},"pr-ext-sum","angles.ext-ext-sum","A triangle has angles 50°, 60° and 70°. Find the **largest** of its three exterior angles (one at each corner).",{"kind":144,"answer":145,"tolerance":146,"unit":147},[263,264],"Each exterior angle = 180° − the interior angle at that corner.","The smallest interior angle gives the largest exterior angle.",{"correct":266,"incorrect":267},"Correct: the exterior angles are 130°, 120° and 110°, which add to 360°.","Exterior angles: 180 − 50 = 130°, 180 − 60 = 120°, 180 − 70 = 110°. The largest is 130°. (They add to 360°, as they should.)",{"id":269,"type":53,"title":270,"eyebrow":271,"navLabel":272},"ch03","Bearings: how pilots and sailors use angles","Chapter 03","3 Bearings",{"id":274,"type":43,"markdown":275},"bearings","Navigators give directions as **bearings**: an angle measured **clockwise from north**, always written with **three figures**. East is **090°**, south is **180°**, west is **270°**, and north-east is **045°**.\n\nThe **back bearing** (the way back) differs by exactly 180°: add 180° if the bearing is less than 180°, otherwise subtract 180°. If a ship sails from Kochi on a bearing of 250°, the bearing from the ship back to Kochi is 250° − 180° = **070°**.\n\nA real example: **airport runways** are numbered from their direction measured from **magnetic** north. ICAO's rule (Annex 14, §5.2.2.4) is that the two-digit number is the whole number nearest one tenth of that bearing: round the bearing to the nearest 10° and drop the final zero. A runway pointing roughly east, about 090°, is called **runway 09**. Its other end points roughly west, 270°, so it is **runway 27**. The two numbers at the ends of any runway always differ by **18**, because the two directions differ by 180°. Many runways carry pairs like 09\u002F27, 14\u002F32 or 10\u002F28.",{"id":277,"type":81,"caption":278,"columns":279,"rows":284},"table-bearings","Compass directions as three-figure bearings and their back bearings",[280,281,282,283],"Direction","Bearing","Back bearing","Runway numbers",[285,290,295,300,305,308,311,314],[286,287,288,289],"North","000°","180°","36\u002F18",[291,292,293,294],"North-east","045°","225°","about 04\u002F22 or 05\u002F23",[296,297,298,299],"East","090°","270°","09\u002F27",[301,302,303,304],"South-east","135°","315°","about 13\u002F31 or 14\u002F32",[306,288,287,307],"South","18\u002F00",[309,293,292,310],"South-west","about 22\u002F04 or 23\u002F05",[312,298,297,313],"West","27\u002F09",[315,303,302,316],"North-west","about 31\u002F13 or 32\u002F14",{"id":318,"type":140,"itemId":319,"prompt":320,"check":321,"hints":323,"feedback":325},"pr-back-bearing","angles.ext-back-bearing","A plane flies from Chennai to a town on a bearing of **063°**. What bearing must it fly to return directly to Chennai?",{"kind":144,"answer":322,"tolerance":146,"unit":147},243,[324],"Add 180° because 063° is less than 180°.",{"correct":326,"incorrect":327},"Correct: 63° + 180° = 243°.","The return trip points the opposite way: 063° + 180° = 243°.",{"id":329,"type":47,"variant":330,"title":331,"markdown":332},"misc-bearing-maths","misconception","“Bearings work like maths angles”","In maths, angles are usually measured **anticlockwise from the east** (the positive x-direction on a graph). Bearings are measured **clockwise from north**. So the direction a mathematician calls 30° (north of east) is a bearing of **060°**, and a bearing of 030° is 60° in maths. To convert either way: bearing = 90° − maths angle, adding 360° if the result is negative.",{"id":334,"type":47,"variant":335,"title":336,"markdown":337},"model-limit-bearing","model_limit","Real navigation is messier","Flat-map bearings are a model. On the round Earth, the shortest route between two distant cities (a *great-circle route*) keeps changing its bearing along the way, which is why flights from Delhi to North America pass over the Arctic. Compasses also point to **magnetic** north, which is a little different from **true** north (the North Pole), and the difference changes from place to place and year to year.",{"id":339,"type":129,"title":340,"problem":341,"steps":342},"we-bearing-turn","Angle between two bearings","From a lighthouse, ship P is on a bearing of **040°** and ship Q is on a bearing of **130°**. What is the angle PLQ at the lighthouse L? Later, a boat sailing on a bearing of 300° turns onto 030°. Through how many degrees did it turn, and which way?",[343,344,345],"Angle PLQ = 130° − 40° = **90°**, a right angle.","Boat: turning clockwise from 300° to 360° is 60°, then on to 030° is another 30°.","Total: **90° clockwise**. (Turning anticlockwise would be 270°, the long way round.)",{"id":347,"type":140,"itemId":348,"prompt":349,"check":350,"hints":352,"feedback":354},"pr-bearing-angle","angles.ext-bearing-angle","From a village, a temple is on a bearing of **075°** and a well is on a bearing of **200°**. What is the angle at the village between the directions to the temple and the well?",{"kind":144,"answer":351,"tolerance":146,"unit":147},125,[353],"Both bearings are measured clockwise from north, so subtract.",{"correct":355,"incorrect":356},"Yes: 200° − 75° = 125°.","Both are measured clockwise from the same north line, so the angle between them is 200° − 75° = 125°.",{"id":358,"type":53,"title":359,"eyebrow":360,"navLabel":361},"ch04","Angles doing real jobs","Chapter 04","4 Real-life angles",{"id":363,"type":81,"caption":364,"columns":365,"rows":369},"table-real-angles","Angles at work (numbers are typical values, rounded)",[366,367,368],"Where","Angle","Why that angle",[370,374,378,382,386,390,394,398],[371,372,373],"Wheelchair ramp, slope 1 in 12","about 4.8°","Gentle enough to push up and safe to roll down. India's Harmonised Guidelines for Universal Accessibility (2021) make 1 : 12 the steepest gradient allowed, with a landing after every 6 m, and prefer 1 : 15 where there is room.",[375,376,377],"Ladder, foot 1 unit out per 4 units up","about 76° with the ground","Too flat and the foot slides out; too steep and it tips backwards. Safety guides usually quote this setting as “about 75°”; worked out exactly it is 76°.",[379,380,381],"Cricket bowling arm","at most 15° elbow straightening","The ICC treats an action as illegal if the elbow straightens by more than 15° between the arm reaching the horizontal and the ball being released.",[383,384,385],"Scissors","blade angle = handle angle","Vertically opposite angles: open the handles 30° and the blades open 30°.",[387,388,389],"Road junctions","close to 90° preferred","When a side road meets a main road near a right angle, drivers can see both ways more easily than at a sharp Y-junction.",[391,392,393],"Roof pitch","steep in heavy rain or snow","Steeper roofs shed water and snow faster; flat roofs suit dry plains and rooftop use.",[395,396,397],"Pie chart","share × 360°","The whole circle stands for the whole group.",[399,400,401],"Stairs","around 30°–40°","Much steeper feels like a ladder; much shallower wastes floor space.",{"id":403,"type":129,"title":404,"problem":405,"steps":406},"we-pie","Angles for a pie chart","In a class of 40 students, the favourite sports are: cricket 18, football 10, badminton 8, kabaddi 4. Find the angle for each slice of a pie chart.",[407,408,409,410],"The whole class (40) is the whole circle (360°), so each student gets 360° ÷ 40 = **9°**.","Cricket: 18 × 9° = **162°**. Football: 10 × 9° = **90°**. Badminton: 8 × 9° = **72°**. Kabaddi: 4 × 9° = **36°**.","Check: 162 + 90 + 72 + 36 = 360°. ✓","Cricket's slice is obtuse (162°), football's is exactly a right angle, and the other two are acute. See the data-handling topic for more.",{"id":412,"type":413,"component":414,"componentVersion":5,"config":415,"objective":483,"textAlternative":484},"lab-sort-pie","interactive","sort-game",{"prompt":416,"bins":417,"items":433,"seconds":482},"Each item is one slice of a pie chart. What type of angle is the slice?",[418,421,424,427,430],{"id":419,"label":420},"acute","Acute",{"id":422,"label":423},"right","Right",{"id":425,"label":426},"obtuse","Obtuse",{"id":428,"label":429},"straight","Straight",{"id":431,"label":432},"reflex","Reflex",[434,438,442,446,450,454,458,462,466,470,474,478],{"id":435,"label":436,"bin":428,"why":437},"s1","Half of the class chose cricket","½ × 360° = 180°.",{"id":439,"label":440,"bin":422,"why":441},"s2","A quarter of the rupees were spent on rent","¼ × 360° = 90°.",{"id":443,"label":444,"bin":425,"why":445},"s3","30% of the rainfall came in July","0.3 × 360° = 108°.",{"id":447,"label":448,"bin":419,"why":449},"s4","10% of trains were late","0.1 × 360° = 36°.",{"id":451,"label":452,"bin":431,"why":453},"s5","75% of the votes went to one candidate","0.75 × 360° = 270°.",{"id":455,"label":456,"bin":425,"why":457},"s6","18 of 40 students chose cricket","18 × 9° = 162°.",{"id":459,"label":460,"bin":419,"why":461},"s7","1 in 6 people walk to school","360° ÷ 6 = 60°.",{"id":463,"label":464,"bin":431,"why":465},"s8","5 of 8 slices of a pizza eaten","5 × 45° = 225°.",{"id":467,"label":468,"bin":422,"why":469},"s9","2 of 8 slices left","2 × 45° = 90°.",{"id":471,"label":472,"bin":425,"why":473},"s10","40% of a shop's sales were rice","0.4 × 360° = 144°.",{"id":475,"label":476,"bin":431,"why":477},"s11","60% of a garden is lawn","0.6 × 360° = 216°.",{"id":479,"label":480,"bin":419,"why":481},"s12","1 in 20 students is left-handed","360° ÷ 20 = 18°.",90,"Work out each pie-chart slice's angle from its share and sort it by type, against a 90-second timer.","A timed sorting game (90 seconds) with bins acute, right, obtuse, straight and reflex. Each card gives a share of a whole; multiply the share by 360°.\n\nAcute: 10% of trains late (36°), 1 in 6 people walk to school (60°), 1 in 20 students left-handed (18°).\nRight: a quarter of the rupees on rent (90°), 2 of 8 pizza slices left (90°).\nObtuse: 30% of rainfall in July (108°), 18 of 40 students chose cricket (162°), 40% of sales were rice (144°).\nStraight: half the class chose cricket (180°).\nReflex: 75% of votes (270°), 5 of 8 pizza slices eaten (225°), 60% of a garden is lawn (216°).\n\nShortcut: less than a quarter is acute, exactly a quarter is right, between a quarter and a half is obtuse, a half is straight, more than a half is reflex.",{"id":486,"type":487,"title":488,"note":489,"scale":490,"rungs":491},"ladder-real","ladder","Angles in the real world, smallest to largest","Log scale: each step up is ten times bigger. Values are typical and rounded.","log",[492,496,500,504,507,509,513,517,520,523],{"label":493,"value":494,"display":495},"Width of the Moon seen from Earth",0.5,"≈ 0.5°",{"label":497,"value":498,"display":499},"Aircraft approach path to a runway",3,"≈ 3°",{"label":501,"value":502,"display":503},"Steepest wheelchair ramp, 1 in 12",4.8,"≈ 4.8°",{"label":505,"value":229,"display":506},"Bowler's allowed elbow straightening","15°",{"label":508,"value":229,"display":506},"Sun's movement across the sky in 1 hour",{"label":510,"value":511,"display":512},"Typical staircase",35,"≈ 30°–40°",{"label":514,"value":515,"display":516},"Safe ladder against a wall",76,"≈ 76°",{"label":518,"value":482,"display":519},"Corner of a page","90°",{"label":521,"value":522,"display":288},"Book lying open flat",180,{"label":524,"value":525,"display":526},"One full turn of a fan blade",360,"360°",{"id":528,"type":129,"title":529,"problem":530,"steps":531},"we-ramp","How long must the ramp be?","A school entrance is **0.6 m** above the playground. A ramp must rise no more than 1 unit for every 12 units along the ground. What is the shortest horizontal length the ramp can have? About what angle will it make with the ground?",[532,533,534,535],"Slope 1 in 12 means: horizontal length = 12 × rise.","Horizontal length = 12 × 0.6 m = **7.2 m**.","The angle depends only on the ratio 1 : 12, not on the size of the ramp, so it is about **4.8°**, a very small acute angle. (A scale drawing, 1 cm up and 12 cm along, measured with a protractor, confirms it.)","If the school could only fit a 4 m ramp, the slope would be 0.6 in 4 = 1 in 6.7, about twice as steep: too steep to push a wheelchair safely.",{"id":537,"type":140,"itemId":538,"prompt":539,"check":540,"hints":542,"feedback":544},"pr-pie-angle","angles.ext-pie-angle","In a survey of **60** families in a colony, **25** use a two-wheeler to go to work. What angle should that slice of the pie chart have?",{"kind":144,"answer":541,"tolerance":146,"unit":147},150,[543],"Each family gets 360° ÷ 60 degrees.",{"correct":545,"incorrect":546},"Correct: 360 ÷ 60 = 6° per family, and 25 × 6° = 150°.","The whole circle, 360°, stands for 60 families, so each family is 6°. The two-wheeler slice is 25 × 6° = 150°.",{"id":548,"type":47,"variant":549,"title":550,"markdown":551},"example-solar","example","Tilting solar panels","Rooftop solar panels in India face **south** (the side of the sky where the Sun spends most of the year, seen from India) and are usually tilted up from the horizontal. A common rule of thumb is to tilt a fixed panel at about the place's **latitude**, so that across the year the Sun's rays strike it as squarely as possible: about 13° in Chennai, 19° in Mumbai, 23° in Bhopal and 28° in Delhi.\n\nIt is only a rule of thumb. Studies that measure the sunlight actually arriving often find the best fixed tilt to be a few degrees **less** than the latitude, and installers change it again for the shape of the roof, for shading, and to keep enough slope for rain to wash dust off the glass.",{"id":553,"type":140,"itemId":554,"prompt":555,"check":556,"hints":558,"feedback":561},"pr-solar","angles.ext-solar","A solar panel in Bengaluru is tilted **13°** up from the flat roof. What angle does the panel make with a vertical wall it leans against at its top edge? (The wall is at 90° to the roof.)",{"kind":144,"answer":557,"tolerance":146,"unit":147},77,[559,560],"The panel, the roof and the wall form a right-angled triangle.","The two acute angles of a right-angled triangle are complementary.",{"correct":562,"incorrect":563},"Correct: 90° − 13° = 77°.","Roof and wall meet at 90°, so the triangle's other two angles add to 90°: the panel meets the wall at 90° − 13° = 77°.",{"id":565,"type":53,"title":566,"eyebrow":567,"navLabel":568},"ch05","A curiosity: the radian","Chapter 05","5 Radians",{"id":570,"type":43,"markdown":571},"radians","Degrees are a human choice. Is there a more **natural** way to measure angles, one that does not depend on the Babylonians? Mathematicians found one.\n\nDraw a circle. Take a piece of string as long as the **radius** and lay it along the edge of the circle. The angle at the centre made by that piece of arc is called **one radian**.\n\nHow many radians fit in a full turn? The distance round a circle (its circumference) is **2π × radius**, where π ≈ 3.14159. So exactly **2π radians** fit in a full turn: about 6.283 radians. That means:\n\n- **360° = 2π radians**, so **180° = π radians**.\n- **1 radian = 180° ÷ π ≈ 57.3°**, a bit less than 60°.\n\nWhy bother? In higher mathematics and physics (the motion of pendulums, waves and planets), formulas become much simpler in radians. For example, the length of an arc is just **radius × angle in radians**. Your calculator has a “RAD” mode for exactly this.",{"id":573,"type":81,"caption":574,"columns":575,"rows":579},"table-radians","Common angles in degrees and radians",[576,577,578],"Degrees","Radians (exact)","Radians (approx.)",[580,584,588,592,595,598,601,604],[581,582,583],"30°","π⁄6","0.524",[585,586,587],"45°","π⁄4","0.785",[589,590,591],"60°","π⁄3","1.047",[519,593,594],"π⁄2","1.571",[288,596,597],"π","3.142",[298,599,600],"3π⁄2","4.712",[526,602,603],"2π","6.283",[605,606,607],"≈ 57.3°","1","1.000",{"id":609,"type":413,"component":610,"componentVersion":5,"config":611,"objective":632,"textAlternative":633},"lab-match-rad","match-pairs",{"prompt":612,"mode":613,"pairs":614},"Match each angle in degrees to the same angle in radians.","memory",[615,617,619,621,623,625,628,630],{"a":288,"b":616},"π radians",{"a":519,"b":618},"π⁄2 radians",{"a":526,"b":620},"2π radians",{"a":589,"b":622},"π⁄3 radians",{"a":585,"b":624},"π⁄4 radians",{"a":626,"b":627},"about 57.3°","1 radian",{"a":581,"b":629},"π⁄6 radians",{"a":298,"b":631},"3π⁄2 radians","A memory game: flip cards to pair each degree measure with the same angle in radians.","A memory game with eight pairs of face-down cards: 180° ↔ π radians; 90° ↔ π⁄2; 360° ↔ 2π; 60° ↔ π⁄3; 45° ↔ π⁄4; about 57.3° ↔ 1 radian; 30° ↔ π⁄6; 270° ↔ 3π⁄2. The key fact is 180° = π radians; every other pair follows by multiplying or dividing. For example, 60° is a third of 180°, so it is π⁄3.",{"id":635,"type":47,"variant":636,"title":637,"markdown":638},"careful-calc","careful","Check your calculator mode","Scientific calculators and spreadsheet software can work in degrees or radians. A calculation done in the wrong mode gives a wildly wrong answer with no warning. Look for a small **D** or **DEG** (degrees) or **R** or **RAD** (radians) on the screen before any angle calculation.",{"id":640,"type":174,"prompt":641,"options":642,"explanation":649},"predict-radian-size","Without calculating: is one radian bigger or smaller than the angle in an equilateral triangle (60°)?",[643,645,647],{"id":178,"label":644},"Bigger",{"id":181,"label":646},"Smaller",{"id":184,"label":648},"Exactly the same","**Slightly smaller** (about 57.3°). Picture an equilateral triangle with one corner at the centre of a circle and two corners on the circle: its straight third side is exactly one radius long. The **curved** arc between those two corners is a little longer than the straight side. An arc of exactly one radius must therefore be a little shorter, so it makes a slightly smaller angle than 60°.",{"id":651,"type":140,"itemId":652,"prompt":653,"check":654,"hints":658,"feedback":661},"pr-radian-frac","angles.ext-radian-frac","120° is what **fraction of π** radians? (For example, 90° is ½ of π radians.)",{"kind":655,"numerator":656,"denominator":498,"acceptEquivalent":657},"fraction",2,true,[659,660],"180° = π radians.","120 ÷ 180 = ?",{"correct":662,"incorrect":663},"Right: 120 ÷ 180 = ⅔, so 120° = 2π⁄3 radians.","Since 180° is π radians, 120° is 120⁄180 = ⅔ of π radians, written 2π⁄3.",{"id":665,"type":53,"title":666,"eyebrow":667,"navLabel":668},"ch06","Olympiad-style puzzles","Chapter 06","6 Puzzles",{"id":670,"type":43,"markdown":671},"puzzles-intro","These puzzles combine several angle facts. The key skill is **angle chasing**: finding one angle after another, each with a reason, until you reach the target. When stuck, try drawing an extra line: a parallel, a diagonal or an extension.",{"id":673,"type":129,"title":674,"problem":675,"steps":676},"we-star","Puzzle 1 (medium): the five-pointed star","Draw any five-pointed star (a pentagram) with five straight lines; it need not be regular. What is the sum of the five angles at its tips?",[677,678,679,680,681,682],"Label the tips A, B, C, D, E in order round the star. The five lines are AC, CE, EB, BD and DA. Tip A is between lines AC and AD.","Line EB crosses AD at Q and AC at P, cutting off a small triangle APQ whose angle at A is the tip angle ∠A.","Triangle QDB is made by lines DA, DB and EB. Its angles at D and B are the tip angles ∠D and ∠B. Its exterior angle at Q is ∠AQP, so **∠AQP = ∠B + ∠D** (exterior angle property).","In the same way, triangle PCE is made by lines CA, CE and EB, with tip angles ∠C and ∠E. Its exterior angle at P is ∠APQ, so **∠APQ = ∠C + ∠E**.","Angle sum of triangle APQ: ∠A + ∠AQP + ∠APQ = 180°, which means **∠A + ∠B + ∠C + ∠D + ∠E = 180°**, for every such star.","For a regular star, each tip is 180° ÷ 5 = **36°**, the same 36° as the apex of the 36°–72°–72° triangle in Deepen.",{"id":684,"type":129,"title":685,"problem":686,"steps":687},"we-min-hand","Puzzle 2 (easy): how far does the minute hand turn?","Through how many degrees does the minute hand turn between 10:15 a.m. and 11:40 a.m.? And the hour hand?",[688,689,690],"Time elapsed: 10:15 to 11:15 is 60 minutes, plus 25 minutes to 11:40 = 85 minutes.","Minute hand: 85 × 6° = **510°** (a full turn plus 150°).","Hour hand: 85 × 0.5° = **42.5°**.",{"id":692,"type":129,"title":693,"problem":694,"steps":695},"we-isos-chase","Puzzle 3 (hard): a chain of isosceles triangles","In triangle ABC, AB = AC and ∠A = 20°. Point D lies on AC with BD = BC. Find ∠ABD.",[696,697,698,699,700],"AB = AC, so the base angles are equal: ∠ABC = ∠ACB = (180° − 20°) ÷ 2 = **80°** (angle sum; isosceles triangle).","In triangle BCD, BD = BC, so the angles opposite these equal sides are equal: ∠BDC = ∠BCD = 80°.","So ∠DBC = 180° − 80° − 80° = **20°** (angle sum of triangle BCD).","∠ABD = ∠ABC − ∠DBC = 80° − 20° = **60°**.","Check in triangle ABD: ∠A = 20°, ∠ABD = 60°, and ∠ADB = 180° − ∠BDC = 100° (linear pair). 20 + 60 + 100 = 180°. ✓",{"id":702,"type":140,"itemId":703,"prompt":704,"check":705,"hints":707,"feedback":710},"pr-puzzle-bisect","angles.ext-puzzle-bisect","In triangle ABC, ∠B = 70° and ∠C = 50°. The bisectors of ∠B and ∠C meet at point I. Find **∠BIC**.",{"kind":144,"answer":706,"tolerance":146,"unit":147},120,[708,709],"In triangle IBC, the angles at B and C are half of 70° and half of 50°.","Then use the angle sum of triangle IBC.",{"correct":711,"incorrect":712},"Correct: ∠IBC = 35°, ∠ICB = 25°, so ∠BIC = 180° − 60° = 120°. (In general ∠BIC = 90° + ∠A ÷ 2, and here ∠A = 60°.)","∠IBC = 70° ÷ 2 = 35° and ∠ICB = 50° ÷ 2 = 25°. In triangle IBC: ∠BIC = 180° − 35° − 25° = 120°.",{"id":714,"type":140,"itemId":715,"prompt":716,"check":717,"hints":718,"feedback":721},"pr-puzzle-clock","angles.ext-puzzle-clock","From 6:00 up to 7:00 (including exactly 6:00), at what time do the clock hands point in exactly **opposite** directions (a straight line)? Give the number of minutes past 6.",{"kind":144,"answer":146,"tolerance":146,"unit":160},[719,720],"Try m = 0 first: at exactly 6:00, |30 × 6 − 5.5 × 0| = ? Minutes past 6 can be 0.","After 6:00 the minute hand catches up with the hour hand (overlap near 6:33) and is only 150° ahead of it at 7:00. Can the angle get back to 180° before 7:00?",{"correct":722,"incorrect":723},"Yes: at exactly 6:00 the hands are opposite. The minute hand then catches up, overlaps near 6:33, and is only 150° ahead by 7:00, so 6:00 is the only time in that hour.","At 6:00 the angle is |180 − 0| = 180°, already a straight line. After that, the minute hand closes the gap (overlap at about 6:32:44) and by 7:00 it is only 150° ahead. So the answer is 0 minutes past 6.",{"id":725,"type":140,"itemId":726,"prompt":727,"check":728,"hints":730,"feedback":733},"pr-quad-ratio","angles.ext-quad-ratio","The angles of a quadrilateral are in the ratio **1 : 2 : 3 : 4**. Find the **largest** angle.",{"kind":144,"answer":729,"tolerance":146,"unit":147},144,[731,732],"The four angles add to 360°.","There are 1 + 2 + 3 + 4 = 10 equal parts.",{"correct":734,"incorrect":735},"Correct: one part is 36°, so the angles are 36°, 72°, 108° and 144°.","The angles total 360°, shared in 10 parts of 36° each. The largest is 4 × 36° = 144°.",{"id":737,"type":140,"itemId":738,"prompt":739,"check":740,"hints":742,"feedback":745},"pr-polygon-5x","angles.ext-polygon-5x","Each interior angle of a regular polygon is **5 times** each exterior angle. How many sides does the polygon have?",{"kind":144,"answer":741,"tolerance":146},12,[743,744],"Interior + exterior = 180° at each corner.","If the exterior angle is e, then 5e + e = 180°.",{"correct":746,"incorrect":747},"Yes: 6e = 180°, e = 30°, and 360 ÷ 30 = 12 sides.","At each corner the interior and exterior angles form a linear pair: 5e + e = 180°, so e = 30°. The exterior angles add to 360°, so there are 360 ÷ 30 = 12 sides.",{"id":749,"type":129,"title":750,"problem":751,"steps":752},"we-lp-olymp","Olympiad-style linear-pair puzzles","Rays stand on a straight line AB at O. (i) A linear pair is in the ratio **5 : 7**. (ii) In another linear pair, one angle is **36° less than twice** the other. (iii) Three rays on the same side of AB make four angles along the line in the ratio **1 : 2 : 3 : 4**. Find the largest angle in each.",[753,754,755,756],"(i) 5 + 7 = 12 parts make 180°, so one part is 15°. The angles are 75° and **105°**.","(ii) Let the other angle be x; this one is 2x − 36. Then x + 2x − 36 = 180, so 3x = 216 and x = 72. The angles are 72° and **108°**.","(iii) All four angles lie along the straight line, so they total 180°: 10 parts, one part 18°. The largest is 4 × 18° = **72°**.","Pattern: count the parts, divide 180° by the number of parts, then multiply back. For “less than twice” or “more than” wording, use one unknown and an equation.",{"id":758,"type":413,"component":759,"componentVersion":5,"config":760,"objective":763,"textAlternative":764},"lab-linear-olymp","angle-pairs",{"scene":761,"initialAngle":762,"challenges":741},"linear-pair",27,"A twelve-challenge speed round on linear pairs: find each missing angle as fast as you can.","A straight line with a ray making the linear pair ∠a and ∠b, starting at 27° and 153°. The live table shows both sizes.\n\nThe twelve challenges each give one angle of the pair and ask for the other; type the number of degrees. Try to answer each in a few seconds using mental subtraction from 180°: 180 − 35 = 145, 180 − 125 = 55. A quick trick: to subtract from 180, first subtract from 200 and then take 20 off.\n\nThe olympiad-style linear-pair puzzles (ratios such as 5 : 7, and “36° less than twice the other”) are worked through in the worked example just above this lab.",{"id":766,"type":129,"title":767,"problem":768,"steps":769},"we-two-transversals","Two transversals make a triangle","Lines l ∥ m. Two transversals start from the same point A on l and cut m at B and C, forming triangle ABC. At B, the angle between m and AB inside the triangle is **50°**; at C, the angle between m and AC inside the triangle is **60°**. Find ∠BAC and the two angles at A between l and each transversal.",[770,771,772,773],"∠BAC = 180° − 50° − 60° = **70°** (angle sum of triangle ABC).","The angle at A between l and AB (on B's side) is alternate to the 50° at B, so it is **50°** (alternate interior angles, l ∥ m).","Similarly, the angle at A between l and AC (on C's side) is **60°**.","Check along line l at A: 50° + 70° + 60° = 180°, a straight angle. ✓ This is the triangle angle-sum proof from Deepen, seen from the other side.",{"id":775,"type":140,"itemId":776,"prompt":777,"check":778,"hints":780,"feedback":783},"pr-corr-alg-ext","angles.ext-corr-alg","Two parallel lines are cut by a transversal. A pair of **corresponding** angles are **(5x − 12)°** and **(3x + 20)°**. Find the angle.",{"kind":144,"answer":779,"tolerance":146,"unit":147},68,[781,782],"Corresponding angles on parallel lines are equal.","5x − 12 = 3x + 20.",{"correct":784,"incorrect":785},"Yes: 2x = 32, x = 16, and the angle is 5 × 16 − 12 = 68°.","Set the corresponding angles equal: 5x − 12 = 3x + 20, so 2x = 32 and x = 16. The angle is 5 × 16 − 12 = 68° (check: 3 × 16 + 20 = 68).",{"id":787,"type":140,"itemId":788,"prompt":789,"check":790,"hints":792,"feedback":795},"pr-cointerior-ext","angles.ext-cointerior-ratio","Co-interior angles between two parallel lines are in the ratio **4 : 5**. What is the **smaller** angle?",{"kind":144,"answer":791,"tolerance":146,"unit":147},80,[793,794],"Co-interior angles on parallel lines add to 180°.","4 + 5 = 9 parts.",{"correct":796,"incorrect":797},"Correct: one part is 20°, so the angles are 80° and 100°.","The 9 parts make 180°, so one part is 20°. The smaller angle is 4 × 20° = 80°.",{"id":799,"type":413,"component":759,"componentVersion":5,"config":800,"objective":803,"textAlternative":804},"lab-transversal-olymp",{"scene":801,"initialAngle":802,"challenges":741},"transversal",48,"Twelve transversal challenges to practise recognising every pair quickly and accurately.","Two parallel lines and a transversal starting at 48° (angles of 48° and 132°). A live table lists all eight angles, and buttons highlight corresponding, alternate interior, alternate exterior, co-interior, vertically opposite and linear pairs.\n\nThe twelve challenges each give one angle and ask for another, anywhere in the picture; type the number of degrees. Aim for twelve in a row with a reason ready for each: equal pairs (F, Z, alternate exterior, vertically opposite) or pairs adding to 180° (co-interior, linear pair). Harder chases with two transversals and algebra are in the worked examples and practice of this chapter.",{"id":806,"type":413,"component":807,"componentVersion":5,"config":808,"objective":812,"textAlternative":813},"lab-estimate-pro","angle-lab",{"modes":809,"allowReflex":657,"rounds":229},[810,811],"estimate","classify","A 15-round estimate-and-classify challenge with randomly tilted angles, including reflex: aim for an average error under 5°.","The longest angle-lab challenge: 15 rounds in estimate mode, then 15 in classify mode, with reflex angles included. Every angle is tilted at random, so you must picture it turned round. In estimate mode your guess is drawn as a dashed arm marked “you” next to the true angle, and points depend on how close you were; the game keeps your streak and best score.\n\nExpert tips: split the angle into known pieces (for example 90° + 45° + a bit), use the clock picture (each hour-gap is 30°), and for reflex angles estimate the small opening and subtract from 360°. Surveyors, pilots and snooker players develop an eye accurate to within a few degrees.",{"id":815,"type":53,"title":816,"eyebrow":817,"navLabel":818},"ch07","Projects to try","Chapter 07","7 Projects",{"id":820,"type":821,"title":822,"prompt":823,"options":824},"explorer-projects","explorer","Five angle projects","Choose a project; each needs only simple materials.",[825,835,844,853,862],{"id":826,"label":827,"chain":828,"note":834},"clinometer","Build a clinometer",[829,830,831,832,833],"Protractor, straw, thread, weight","Sight the top of a tree","Read the angle","Pace the distance","Scale drawing gives the height","Tape a drinking straw along the straight edge of a protractor and hang a thread with a small weight from its centre. Look through the straw at the top of a tree or building; the thread hangs straight down and shows the angle of elevation. Pace out your distance from the base, then draw a scale diagram (for example 1 cm for 1 m) with that angle to find the height. The measuring skills are in the constructing angles topic.",{"id":836,"label":837,"chain":838,"note":843},"shadow","Sun and shadows",[839,840,841,842],"Stick upright in the ground","Mark the shadow tip each hour","Measure the shadow angles","Compare with the Sun's direction","Stand a stick straight up on flat ground on a sunny day. Every hour, mark the tip of its shadow and draw a line from the stick to the mark. Measure the angle turned by the shadow between readings. The Sun appears to move about 15° per hour across the sky (360° ÷ 24 hours), but the shadow's angle on the ground does not change at exactly the same steady rate. Why might that be?",{"id":845,"label":846,"chain":847,"note":852},"rangoli","Rotation rangoli",[848,849,850,851],"Choose a motif","Choose n copies","Rotate by 360° ÷ n","Complete the pattern","Design a rangoli with rotational symmetry. Draw one motif, then repeat it around a centre point, turning each copy by the same angle. For 8 copies turn by 45°, for 6 copies by 60°, for 12 copies by 30°. Which numbers of copies give whole-number angles? (Any divisor of 360.) Link: the patterns topic.",{"id":854,"label":855,"chain":856,"note":861},"junction","Road junction survey",[857,858,859,860],"Pick 5 junctions near home","Sketch each from above","Estimate the angles","Rate how safe it feels","With an adult, look at road junctions near your home or school. Sketch each from above, estimate the angles where roads meet and check with a protractor on a map. Note whether drivers seem to have a clear view. Do sharper junctions seem harder to see around? Present your findings as a table and a pie chart of angle types.",{"id":863,"label":864,"chain":865,"note":870},"pie-survey","Class pie chart",[866,867,868,869],"Ask a survey question","Count the answers","Angle = share × 360°","Draw with a protractor","Ask your class a question with 3 to 6 possible answers (favourite fruit, how you come to school). Work out each answer's angle as its share of 360°, check the angles add to 360°, and draw the pie chart with a protractor and compass. Discuss: when is a pie chart a good choice, and when is a bar chart clearer? See the data-handling topic.",{"id":872,"type":53,"title":873,"eyebrow":874,"navLabel":875},"ch08","Who uses angles at work?","Chapter 08","8 Careers",{"id":877,"type":81,"caption":878,"columns":879,"rows":882},"table-careers","Careers where angles matter every day",[880,881],"Career","How angles are used",[883,886,889,892,895,898,901,904],[884,885],"Architect and civil engineer","Roof pitches, ramps, stair angles, and making sure walls meet at right angles; bridge trusses use triangles because their angles cannot change without the sides changing.",[887,888],"Surveyor","Measures angles between landmarks with a theodolite or total station to map land, lay out roads and railway lines, and fix property boundaries.",[890,891],"Pilot and air traffic controller","Headings and bearings, runway directions, and the angle of climb and descent (the standard approach slope to a runway is 3°, the angle ICAO uses to set up the lights and glide path).",[893,894],"Physiotherapist and doctor","Measures joint angles (range of motion) with a goniometer to track recovery after injury.",[896,897],"Sports analyst and coach","Tests bowling actions for elbow extension, studies bat swings, launch angles and the angles of a javelin throw.",[899,900],"Game designer and animator","Rotates characters and cameras by angles every frame; game engines use both degrees and radians.",[902,903],"Astronomer","Measures positions of stars in degrees, minutes and seconds of arc; the Moon looks about half a degree wide from Earth.",[905,906],"Carpenter and tailor","Mitre joints at 45° to make a 90° frame corner; cutting fabric at angles for collars and pleats.",{"id":908,"type":47,"variant":549,"title":909,"markdown":910},"example-mitre","The carpenter's 45°","A photo frame has four corners of 90°. The carpenter cuts the end of each piece at **45°**, so that two pieces meeting at a corner make 45° + 45° = 90°. For a hexagonal frame, each corner is 120°, so each cut is 60°. For an octagonal frame, each corner is 135°, so each cut is 67.5°.",{"id":912,"type":53,"title":913,"eyebrow":914,"navLabel":915},"ch09","Open questions","Chapter 09","9 Open questions",{"id":917,"type":47,"variant":918,"title":919,"markdown":920},"open-questions","question","Questions to keep thinking about","Some of these have known answers you could look up; some are still being studied. Pick one and dig in.\n\n- If a full turn had been divided into 100 parts, or 1,000, would anything important in the world be different?\n- Which other planets would give a different \"natural\" number of degrees if astronomers there used days in a year? (Mars has about 669 of its days in a year.)\n- Bees build hexagonal cells with 120° corners. Is that the best shape for storing honey with the least wax? Mathematicians proved a version of this, the honeycomb conjecture, only in 1999.\n- Why do so many sports (cricket, football, javelin, basketball) depend on angles, and what is the best launch angle when air resistance matters?\n- Could a triangle ever have angles adding to *less* than 180°? (Yes, on a saddle-shaped surface. What would that look like?)\n- How do robot vacuum cleaners and self-driving cars keep track of which way they are facing after thousands of turns?",{"id":922,"type":923,"prompt":924},"reflect-extend","reflection","Choose one real-life angle from this layer (a ramp, a ladder, a runway, a roof, a bowling arm, a pie chart). Explain what would go wrong if that angle were much bigger or much smaller.",{"id":926,"type":927,"title":928,"terms":929},"glossary-extend","glossary","Extension words",[930,934,938,942,946,950,954,958,962,966,970,974],{"term":931,"meaning":932,"example":933},"absolute value | |","The size of a number without its sign.","|−35| = 35",{"term":935,"meaning":936,"example":937},"exterior angle of a triangle","The angle between one side of a triangle and the extension of the next side.","Extending BC to D gives exterior angle ∠ACD.",{"term":939,"meaning":940,"example":941},"interior opposite angles","For an exterior angle of a triangle, the two inside angles at the other corners.","For ∠ACD, they are ∠A and ∠B.",{"term":943,"meaning":944,"example":945},"exterior angle property","An exterior angle of a triangle equals the sum of the two interior opposite angles.","∠ACD = ∠A + ∠B",{"term":947,"meaning":948,"example":949},"bearing","A direction given as an angle measured clockwise from north, written with three figures.","East is 090°.",{"term":951,"meaning":952,"example":953},"back bearing","The bearing for the return journey, differing by 180°.","The back bearing of 063° is 243°.",{"term":955,"meaning":956,"example":957},"radian","The angle at the centre of a circle made by an arc as long as the radius. 1 radian ≈ 57.3°.","180° = π radians",{"term":959,"meaning":960,"example":961},"angle of elevation","The angle you look up through, from the horizontal, to see an object above you.","Measured with a clinometer.",{"term":963,"meaning":964,"example":965},"angle chasing","Solving a geometry problem by finding one angle after another, with reasons.","Olympiad problems",{"term":967,"meaning":968,"example":969},"pentagram","A five-pointed star drawn with five straight lines.","Its tip angles add to 180°.",{"term":971,"meaning":972,"example":973},"goniometer","An instrument for measuring angles, especially joint angles in the body.","Used by physiotherapists.",{"term":975,"meaning":976,"example":977},"mitre joint","A corner joint made by cutting two pieces at equal angles.","Two 45° cuts make a 90° corner.",{"id":979,"type":980,"title":981,"questions":982},"quiz-extend","quiz","Extension challenge",[983,995,1006,1018,1031,1041,1053,1064,1073,1084],{"itemId":984,"prompt":985,"options":986,"correct":184,"why":994},"angles.ext-q-clock","Using |30h − 5.5m|, what is the smaller angle at 2:30?",[987,989,990,991],{"id":178,"label":988},"75°",{"id":181,"label":519},{"id":184,"label":110},{"id":992,"label":993},"d","165°","|60 − 165| = 105°.",{"itemId":996,"prompt":997,"options":998,"correct":181,"why":1005},"angles.ext-q-clock-reflex","At 10:10 the formula gives |300 − 55| = 245°. What is the smaller angle?",[999,1001,1002,1003],{"id":178,"label":1000},"55°",{"id":181,"label":105},{"id":184,"label":104},{"id":992,"label":1004},"300°","245° is more than 180°, so the smaller angle is 360° − 245° = 115°.",{"itemId":1007,"prompt":1008,"options":1009,"correct":181,"why":1017},"angles.ext-q-exterior","An exterior angle of a triangle is 110° and one interior opposite angle is 45°. The other interior opposite angle is…",[1010,1012,1014,1016],{"id":178,"label":1011},"25°",{"id":181,"label":1013},"65°",{"id":184,"label":1015},"70°",{"id":992,"label":302},"Exterior angle = sum of interior opposite angles: 110° − 45° = 65°.",{"itemId":1019,"prompt":1020,"options":1021,"correct":184,"why":1030},"angles.ext-q-polygon-ext","Each exterior angle of a regular polygon is 30°. How many sides does it have?",[1022,1024,1026,1028],{"id":178,"label":1023},"6",{"id":181,"label":1025},"10",{"id":184,"label":1027},"12",{"id":992,"label":1029},"30","Exterior angles sum to 360°: 360 ÷ 30 = 12.",{"itemId":1032,"prompt":1033,"options":1034,"correct":181,"why":1040},"angles.ext-q-bearing","What is the back bearing of 315°?",[1035,1036,1037,1038],{"id":178,"label":292},{"id":181,"label":302},{"id":184,"label":293},{"id":992,"label":1039},"495°","315° − 180° = 135°.",{"itemId":1042,"prompt":1043,"options":1044,"correct":184,"why":1052},"angles.ext-q-runway","One end of a runway is numbered 12. The other end is numbered…",[1045,1047,1049,1050],{"id":178,"label":1046},"06",{"id":181,"label":1048},"24",{"id":184,"label":1029},{"id":992,"label":1051},"36","The directions differ by 180°, so the numbers differ by 18: 12 + 18 = 30.",{"itemId":1054,"prompt":1055,"options":1056,"correct":184,"why":1063},"angles.ext-q-radian","About how many degrees is 1 radian?",[1057,1059,1060,1062],{"id":178,"label":1058},"1°",{"id":181,"label":585},{"id":184,"label":1061},"57.3°",{"id":992,"label":519},"1 radian = 180° ÷ π ≈ 57.3°.",{"itemId":1065,"prompt":1066,"options":1067,"correct":992,"why":1072},"angles.ext-q-pie","In a pie chart of 60 people, 15 chose tea. What angle is the tea slice?",[1068,1069,1070,1071],{"id":178,"label":506},{"id":181,"label":585},{"id":184,"label":589},{"id":992,"label":519},"15 out of 60 is a quarter, and a quarter of 360° is 90°.",{"itemId":1074,"prompt":1075,"options":1076,"correct":178,"why":1083},"angles.ext-q-star","What is the sum of the tip angles of any five-pointed star drawn with straight lines?",[1077,1078,1079,1081],{"id":178,"label":288},{"id":181,"label":526},{"id":184,"label":1080},"540°",{"id":992,"label":1082},"It depends on the star","Angle chasing with the exterior angle property turns the five tips into the three angles of one triangle: 180°.",{"itemId":1085,"prompt":1086,"options":1087,"correct":184,"why":1093},"angles.ext-q-mitre","A carpenter makes a regular hexagonal frame. At what angle is each piece cut?",[1088,1089,1090,1091],{"id":178,"label":581},{"id":181,"label":585},{"id":184,"label":589},{"id":992,"label":1092},"120°","Each corner of a regular hexagon is 120°, split equally between two pieces: 60°.",{"id":1095,"type":1096,"title":1097,"points":1098},"cheat-sheet","summary","Cheat sheet",[1099,1100,1101,1102,1103,1104,1105,1106],"**Clock:** angle = |30h − 5.5m|; if over 180°, use 360° minus it. Hands overlap every 65 5⁄11 minutes.","**Exterior angle of a triangle** = sum of the two interior opposite angles (linear pair + angle sum).","**Exterior angles of any convex polygon add to 360°.** Regular n-gon: exterior 360° ÷ n, interior 180° − 360° ÷ n.","**Bearings:** clockwise from north, three figures (090° = east). Back bearing: ± 180°. Runway numbers at the two ends differ by 18.","**Real life:** ramp 1 : 12 ≈ 4.8°; ladder 4 : 1 ≈ 76°; bowlers may straighten the elbow at most 15°; pie slice = share × 360°.","**Radian:** arc = radius. 180° = π radians; 1 radian ≈ 57.3°.","**Puzzles:** chase angles with reasons; add a parallel, a diagonal or an extension when stuck. Star tips sum to 180°.","Angles are used by architects, surveyors, pilots, physiotherapists, coaches, game designers, astronomers and carpenters.",{"id":1108,"type":1109,"conceptId":1110,"relation":1111,"explanation":1112},"conn-data","connection","data-handling","applied_in","Pie charts turn shares of data into angles: each slice is its share of 360°.",{"id":1114,"type":1109,"conceptId":1115,"relation":1111,"explanation":1116},"conn-construct","constructing-angles","Clinometer and pie-chart projects need accurate measuring and drawing of angles with a protractor and compass.",{"id":1118,"type":1109,"conceptId":1119,"relation":1120,"explanation":1121},"conn-patterns","patterns","related_to","Rotation rangoli and the times when clock hands meet are patterns governed by angles.",{"id":1123,"type":1109,"conceptId":1124,"relation":1120,"explanation":1125},"conn-electricity","electricity","Generators spin through 360° again and again; one full turn of a two-pole generator makes one cycle of alternating current.",{"id":1127,"type":1128,"sourceIds":1129},"sources-extend","sources",[1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141],"angles-wiki-clock-angle","angles-ncert-class7-triangle","angles-euclid-i32","angles-wiki-radian","angles-wiki-degree","angles-mathsisfun-angles","angles-khan-angle-basics","angles-mohua-accessibility-ramps","angles-niosh-ladder","angles-icc-illegal-bowling","angles-icao-annex14-runways","angles-wiki-pv-tilt",[1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141],"needs_review",{"generatedBy":1145,"notes":1146},"claude-code","Draft generated with Python generator scripts; every angle computed and asserted. Pending owner review.","ce7b7377e2a3cffd73f3b7672efef4e777b8cec2499d962885b15783d25ca5fd",{"logic:practice":1149,"component:sort-game@1":1150,"component:match-pairs@1":1151,"component:angle-pairs@1":1152,"component:angle-lab@1":1153,"source:angles-euclid-i32":1154,"source:angles-icao-annex14-runways":1155,"source:angles-icc-illegal-bowling":1156,"source:angles-khan-angle-basics":1157,"source:angles-mathsisfun-angles":1158,"source:angles-mohua-accessibility-ramps":1159,"source:angles-ncert-class7-triangle":1160,"source:angles-niosh-ladder":1161,"source:angles-wiki-clock-angle":1162,"source:angles-wiki-degree":1163,"source:angles-wiki-pv-tilt":1164,"source:angles-wiki-radian":1165},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","cbb509e9a575a1b2ef133804e3450487de14952c1df934747fe56ff5d10d847e","1916502bd0021560b90784e9e612bc92ea263e6ed383d55cff0b75753b92213f","336834da6f32f1f0c454d6e5e6a714b7a3f52963b223288563bc48c1085bbc46","f6daf5774bce21c1fcf48eb4345a786bf4015857be9bcb6fbe978197eb2c2ecc","c4487affbd8ca1c7cbfd54d9293933b40e3c24201a82aaeb13cce9bb41d81a26","68741673c0dc6c96760fde888327818c508788257b5306bad48ab148615541a4","2dab8c7c8736bbad61c4a7223c297c3b250b20399d5b5544c02434cab39a0c9c","26708f3ab1210e6438207954ba955d647d14dc15ef1ddc44511e1608d988e5a7","286f44f488c463767261100625e631e8edb823f1f296976d251359a4bc9677ee","dd85dfd2c7ee127db700b8721af5975437c999a7625b759ffb988d5f7af0c2e2","141f5516f8832cc0eea67b34a5726cd12ed7309036b178da74f753ca6e4ae706","79b5ead855ebdf6f15ad061e87052809bfff58b83c8dfb9f7a1f592debdf62e9","79f95117866461cf4beb6f35eabd99109d5b05cecffd26271b1c18faee3706c0","d3ba163093746b28c02ebab9f522ab7f88f2bf57f4b01e73547910f83286d311",{"state":1167,"reviewer":1168,"selfReview":657,"reviewedAt":1169,"method":1170},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899596942]