[{"data":1,"prerenderedAt":1032},["ShallowReactive",2],{"layer:constructing-angles:understand":3},{"layer":4,"contentHash":1009,"dependencyHashes":1010,"approval":1025,"releaseId":1031},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":17,"plate":18,"blocks":39,"sourceIds":1004,"reviewStatus":1005,"authoring":1006},1,"constructing-angles","en","understand","Reading the protractor and the compass constructions","Why the two scales exist, how to measure and draw any angle, and why 60°, 90°, 30° and 45° constructions work","Learn the precise protractor method (and the wrong-scale trap), measure and draw reflex angles, copy lengths with a compass, and construct 60°, 120°, 90°, 30° and 45° angles and perpendicular bisectors with the reason each one works.",[13,14,15,16],"Measure and draw any angle from 0° to 360° with a protractor, choosing the correct scale and checking with an estimate.","Explain why the two protractor scales add to 180° and how to avoid reading the wrong one.","Construct 60°, 120°, 90°, 30° and 45° angles, a perpendicular bisector and an angle bisector with ruler and compass.","Give the reason each construction works, and check constructions with a protractor to within 1°.",40,{"title":19,"rows":20},"Lesson plate",[21,24,27,30,33,36],{"label":22,"value":23},"Depth","Understand",{"label":25,"value":26},"Reading time","≈ 40 minutes",{"label":28,"value":29},"Prior knowledge","Geometry box and degrees (Discover)",{"label":31,"value":32},"Chapters","11",{"label":34,"value":35},"Labs","Protractor ×2, scale sort, 4 constructions",{"label":37,"value":38},"Accuracy goal","within ± 1°",[40,44,50,56,59,64,69,102,107,112,115,144,148,153,156,172,183,193,254,281,286,319,324,334,353,358,361,371,381,397,401,406,409,421,430,440,453,458,461,528,539,543,548,555,575,580,584,604,609,612,617,626,629,639,659,664,668,673,678,704,713,722,725,730,736,740,745,750,786,797,809,813,974,990],{"id":41,"type":42,"markdown":43},"intro","prose","In Discover you met the geometry box, measured a few angles and made a 60° angle with a compass. This layer slows everything down and asks **how** each method works and **why** it gives the right answer, so you can do it confidently every time, including the tricky cases: the wrong scale, short arms, angles bigger than 180°, and constructions that go wrong because the compass slipped.\n\nThere are two families of methods in this topic:\n\n1. **Measuring and drawing with a protractor.** Fast and flexible: any whole number of degrees. Accuracy depends on your eye and your pencil.\n2. **Constructing with a ruler and compass only.** Slower, and only certain angles are possible (60°, 120°, 90°, 30°, 45° and their relatives), but the result is **exact in principle**, because it comes from reasoning rather than reading a scale.",{"id":45,"type":46,"variant":47,"title":48,"markdown":49},"how-to-read","callout","observation","How to use this layer","Work with paper and your geometry box beside you. Every worked example is something you can do on paper as you read. The protractor labs let you practise the reading and drawing steps as often as you like; the construction animations show each compass move one at a time.",{"id":51,"type":52,"title":53,"eyebrow":54,"navLabel":55},"ch1","chapter","What a degree measures, exactly","Chapter 01","1 Degrees precisely",{"id":57,"type":42,"markdown":58},"degree-def","An **angle** is formed by two rays, called its **arms**, that start from the same point, its **vertex**. We name an angle with three letters, the vertex in the middle: **∠AOB** has vertex O and arms OA and OB.\n\nThe **size** of an angle is the amount of **turn** needed to rotate one arm onto the other, about the vertex. We measure turn in **degrees**: one complete turn is **360°**, so **1° is 1\u002F360 of a full turn**.\n\nNotice what is *not* in that definition: the **length of the arms**. The arms of an angle are rays, which go on for ever. When we draw them, we draw only a piece, and the piece can be short or long. Drawing the arms longer does not change the amount of turn, so it does not change the angle.",{"id":60,"type":46,"variant":61,"title":62,"markdown":63},"def-angle","definition","Angle and degree","An **angle** is the figure formed by two rays with a common starting point (the vertex). Its **measure** is the amount of turn from one arm to the other. **1 degree (1°)** is 1\u002F360 of a complete turn. A right angle is 90°, a straight angle 180°, a complete angle 360°.",{"id":65,"type":46,"variant":66,"title":67,"markdown":68},"misc-arm-length","misconception","“Longer arms make a bigger angle”","Two angles are drawn: one with 2 cm arms, one with 10 cm arms. The long one *looks* bigger, but if both arms turn by the same amount they are the **same angle**. The size of an angle is about the **opening**, not the length. This is also why you may **extend** the arms of a small drawing with a ruler before measuring: extending changes the length, not the angle.",{"id":70,"type":71,"caption":72,"columns":73,"rows":77},"table-types","table","The angle families and their ranges",[74,75,76],"Type","Size","Example",[78,82,86,90,94,98],[79,80,81],"Acute","more than 0°, less than 90°","35°, 60°, 89°",[83,84,85],"Right","exactly 90°","The corner of a page",[87,88,89],"Obtuse","more than 90°, less than 180°","100°, 135°, 170°",[91,92,93],"Straight","exactly 180°","A straight line through the vertex",[95,96,97],"Reflex","more than 180°, less than 360°","200°, 270°, 330°",[99,100,101],"Complete","exactly 360°","One full turn",{"id":103,"type":46,"variant":104,"title":105,"markdown":106},"nuance-two-angles","nuance","Every pair of arms makes two angles","Two arms from the same vertex split the full turn into **two** angles that add up to 360°. If one is 110°, the other is 360° − 110° = 250°. Usually \"the angle\" means the smaller one, but when a problem asks for the **reflex angle**, it means the bigger one, the one more than 180°. An arc drawn near the vertex shows which one is meant.",{"id":108,"type":52,"title":109,"eyebrow":110,"navLabel":111},"ch2","Anatomy of a protractor","Chapter 02","2 The protractor",{"id":113,"type":42,"markdown":114},"anatomy","A standard school protractor is a **semicircle** of clear plastic, graduated from 0° to 180°. Four features matter:\n\n- **The centre point** (also called the **origin** or **reference point**) is the centre of the semicircle. It is marked with a small hole, a cross or a short line on the straight edge. Every angle is measured from here.\n- **The base line** (the **zero line**) is the straight line through the centre point joining the two 0° marks. It is usually a few millimetres above the plastic edge.\n- **The two scales** run round the curved edge in opposite directions: one reads 0° at the **right** end and 180° at the left, the other reads 0° at the **left** end and 180° at the right. Which of them is printed on the inside and which on the outside **varies from protractor to protractor**, so the rule below never mentions inner or outer. It only asks where the **0** is.\n- **The graduations**: long marks every 10°, medium marks every 5°, short marks every 1°.",{"id":116,"type":117,"tone":118,"items":119},"spec-anatomy","spec","amber",[120,124,128,132,136,140],{"label":121,"big":122,"value":123},"Shape","semicircle","Covers half a turn, 0° to 180°. Full-circle protractors cover 0° to 360°.",{"label":125,"big":126,"value":127},"Centre point","the vertex","The centre of the semicircle; the angle's vertex must sit exactly on it.",{"label":129,"big":130,"value":131},"Base line","0° line","Joins the two zeros through the centre. One arm lies along it.",{"label":133,"big":134,"value":135},"Two scales","x and 180 − x","Opposite directions, 0 at opposite ends. At any point the two numbers add to 180.",{"label":137,"big":138,"value":139},"Smallest mark","1°","Short marks each 1°, medium each 5°, long each 10°.",{"label":141,"big":142,"value":143},"Good accuracy","± 1°","A careful reading with sharp lines is within 1° of the true value.",{"id":145,"type":46,"variant":66,"title":146,"markdown":147},"misc-edge-u","“Put the arm along the bottom edge”","The plastic edge is not the base line on most protractors. If the arm lies along the edge, the vertex sits a few millimetres below the centre point. The arm then crosses the scale at the wrong place, and the error is bigger the shorter the arms are. Always look for the printed line through the centre point.",{"id":149,"type":52,"title":150,"eyebrow":151,"navLabel":152},"ch3","Two scales and the number one mistake","Chapter 03","3 Which scale?",{"id":154,"type":42,"markdown":155},"two-scales","The most common protractor mistake in the world is **reading the wrong scale**. It is easy to make, because both numbers are printed right next to each other where the second arm crosses, and they are both perfectly real numbers.\n\nHere is why there are two. Suppose the arm you lined up points to the right. You must count the turn starting from **that** arm, so you need a scale that starts at 0 on the right. If the arm you lined up points to the left, you need a scale that starts at 0 on the left. A single protractor carries both, so it works either way round.\n\n**The rule:** read the scale whose **0 lies on the arm you lined up with the base line**. Then count upwards along that scale to the second arm.",{"id":157,"type":158,"items":159},"formulas-scales","formulas",[160,163,166,169],{"expression":161,"caption":162},"one scale + other scale = 180","At any point on the edge, the two printed numbers add up to 180.",{"expression":164,"caption":165},"wrong scale gives 180 − x","Reading the other scale gives the supplement of the true angle.",{"expression":167,"caption":168},"acute → answer under 90","Use your estimate to pick between x and 180 − x.",{"expression":170,"caption":171},"90 on both scales","At the top, both scales read 90, so a right angle cannot be misread.",{"id":173,"type":174,"title":175,"problem":176,"steps":177},"we-40-140","worked_example","Two numbers, one answer","∠PQR has its vertex Q on the centre point and arm QR along the base line, pointing to the **right**. Arm QP crosses the protractor where the scales show **40** and **140**. The angle looks clearly smaller than a right angle. Find ∠PQR.",[178,179,180,181,182],"Estimate first: acute, so the answer is less than 90°.","QR points to the right, so read the scale that has 0 at the right-hand end.","Count up that scale from 0: 10, 20, 30, 40. Arm QP crosses at **40**.","Check the pair: 40 + 140 = 180, so 140 is the reading on the other scale.","40° agrees with the estimate (acute). So **∠PQR = 40°**.",{"id":184,"type":174,"title":185,"problem":186,"steps":187},"we-115","An obtuse angle opening the other way","∠XYZ has vertex Y on the centre point and arm YX along the base line, pointing to the **left**. Arm YZ crosses where the scales show **65** and **115**. The angle looks a bit wider than a right angle. Find ∠XYZ.",[188,189,190,191,192],"Estimate: obtuse, somewhat more than 90°.","YX points left, so read the scale with 0 at the left-hand end.","Counting up that scale from 0 you pass 90 and reach YZ at **115**.","The other scale reads 180 − 115 = 65 at the same point.","115° is obtuse, as estimated, so **∠XYZ = 115°**.",{"id":194,"type":195,"component":196,"componentVersion":5,"config":197,"objective":247,"textAlternative":248,"help":249},"lab-scale-sort","interactive","sort-game",{"prompt":198,"bins":199,"items":206,"seconds":246},"Which scale gives the right answer: the one whose 0 is at the left-hand end of the base line, or the one whose 0 is at the right-hand end?",[200,203],{"id":201,"label":202},"left","The scale with 0 on the left",{"id":204,"label":205},"right","The scale with 0 on the right",[207,211,215,219,223,227,230,234,238,242],{"id":208,"label":209,"bin":204,"why":210},"s1","The arm you lined up points right from the vertex","Always read the scale whose 0 sits on the lined-up arm, and that arm is on the right.",{"id":212,"label":213,"bin":201,"why":214},"s2","The arm you lined up points left from the vertex","The 0 on the lined-up arm is the left-hand 0, so count up from there.",{"id":216,"label":217,"bin":201,"why":218},"s3","Drawing 35° from an arm that points left","Count 35 from the left-hand 0. The other scale's 35 would give 145°.",{"id":220,"label":221,"bin":204,"why":222},"s4","Drawing 140° from an arm that points right","Count 140 from the right-hand 0. The other scale's 140 would give 40°.",{"id":224,"label":225,"bin":201,"why":226},"s5","The lined-up arm passes through the 0 printed at the left-hand end","The 0 on the lined-up arm decides the scale.",{"id":228,"label":229,"bin":204,"why":226},"s6","The lined-up arm passes through the 0 printed at the right-hand end",{"id":231,"label":232,"bin":204,"why":233},"s7","Arm points right; obtuse angle; the second arm crosses at 60 on one scale and 120 on the other","Count up from the right-hand 0 and you pass 90 to reach 120. Obtuse agrees: the answer is 120°.",{"id":235,"label":236,"bin":201,"why":237},"s8","Arm points left; acute angle; the second arm crosses at 110 on one scale and 70 on the other","Counting up from the left-hand 0 reaches 70. Acute agrees: the answer is 70°.",{"id":239,"label":240,"bin":201,"why":241},"s9","Arm points left; obtuse angle; the second arm crosses at 150 on one scale and 30 on the other","Counting up from the left-hand 0 reaches 150, and 150° is obtuse, as the picture shows.",{"id":243,"label":244,"bin":204,"why":245},"s10","Arm points right; acute angle; the second arm crosses at 25 on one scale and 155 on the other","Counting up from the right-hand 0 reaches 25, and 25° is acute, as the picture shows.",0,"Decide, in each situation, which of the protractor's two scales gives the correct reading.","This sort game has ten cards and two bins: read the scale whose 0 is on the left, or the scale whose 0 is on the right. It never says “inner” or “outer”, because which row is printed inside differs from protractor to protractor.\n\n**0 on the left:** the lined-up arm points left; the lined-up arm passes through the left-hand 0; drawing 35° from an arm pointing left; an acute angle crossing at 110\u002F70 with the arm on the left (answer 70°); an obtuse angle crossing at 150\u002F30 with the arm on the left (answer 150°).\n\n**0 on the right:** the lined-up arm points right; the lined-up arm passes through the right-hand 0; drawing 140° from an arm pointing right; an obtuse angle crossing at 60\u002F120 with the arm on the right (answer 120°); an acute angle crossing at 25\u002F155 with the arm on the right (answer 25°).\n\nTwo ways to decide, and they always agree: find the 0 on the lined-up arm, or use your estimate to choose between x and 180 − x.",{"simplerExplanation":250,"hints":251},"Find the 0 that sits on the arm lying along the base line. Read that row of numbers.",[252,253],"Acute answers are under 90; obtuse answers are over 90.","Both numbers at the crossing add to 180.",{"id":255,"type":256,"itemId":257,"prompt":258,"check":259,"hints":275,"feedback":278},"practice-scale","practice","constructing-angles.und-which-scale","Aarav measures an angle and writes **155°**. His friend says the angle looks small and sharp, like the tip of a slice of pizza. What most likely went wrong, and what is the angle?",{"kind":260,"options":261,"correct":274},"choice",[262,265,268,271],{"id":263,"label":264},"a","He read the wrong scale; the angle is 25°",{"id":266,"label":267},"b","He misplaced the centre; the angle is 55°",{"id":269,"label":270},"c","Nothing: 155° is sharp",{"id":272,"label":273},"d","He should have used 360° − 155° = 205°",[263],[276,277],"A sharp angle is acute: less than 90°.","Reading the other scale gives 180 − x.",{"correct":279,"incorrect":280},"Yes. A sharp angle is acute, so 155° must be the other scale's reading. 180 − 155 = 25°.","A pizza-tip angle is acute (under 90°), so 155° cannot be right. The two scales add to 180, so the true reading is 180 − 155 = **25°**: he read the wrong scale.",{"id":282,"type":52,"title":283,"eyebrow":284,"navLabel":285},"ch4","Measuring an angle, step by step","Chapter 04","4 Measuring",{"id":287,"type":288,"title":289,"items":290},"steps-measure-u","steps","Measuring an angle: the full method",[291,295,299,303,307,311,315],{"title":292,"tag":293,"text":294},"Estimate","acute or obtuse?","Compare with a right angle and with 45° or 135°. Write down a rough value.",{"title":296,"tag":297,"text":298},"Extend short arms","ruler","If an arm will not reach the scale, extend it with a ruler and a light line. This does not change the angle.",{"title":300,"tag":301,"text":302},"Centre on the vertex","exactly","Place the centre point exactly on the vertex. Look straight down, not from the side.",{"title":304,"tag":305,"text":306},"Base line on one arm","all along","Rotate the protractor, keeping the centre fixed, until the base line lies along one arm.",{"title":308,"tag":309,"text":310},"Choose the scale","0 on that arm","Find the scale with 0 on the lined-up arm.",{"title":312,"tag":313,"text":314},"Count up and read","to the other arm","Count up from that 0 to where the second arm crosses. Read to the nearest degree.",{"title":316,"tag":317,"text":318},"Compare","sanity check","Compare with your estimate. If it is far off, check the scale and the centre.",{"id":320,"type":46,"variant":321,"title":322,"markdown":323},"careful-parallax","careful","Look straight down","If you look at the protractor from the side, the scale and the line under it seem to shift apart, and you can read a degree or two wrong. This is called **parallax error**. Put your eye directly above the point where the arm crosses the scale.",{"id":325,"type":174,"title":326,"problem":327,"steps":328},"we-short-arms","When the arms are too short","∠ABC is drawn with arms only 1.5 cm long. A standard protractor has a radius of about 5 cm, so arm BC ends far inside the scale. How do you measure the angle?",[329,330,331,332,333],"Do not guess where the arm would reach. Take a ruler and extend BA and BC lightly as straight lines, well past 5 cm.","Extending is allowed because the angle depends only on the turn between the arms, not on their length.","Now place the centre on B and the base line along the extended BA.","Read the scale that has 0 on BA at the point where extended BC crosses the edge.","Compare with your estimate. Suppose it reads 72° and you estimated \"a bit less than a right angle\". ✓",{"id":335,"type":195,"component":336,"componentVersion":5,"config":337,"objective":346,"textAlternative":347,"help":348},"lab-measure-u","protractor",{"mode":338,"targets":339,"tolerance":5},"measure",[340,341,342,343,344,345],25,70,115,140,165,55,"Measure six angles to within 1° by placing the protractor correctly and choosing the right scale.","In this lab a drawn angle appears and you drag and rotate a virtual protractor onto it, then type the reading. Answers within 1° score.\n\nThe six angles are 25°, 70°, 115°, 140°, 165° and 55°. For each, the wrong-scale reading would be 180 minus the true value: 155, 110, 65, 40, 15 and 125. Notice that for every acute angle the trap number is obtuse, and for every obtuse angle the trap number is acute. An estimate made before measuring always tells the two apart.\n\nThe angles come in different orientations, so sometimes the lined-up arm is on the right (count from the right-hand 0) and sometimes on the left (count from the left-hand 0).",{"simplerExplanation":349,"hints":350},"Dot on corner, line on arm, count up from the 0 on that arm, then check against your guess.",[351,352],"Before reading, say out loud: acute or obtuse?","If the reading and your estimate disagree by a lot, you are probably on the wrong scale.",{"id":354,"type":52,"title":355,"eyebrow":356,"navLabel":357},"ch5","Drawing an angle of a given measure","Chapter 05","5 Drawing angles",{"id":359,"type":42,"markdown":360},"draw-u","To **draw** an angle of a given measure is to reverse the measuring process. You create one arm, then use the protractor to find where the other arm must go.\n\nThe same rule decides the scale: count from the **0 on the arm you have already drawn**. The same safety net catches mistakes: an estimate of what the angle should look like.",{"id":362,"type":174,"title":363,"problem":364,"steps":365},"we-draw-75","Draw ∠ABC = 75°","Draw an angle ABC of measure 75°, with BC as the first arm.",[366,367,368,369,370],"Draw a ray BC with a ruler, about 6 cm long, pointing right from B.","Place the protractor with its centre point on B and its base line along BC.","BC points right, so use the scale with 0 on the right. Count up to **75** and mark a small dot A at the edge.","Remove the protractor and draw ray BA with the ruler.","Check the look: 75° is acute, just less than a right angle. Measure it again to confirm 75°.",{"id":372,"type":174,"title":373,"problem":374,"steps":375},"we-draw-130","Draw ∠PQR = 130° with the first arm pointing left","Draw ∠PQR = 130°, where the first arm QR points to the **left** from Q.",[376,377,378,379,380],"Draw ray QR pointing left from Q.","Centre point on Q, base line along QR.","QR points left, so use the scale with 0 on the left. Count past 90 up to **130** and mark P.","Join QP. The angle opens wide, more than a right angle, as a 130° angle should.","Warning sign: if your angle looked acute, you used the other scale and drew 180° − 130° = 50°.",{"id":382,"type":195,"component":336,"componentVersion":5,"config":383,"objective":390,"textAlternative":391,"help":392},"lab-construct-u",{"mode":384,"targets":385,"tolerance":5},"construct",[17,386,387,388,389],75,110,135,160,"Use a virtual protractor to draw five angles of given measure, each within 1°.","This lab gives one arm and a target angle. You place the protractor, pick the right scale and set the second arm. Your angle is scored if it is within 1° of the target.\n\nTargets: 40°, 75°, 110°, 135° and 160°. The wrong-scale versions would be 140°, 105°, 70°, 45° and 20°, which look completely different: acute targets would come out obtuse and obtuse targets acute. Picture the target before you start: 40° is a little less than half a right angle; 75° is a little less than a right angle; 110° is a right angle plus a bit; 135° is a right angle plus half a right angle; 160° is almost a straight line.",{"simplerExplanation":393,"hints":394},"Count up from the 0 that sits on your first arm, and stop at the target number.",[395,396],"135° is exactly halfway between 90° and 180°.","Is your target more or less than 90? Does your drawing agree?",{"id":398,"type":46,"variant":66,"title":399,"markdown":400},"misc-dot","“Any 75 on the protractor will do”","There are **two** 75 marks on a protractor, one on each scale, at opposite sides. One gives 75°; the other gives 105°. Only the 75 counted from the **0 on your drawn arm** is right.",{"id":402,"type":52,"title":403,"eyebrow":404,"navLabel":405},"ch6","Reflex angles with a half-circle protractor","Chapter 06","6 Reflex angles",{"id":407,"type":42,"markdown":408},"reflex-intro","A **reflex angle** is more than 180° and less than 360°. A half-circle protractor only reaches 180°, so it cannot measure a reflex angle in one go. There are two simple methods.\n\n**Method 1: measure the other angle and subtract from 360°.** The two angles at a vertex add up to a full turn. Measure the ordinary (non-reflex) angle, x, then the reflex angle is **360° − x**.\n\n**Method 2: split at a straight line.** Extend one arm backwards through the vertex to make a straight line. The reflex angle is then **180° plus** the extra piece beyond the straight line, which you can measure normally.",{"id":410,"type":158,"items":411},"formulas-reflex",[412,415,418],{"expression":413,"caption":414},"reflex = 360° − x","x is the ordinary angle between the same two arms.",{"expression":416,"caption":417},"reflex = 180° + y","y is the part beyond the straight line made by extending one arm.",{"expression":419,"caption":420},"x + reflex = 360°","The two angles at a vertex make a complete turn.",{"id":422,"type":174,"title":423,"problem":424,"steps":425},"we-reflex-measure","Measuring a reflex angle two ways","At vertex O, the ordinary angle between arms OA and OB measures **110°**. Find the reflex angle AOB.",[426,427,428,429],"Method 1: reflex ∠AOB = 360° − 110° = **250°**.","Method 2: extend AO backwards through O to a point A′, making the straight line AOA′ (180°).","Then ∠A′OB is the part of the ordinary angle's partner beyond the straight line: 180° − 110° = 70°. Measure it: it reads 70°.","Reflex ∠AOB = 180° + 70° = **250°**. Both methods agree. ✓",{"id":431,"type":174,"title":432,"problem":433,"steps":434},"we-reflex-draw","Drawing an angle of 250°","Draw ∠POQ = 250°.",[435,436,437,438,439],"Method 1: 360° − 250° = 110°. Draw an ordinary 110° angle POQ with the protractor.","The reflex angle on the **outside** of that 110° is 250°. Mark it with a big arc going the long way round, so everyone knows which angle you mean.","Method 2: draw OP and extend it backwards through O to make a straight line. That is 180°.","250° − 180° = 70°, so from the backwards extension, measure 70° further round (on the same side you are turning) and draw OQ.","Check: the ordinary angle between OP and OQ is 110°, and 110 + 250 = 360. ✓",{"id":441,"type":256,"itemId":442,"prompt":443,"check":444,"hints":448,"feedback":450},"practice-reflex","constructing-angles.und-reflex-300","The ordinary angle between two arms is **60°**. What is the reflex angle between the same arms?",{"kind":445,"answer":446,"tolerance":246,"unit":447},"number",300,"°",[449],"The two angles at the vertex add up to 360°.",{"correct":451,"incorrect":452},"Right: 360° − 60° = 300°.","Subtract from a full turn: 360° − 60° = **300°**.",{"id":454,"type":52,"title":455,"eyebrow":456,"navLabel":457},"ch7","Ruler-and-compass basics","Chapter 07","7 Compass basics",{"id":459,"type":42,"markdown":460},"construct-meaning","In geometry, **to construct** means to draw a figure using only two tools:\n\n- a **straightedge** (a ruler used only for drawing straight lines through two points, never for reading its markings), and\n- a **compass** (for drawing circles and arcs with a chosen centre and radius, and for copying a length).\n\nWhy bother, when a protractor exists? Because a construction is **exact in principle**. A protractor reading is only as good as your eyesight. A construction's correctness comes from a reason, such as \"these three lengths are equal, so this triangle is equilateral\". If your drawing is careful, the result is as accurate as your pencil line allows.",{"id":462,"type":463,"title":464,"terms":465},"glossary-understand","glossary","Construction and measurement vocabulary",[466,470,474,477,480,484,488,491,494,497,500,502,506,509,512,516,519,522,525],{"term":467,"meaning":468,"example":469},"vertex","The common starting point of the two arms of an angle.","In ∠AOB the vertex is O.",{"term":471,"meaning":472,"example":473},"arm","Each of the two rays that form an angle.","OA and OB are the arms of ∠AOB.",{"term":475,"meaning":476},"centre point (protractor)","The point at the centre of the protractor's semicircle, placed on the vertex when measuring.",{"term":478,"meaning":479},"base line","The protractor's zero line through the centre point; one arm lies along it.",{"term":481,"meaning":482,"example":483},"the two scales","The two rows of numbers on a protractor, running in opposite directions with their 0s at opposite ends; readings at one point add to 180. Which row is the inner one varies between protractors.","40 on one scale, 140 on the other.",{"term":485,"meaning":486,"example":487},"reflex angle","An angle of more than 180° and less than 360°.","250°",{"term":489,"meaning":490},"parallax error","A reading error caused by looking at a scale from an angle instead of straight on.",{"term":492,"meaning":493},"arc","A part of the curve of a circle.",{"term":495,"meaning":496},"radius","The distance from the centre of a circle to its edge; for a compass, the gap between the point and the pencil.",{"term":498,"meaning":499},"straightedge","A ruler used only to draw straight lines, not to measure.",{"term":384,"meaning":501},"To draw a figure exactly using only a straightedge and a compass.",{"term":503,"meaning":504,"example":505},"bisect","To cut into two equal parts.","Bisecting 60° gives two 30° angles.",{"term":507,"meaning":508},"angle bisector","The ray that divides an angle into two equal angles.",{"term":510,"meaning":511},"midpoint","The point that divides a line segment into two equal parts.",{"term":513,"meaning":514,"example":515},"perpendicular","Meeting at a right angle (90°).","The sides of a page are perpendicular.",{"term":517,"meaning":518},"perpendicular bisector","The line that passes through the midpoint of a segment at 90° to it.",{"term":520,"meaning":521},"equidistant","At equal distances from two points or lines.",{"term":523,"meaning":524},"equilateral triangle","A triangle with all three sides equal; each of its angles is 60°.",{"term":526,"meaning":527},"tolerance","How far from the exact value a drawing may be and still count as accurate, such as ± 1°.",{"id":529,"type":174,"title":530,"problem":531,"steps":532,"help":537},"we-copy-segment","Copying a line segment with a compass","A segment AB is drawn on the board. Draw a segment PQ of exactly the same length without reading any ruler marks.",[533,534,535,536],"Place the compass point on A and open the compass until the pencil tip is exactly on B.","Draw a straight line with the ruler and mark a point P on it.","Without changing the compass width, put the point on P and draw an arc that crosses the line. Call the crossing point Q.","PQ = AB, because both equal the compass width. A **divider** does the same job: open it on AB and prick the two points onto the new line.",{"simplerExplanation":538},"The compass remembers the length for you. Open it to fit AB, then use that same opening on your new line.",{"id":540,"type":46,"variant":104,"title":541,"markdown":542},"nuance-straightedge","Why not just measure with the ruler?","Copying with the compass avoids reading marks. If AB is 5.37 cm, you would struggle to read that off a ruler, but the compass copies it exactly. In constructions, the ruler's numbers are deliberately ignored.",{"id":544,"type":52,"title":545,"eyebrow":546,"navLabel":547},"ch8","Constructing 60° and 120°","Chapter 08","8 60° and 120°",{"id":549,"type":550,"component":551,"componentVersion":5,"config":552,"textAlternative":554},"anim-60u","animation","compass-construction",{"construction":553},"angle-60","This construction makes an angle of exactly 60° at a point O on a ray OA.\n\n1. Draw the ray OA with a ruler.\n2. With the compass point on O and any convenient radius (say 4 cm), draw a large arc that cuts OA at P. Continue the arc well above the ray.\n3. **Keeping exactly the same radius**, place the compass point on P and draw an arc that cuts the first arc at Q.\n4. Draw the ray OQ with the ruler. ∠AOQ = 60°.\n\n**Why it is exactly 60°.** OP and OQ are both radii of the first arc, so OP = OQ. PQ was drawn with the same radius, so PQ = OP. All three sides of triangle OPQ are equal, so it is equilateral. The three angles of any triangle add up to 180°, and in an equilateral triangle they are all equal, so each one is 180° ÷ 3 = 60°. In particular ∠POQ = 60°.\n\nCheck your drawing with a protractor: a careful construction reads 60° within 1°.",{"id":556,"type":288,"title":557,"items":558},"steps-120","Constructing 120°: two 60° steps",[559,563,567,571],{"title":560,"tag":561,"text":562},"Ray and first arc","centre O","Draw ray OA. With centre O and any radius, draw a big arc cutting OA at P.",{"title":564,"tag":565,"text":566},"First step","centre P","With the same radius and centre P, cut the arc at Q. ∠AOQ = 60°.",{"title":568,"tag":569,"text":570},"Second step","centre Q","With the same radius and centre Q, cut the arc again at R.",{"title":572,"tag":573,"text":574},"Draw the arm","ray OR","Join OR. ∠AOR = ∠AOQ + ∠QOR = 60° + 60° = 120°.",{"id":576,"type":46,"variant":577,"title":578,"markdown":579},"aha-why-120","aha","Why stepping twice is safe","Triangle OQR is also equilateral (OQ = OR are radii, QR is the compass width), so ∠QOR = 60° too. The two 60° angles sit side by side, so together they make 120°. Stepping a third time would give 180°, a straight line, which is a neat way to check your compass has not slipped: the third point should land exactly on the backwards extension of OA.",{"id":581,"type":46,"variant":321,"title":582,"markdown":583},"careful-radius","Never change the radius in the middle","The whole argument depends on OP = PQ. If the compass opens a little while you move it from O to P, PQ is longer than OP and the triangle is no longer equilateral: the angle comes out bigger than 60°. Hold the compass by its top knob and check the radius against the first arc before drawing the second.",{"id":585,"type":256,"itemId":586,"prompt":587,"check":588,"hints":599,"feedback":601},"practice-60","constructing-angles.und-why-60","Why is ∠AOQ in the 60° construction exactly 60° and not just \"about 60°\"?",{"kind":260,"options":589,"correct":598},[590,592,594,596],{"id":263,"label":591},"Because triangle OPQ has three equal sides, so all its angles are 180° ÷ 3",{"id":266,"label":593},"Because the compass was opened to 6 cm",{"id":269,"label":595},"Because P is 60 mm from O",{"id":272,"label":597},"Because Q lies on the protractor's 60 mark",[263],[600],"Which three lengths are equal?",{"correct":602,"incorrect":603},"Exactly. OP = OQ = PQ, so the triangle is equilateral and each angle is 60°.","The reason is the equal lengths: OP and OQ are radii, PQ is the same radius, so triangle OPQ is equilateral and every angle is 180° ÷ 3 = **60°**. The size of the radius does not matter.",{"id":605,"type":52,"title":606,"eyebrow":607,"navLabel":608},"ch9","Perpendicular bisectors and right angles","Chapter 09","9 90° and bisectors",{"id":610,"type":42,"markdown":611},"pb-intro","A **perpendicular bisector** of a segment AB is a line that does two jobs at once: it cuts AB into two equal halves (it passes through the **midpoint**) and it crosses AB at a **right angle**. It is one of the most useful constructions, because it gives you both a midpoint and a 90° angle.",{"id":613,"type":550,"component":551,"componentVersion":5,"config":614,"textAlternative":616},"anim-pb",{"construction":615},"perpendicular-bisector","This construction draws the perpendicular bisector of a segment AB.\n\n1. Draw the segment AB, say 8 cm long.\n2. Open the compass to a radius **more than half of AB** (more than 4 cm here, say 5 cm). With the point on A, draw an arc above AB and an arc below AB.\n3. With the **same radius** and the point on B, draw arcs that cut the first two arcs. Call the crossing points P (above) and Q (below).\n4. Draw the straight line PQ. It crosses AB at a point M.\n\n**Result:** AM = MB, so M is the midpoint of AB, and PQ is perpendicular to AB: all four angles at M are 90°.\n\n**Why the radius must be more than half of AB:** if the radius is less than half, the arcs from A and B never meet, because the two circles do not reach each other. **Why it works:** P is the same distance from A as from B (both are the radius), and so is Q. Every point that is the same distance from A and B lies on the perpendicular bisector, so the line through P and Q is that bisector. Check with a ruler (AM = MB) and a protractor (90°).",{"id":618,"type":174,"title":619,"problem":620,"steps":621},"we-pb","Finding the midpoint of a 7.4 cm line without measuring halves","A segment AB is 7.4 cm long. Find its midpoint using a construction, and state the smallest sensible compass radius.",[622,623,624,625],"Half of 7.4 cm is 3.7 cm, so the radius must be **more than 3.7 cm**; 5 cm is comfortable.","With centre A, radius 5 cm, draw arcs above and below AB.","With centre B, the same radius, cut those arcs at P and Q.","Join PQ; it meets AB at M. M is the midpoint: AM = MB = 3.7 cm, and ∠AMP = 90°.",{"id":627,"type":42,"markdown":628},"ninety-two-ways","There are two standard ways to make a **90° angle at a point O on a ray OA**:\n\n**Method A: between 60° and 120°.** Make the 60° and 120° marks, Q and R, on the big arc as in Chapter 8. Then bisect the angle between them: with centres Q and R and equal radii, draw arcs that meet at S. The ray OS makes 90° with OA, because it sits exactly halfway between 60° and 120°: (60° + 120°) ÷ 2 = 90°.\n\n**Method B: perpendicular at a point on a line.** Extend OA backwards through O to make a straight line. With centre O, draw an arc cutting the line on both sides of O, at X and Y. Now draw the perpendicular bisector of XY: it passes through O (because OX = OY) and makes 90° with the line.",{"id":630,"type":174,"title":631,"problem":632,"steps":633},"we-perp-point","A right angle at a point in the middle of a line (method B)","A straight line XY has a point O somewhere in the middle. Construct a line through O perpendicular to XY.",[634,635,636,637,638],"With centre O and any radius (say 3 cm), draw an arc that cuts the line on both sides of O, at C and D. Now OC = OD.","Open the compass wider than OC (say 5 cm). With centre C, draw an arc above the line.","With centre D and the same radius, draw an arc that cuts the previous one at E.","Draw the line OE. It is perpendicular to XY: ∠EOY = 90°.","Why: E is equally far from C and D, and so is O, so OE is the perpendicular bisector of CD. It is really the straight angle COD (180°) being bisected: 180° ÷ 2 = 90°.",{"id":640,"type":256,"itemId":641,"prompt":642,"check":643,"hints":654,"feedback":656},"practice-pb","constructing-angles.und-pb-midpoint","After drawing the perpendicular bisector PQ of a segment AB, which statement is **always** true about the point M where PQ crosses AB?",{"kind":260,"options":644,"correct":653},[645,647,649,651],{"id":263,"label":646},"AM = MB and ∠AMP = 90°",{"id":266,"label":648},"AM = 2 × MB",{"id":269,"label":650},"∠AMP = 60°",{"id":272,"label":652},"M is always 4 cm from A",[263],[655],"The word bisector means cutting into two equal parts; perpendicular means at 90°.",{"correct":657,"incorrect":658},"Right: M is the midpoint (AM = MB) and the bisector meets AB at a right angle.","The perpendicular bisector passes through the **midpoint**, so AM = MB, and it is **perpendicular**, so ∠AMP = 90°.",{"id":660,"type":550,"component":551,"componentVersion":5,"config":661,"textAlternative":663},"anim-90",{"construction":662},"angle-90","This construction makes an angle of 90° at O on the ray OA, using the 60° and 120° marks.\n\n1. Draw ray OA. With centre O and any radius, draw a large arc cutting OA at P.\n2. Same radius, centre P: cut the arc at Q (this is the 60° mark).\n3. Same radius, centre Q: cut the arc at R (the 120° mark).\n4. With centres Q and R in turn, and any equal radius more than half of QR, draw two arcs that cross at S, above the big arc.\n5. Draw ray OS. ∠AOS = 90°.\n\n**Why:** ray OS bisects ∠QOR, which is 60°, so it is 30° beyond OQ. ∠AOS = 60° + 30° = 90°. Equivalently, OS is halfway between the 60° and 120° marks: (60° + 120°) ÷ 2 = 90°. Check with a protractor or with the corner of a set square.",{"id":665,"type":46,"variant":66,"title":666,"markdown":667},"misc-set-square","“A set square is a construction”","Tracing the corner of a set square gives a very good right angle, but it is not a **construction** in the geometric sense: it relies on the set square being made accurately. A construction uses only the straightedge and compass, and its correctness comes from reasoning. Both are useful; they are just different methods.",{"id":669,"type":52,"title":670,"eyebrow":671,"navLabel":672},"ch10","Bisecting angles: 30°, 45° and checking","Chapter 10","10 Bisecting angles",{"id":674,"type":550,"component":551,"componentVersion":5,"config":675,"textAlternative":677},"anim-bis",{"construction":676},"bisector","This construction bisects a given angle AOB (70° in the animation), cutting it into two equal angles.\n\n1. With the compass point on the vertex O and any convenient radius, draw an arc that cuts arm OA at P and arm OB at Q.\n2. With centre P and a radius more than half of PQ, draw an arc inside the angle.\n3. With centre Q and the **same radius**, draw an arc that cuts the previous one at T.\n4. Draw the ray OT. It is the **angle bisector**: ∠AOT = ∠TOB = half of ∠AOB.\n\n**Why it works:** OP = OQ (radii of the first arc) and PT = QT (the same radius again), and OT is shared. So triangles OPT and OQT have all three sides equal in pairs, which means they are identical in shape and size (congruent), and their angles at O are equal. In the animation ∠AOB = 70°, so each half is 70° ÷ 2 = 35°. Check with a protractor.",{"id":679,"type":71,"caption":680,"columns":681,"rows":685},"table-bisect","Angles you get by bisecting constructed angles",[682,683,684],"Start with","Bisect it","Result",[686,690,693,696,698,701],[687,688,689],"60°","bisect once","30°",[691,688,692],"90°","45°",[689,694,695],"bisect again","15°",[692,694,697],"22.5° (22½°)",[699,688,700],"120°","60° (a second route to 60°)",[702,688,703],"180° (a straight line)","90° (method B)",{"id":705,"type":174,"title":706,"problem":707,"steps":708},"we-30","Constructing 30°","Construct ∠AOT = 30° at the point O on ray OA.",[709,710,711,712],"Construct 60° as usual: arc with centre O cuts OA at P; same radius from P cuts the arc at Q.","Now bisect ∠AOQ: with centres P and Q and equal radii (more than half of PQ), draw arcs meeting at T.","Draw ray OT. ∠AOT = 60° ÷ 2 = **30°**.","Check: measure with a protractor (30°), or compare with the 30° corner of a set square.",{"id":714,"type":174,"title":715,"problem":716,"steps":717},"we-45","Constructing 45°","Construct ∠AOV = 45° at O on ray OA.",[718,719,720,721],"Construct 90° first (for example, between the 60° and 120° marks) and draw ray OS.","The first big arc crosses OS at a point U, and OA at P.","Bisect ∠AOS: with centres P and U and equal radii, draw arcs meeting at V.","Draw ray OV. ∠AOV = 90° ÷ 2 = **45°**.",{"id":723,"type":42,"markdown":724},"checking","**Always check a construction with a protractor.** A careful construction should read within **± 1°** of the target. If it is further off, the usual culprits are: the compass width changed between arcs; the compass point slipped off the exact point; arcs that meet at a very shallow crossing, so the crossing point is blurry; or a blunt pencil. Keep your construction arcs on the page: they are your working and they let a teacher see where a slip happened.",{"id":726,"type":46,"variant":727,"title":728,"markdown":729},"ex-tolerance","example","How big is a 1° error on paper?","On a protractor of radius 5 cm, the edge moves only about 0.9 mm for each degree (5 × π ÷ 180 ≈ 0.087 cm). A pencil line can be half a millimetre wide. So being within 1° is genuinely careful work, and it is the usual standard for school constructions.",{"id":731,"type":732,"conceptId":733,"relation":734,"explanation":735},"conn-angles-u","connection","angles","helps_understand","Angle types and pairs (supplementary angles add to 180°) explain why the two protractor scales always add to 180.",{"id":737,"type":732,"conceptId":738,"relation":734,"explanation":739},"conn-lines-u","lines","Perpendicular lines, rays and segments are the raw materials of every construction.",{"id":741,"type":732,"conceptId":742,"relation":743,"explanation":744},"conn-shape-u","shape-and-space","applied_in","Equilateral triangles, squares and regular hexagons are drawn with the 60°, 90° and 120° constructions.",{"id":746,"type":52,"title":747,"eyebrow":748,"navLabel":749},"ch11","Check your understanding","Chapter 11","11 Wrap-up",{"id":751,"type":71,"caption":752,"columns":753,"rows":757},"table-mistakes","Common slips and how to fix them",[754,755,756],"What went wrong","What you see","Fix",[758,762,766,770,774,778,782],[759,760,761],"Read the wrong scale","Answer is 180° − the true angle (acute looks obtuse)","Estimate first; read from the 0 on the lined-up arm",[763,764,765],"Vertex not on the centre point","Reading off by several degrees","Put the centre mark exactly on the vertex",[767,768,769],"Arm along the plastic edge","Small, steady error","Use the printed base line through the centre",[771,772,773],"Arms too short","Cannot see where the arm crosses","Extend the arms lightly with a ruler",[775,776,777],"Compass width changed","60° comes out as 64° or 57°","Hold the compass by the top; re-check the width",[779,780,781],"Arcs cross at a shallow angle","Blurry crossing point","Use a larger radius so arcs cross more steeply",[783,784,785],"Blunt pencil","Thick lines, ± 2° uncertainty","Sharpen; draw arcs lightly and thinly",{"id":787,"type":256,"itemId":788,"prompt":789,"check":790,"hints":792,"feedback":794},"practice-draw-other","constructing-angles.und-wrong-scale-draw","Meera meant to draw **35°** but used the wrong scale. What size of angle did she actually draw, in degrees?",{"kind":445,"answer":791,"tolerance":246,"unit":447},145,[793],"The two scales add up to 180.",{"correct":795,"incorrect":796},"Right: the wrong scale gives 180° − 35° = 145°.","The other scale's 35 mark is where the correct scale reads 180 − 35, so she drew **145°**, an obtuse angle.",{"id":798,"type":799,"prompt":800,"options":801,"explanation":808},"predict-bisect-120","prediction","You construct 120° and then bisect it. Then you bisect **one of the halves**. What angle is each of the smallest pieces?",[802,803,805,806],{"id":263,"label":687},{"id":266,"label":804},"40°",{"id":269,"label":689},{"id":272,"label":807},"20°","**30°.** Bisecting 120° gives two 60° angles; bisecting one of them gives two 30° angles. Bisecting always halves: 120 → 60 → 30. Notice you can never get 40° or 20° by halving 120° this way, because 120 ÷ 3 = 40 is a *third*, not a half. Cutting an angle into three equal parts with ruler and compass is a famous problem you will meet in later layers.",{"id":810,"type":811,"prompt":812},"reflect-understand","reflection","Explain to a younger student, in three or four sentences, how to decide which protractor scale to read. Then explain why the 60° construction gives *exactly* 60°, using the words \"equal\" and \"triangle\".",{"id":814,"type":815,"title":816,"questions":817},"quiz-understand","quiz","Protractors and constructions",[818,831,844,857,870,883,896,909,922,935,948,961],{"itemId":819,"prompt":820,"options":821,"correct":266,"why":830},"constructing-angles.und-q-arms","Two angles have the same opening, but one has arms 3 cm long and the other 9 cm. How do their sizes compare?",[822,824,826,828],{"id":263,"label":823},"The 9 cm one is 3 times bigger",{"id":266,"label":825},"They are equal",{"id":269,"label":827},"The 3 cm one is bigger",{"id":272,"label":829},"You cannot tell","An angle's size is the amount of turn between the arms, not their length.",{"itemId":832,"prompt":833,"options":834,"correct":266,"why":843},"constructing-angles.und-q-other-scale","The arm crosses at 72 on the correct scale. What is on the other scale at that point?",[835,837,839,841],{"id":263,"label":836},"72",{"id":266,"label":838},"108",{"id":269,"label":840},"288",{"id":272,"label":842},"18","The two scales always add to 180: 180 − 72 = 108.",{"itemId":845,"prompt":846,"options":847,"correct":266,"why":856},"constructing-angles.und-q-baseline","What is the base line of a protractor?",[848,850,852,854],{"id":263,"label":849},"Its bottom plastic edge",{"id":266,"label":851},"The line through the centre point joining the two zeros",{"id":269,"label":853},"The 90° line",{"id":272,"label":855},"The curved edge","The base line passes through the centre point; the plastic edge is often a few mm below it.",{"itemId":858,"prompt":859,"options":860,"correct":266,"why":869},"constructing-angles.und-q-short","The arms of a drawn angle are too short to reach the scale. What should you do?",[861,863,865,867],{"id":263,"label":862},"Guess where they would cross",{"id":266,"label":864},"Extend them lightly with a ruler",{"id":269,"label":866},"Use the smaller number",{"id":272,"label":868},"Measure the reflex angle instead","Extending the arms does not change the angle and lets you read the scale.",{"itemId":871,"prompt":872,"options":873,"correct":266,"why":882},"constructing-angles.und-q-reflex","The ordinary angle between two arms is 145°. What is the reflex angle?",[874,876,878,880],{"id":263,"label":875},"35°",{"id":266,"label":877},"215°",{"id":269,"label":879},"235°",{"id":272,"label":881},"325°","360° − 145° = 215°.",{"itemId":884,"prompt":885,"options":886,"correct":266,"why":895},"constructing-angles.und-q-reflex-split","A reflex angle is split into a straight angle plus 40°. How big is it?",[887,889,891,893],{"id":263,"label":888},"140°",{"id":266,"label":890},"220°",{"id":269,"label":892},"320°",{"id":272,"label":894},"260°","180° + 40° = 220°.",{"itemId":897,"prompt":898,"options":899,"correct":266,"why":908},"constructing-angles.und-q-120","In the 120° construction, how many times do you step the same radius along the arc from P?",[900,902,904,906],{"id":263,"label":901},"Once",{"id":266,"label":903},"Twice",{"id":269,"label":905},"Three times",{"id":272,"label":907},"Four times","Each step adds 60°: P → Q is 60°, Q → R is another 60°, so ∠AOR = 120°.",{"itemId":910,"prompt":911,"options":912,"correct":266,"why":921},"constructing-angles.und-q-pb-radius","To draw the perpendicular bisector of a 10 cm segment, the compass radius must be:",[913,915,917,919],{"id":263,"label":914},"exactly 5 cm",{"id":266,"label":916},"more than 5 cm",{"id":269,"label":918},"less than 5 cm",{"id":272,"label":920},"exactly 10 cm","With a radius of 5 cm or less, the arcs from the two ends do not cross above and below the segment.",{"itemId":923,"prompt":924,"options":925,"correct":263,"why":934},"constructing-angles.und-q-90","Which construction gives 90° at O on ray OA?",[926,928,930,932],{"id":263,"label":927},"Bisect the angle between the 60° and 120° marks",{"id":266,"label":929},"Step the radius three times",{"id":269,"label":931},"Bisect the 60° angle",{"id":272,"label":933},"Bisect the 120° angle","Halfway between 60° and 120° is (60 + 120) ÷ 2 = 90°.",{"itemId":936,"prompt":937,"options":938,"correct":266,"why":947},"constructing-angles.und-q-45","How do you construct 45°?",[939,941,943,945],{"id":263,"label":940},"Bisect 60°",{"id":266,"label":942},"Bisect 90°",{"id":269,"label":944},"Bisect 120°",{"id":272,"label":946},"Step the radius once","90° ÷ 2 = 45°.",{"itemId":949,"prompt":950,"options":951,"correct":263,"why":960},"constructing-angles.und-q-bisector-why","In the angle-bisector construction, why are the two halves equal?",[952,954,956,958],{"id":263,"label":953},"The two triangles OPT and OQT have all three sides equal in pairs",{"id":266,"label":955},"Because the arcs are curved",{"id":269,"label":957},"Because T is on the protractor",{"id":272,"label":959},"Because OP is longer than PT","OP = OQ, PT = QT and OT is shared, so the triangles are congruent and the angles at O are equal.",{"itemId":962,"prompt":963,"options":964,"correct":263,"why":973},"constructing-angles.und-q-check","A student's constructed 30° measures 34° with a protractor. What is the most likely cause?",[965,967,969,971],{"id":263,"label":966},"The compass width changed between arcs",{"id":266,"label":968},"30° cannot be constructed",{"id":269,"label":970},"The protractor has two scales",{"id":272,"label":972},"The arms were too long","A slipping compass breaks the equal-length reasoning; 30° is certainly constructible, and long arms help accuracy.",{"id":975,"type":976,"title":977,"points":978},"cheat-understand","summary","Cheat sheet",[979,980,981,982,983,984,985,986,987,988,989],"**1° = 1\u002F360 of a full turn.** An angle's size is the turn between its arms; arm length does not matter, so short arms may be extended.","**Protractor:** centre point on the vertex, base line (not the plastic edge) along one arm, read the scale whose **0 is on that arm**.","The two scales add to **180**: the wrong scale gives 180° − x. An estimate (acute or obtuse?) always catches it.","**Drawing:** draw one arm, count up from its 0 to the target, dot, join. Check it looks right.","**Reflex angles:** 360° − (the ordinary angle), or 180° + (the part beyond the straight line).","**Construct** = straightedge (no measuring) + compass. The result is exact in principle because it rests on a reason.","**60°:** equal arcs from O and from P give an equilateral triangle OPQ. **120°:** step the radius twice.","**Perpendicular bisector of AB:** equal arcs (radius more than ½AB) from A and B meet at P and Q; PQ passes through the midpoint at 90°.","**90°:** bisect between the 60° and 120° marks, or draw the perpendicular at a point on a line.","**Angle bisector:** arc from the vertex cuts the arms at P and Q; equal arcs from P and Q meet at T; OT halves the angle. 60° → **30°**, 90° → **45°**.","**Check every construction** with a protractor: within ± 1° is careful work.",{"id":991,"type":992,"sourceIds":993},"sources-understand","sources",[994,995,996,997,998,999,1000,1001,1002,1003],"constructing-angles-ncert-math-6-practical-geometry","constructing-angles-ncert-ganita-prakash-6","constructing-angles-ncert-math-7-practical-geometry","constructing-angles-mathsisfun-protractor","constructing-angles-mathsisfun-constructions","constructing-angles-mathsisfun-degrees","constructing-angles-wikipedia-straightedge-compass","constructing-angles-wikipedia-angle-trisection","constructing-angles-wikipedia-shulba-sutras","constructing-angles-britannica-euclid-elements",[994,995,996,997,998,999,1000,1001,1002,1003],"needs_review",{"generatedBy":1007,"notes":1008},"claude-code","Draft generated with Python; every angle fact was computed and asserted. Pending owner review.","9427b0f683b78984e21e078a598e1e426805b2105149466a4da911dae7eb5ec0",{"component:sort-game@1":1011,"logic:practice":1012,"component:protractor@1":1013,"component:compass-construction@1":1014,"source:constructing-angles-britannica-euclid-elements":1015,"source:constructing-angles-mathsisfun-constructions":1016,"source:constructing-angles-mathsisfun-degrees":1017,"source:constructing-angles-mathsisfun-protractor":1018,"source:constructing-angles-ncert-ganita-prakash-6":1019,"source:constructing-angles-ncert-math-6-practical-geometry":1020,"source:constructing-angles-ncert-math-7-practical-geometry":1021,"source:constructing-angles-wikipedia-angle-trisection":1022,"source:constructing-angles-wikipedia-shulba-sutras":1023,"source:constructing-angles-wikipedia-straightedge-compass":1024},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","7407db21456592711dd16c6bdad23f042e85ebab9c314b072b17b7065eaf8ae3","0eba81381f31d8d78008911eb4c3d62745efd33b4ffadd94aef0851d0a2ad5f3","ad1a9a6a227fda5d3c1569f37efbe35e448ebaceba8cba872821fd48e2e00ed6","b93faffbd9c4d40f5fce2bc4b2ea0ab5ac64bb8c176f5e2bba3f37444df5e400","216eb0db510461864a47157f14054a39e15b1b0fc461b0fbc77664d9eb28b91d","4d3f50c07f44df57c80455dc39e01b2aa11bb0ee40811fca3b0f12d16e0b3f5e","acad5a4d56d24a5c1ad6e908f3809f2e7b3978f7c2810a9b33cdf82a65c4bb47","4aedaa1be389589b6e840923ef4e92fd15d03eda0b0ba0302d57b7e71bcb3dc7","21119e12648b9efd4cc82b11c59d626f2a53eace3a70552041f26c311b77ba2d","7f00387dc29d17172d25b6aa96420e2544a8bc59edf939af3dce91d515300a1f","f7559697a2f963f9cb1e02a93fc5697840f583f22605d7483d688664862d70f9","90d6cc756bc1bdd6cde0d5e4ed2000c88c3e2f3a8d91fdaf0ec4ab73af72b434",{"state":1026,"reviewer":1027,"selfReview":1028,"reviewedAt":1029,"method":1030},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597613]