[{"data":1,"prerenderedAt":1074},["ShallowReactive",2],{"questions:constructing-angles":3},{"bank":4,"contentHash":1060,"dependencyHashes":1061,"releaseId":1073},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1045,"reviewStatus":1056,"authoring":1057},1,"constructing-angles","Measuring and constructing angles: question bank","Practise every skill in this topic: choosing tools from the geometry box, reading the correct protractor scale, measuring and drawing angles (including reflex angles), ruler-and-compass constructions and why they work, building angles such as 75° and 135°, set-square combinations, and accuracy. Keep your geometry box handy: many questions are best checked by actually drawing. Every question has a full worked solution.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"tools","The geometry box","What each tool is for, and using it with care and accuracy.",{"id":15,"title":16,"description":17},"reading","Reading a protractor","The two scales, why they add to 180, and avoiding the wrong-scale mistake.",{"id":19,"title":20,"description":21},"measuring","Measuring angles","Step-by-step measuring, extending short arms, and checking readings with angle sums.",{"id":23,"title":24,"description":25},"drawing","Drawing angles","Drawing an angle of a given measure with a protractor.",{"id":27,"title":28,"description":29},"reflex","Reflex angles","Measuring and drawing angles greater than 180°.",{"id":31,"title":32,"description":33},"compass","Compass constructions","Steps, order and reasons for 60°, 120°, 90°, bisectors, perpendicular bisectors and copying angles.",{"id":35,"title":36,"description":37},"constructible","Constructible angles and combinations","Building 15°, 22.5°, 45°, 75°, 105°, 135°, 150°, 165° and seeing why 20° is impossible.",{"id":39,"title":40,"description":41},"setsquares","Set squares","Angles from the two set squares and their sums and differences.",{"id":43,"title":44,"description":45},"accuracy","Accuracy and reasoning","Tolerance, how small a degree is, and checking constructions.",[47,72,90,108,123,139,157,177,188,197,206,224,234,244,253,263,280,290,307,317,326,344,356,374,383,392,410,419,427,439,447,454,464,472,480,489,506,517,534,543,562,572,590,598,606,624,642,659,667,684,702,719,727,743,752,767,786,803,811,820,838,856,871,878,895,912,921,938,954,971,988,1001,1018,1028],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":67,"solution":69,"skills":70},"constructing-angles.q001","foundation","Which tool in the geometry box is used to draw circles and arcs?",{"kind":52,"options":53,"correct":66},"choice",[54,57,60,63],{"id":55,"label":56},"a","Divider",{"id":58,"label":59},"b","Compass",{"id":61,"label":62},"c","Protractor",{"id":64,"label":65},"d","Set square",[58],[68],"Which tool holds a pencil in one leg?","A **compass** has a sharp point that stays at the centre and a pencil leg that swings round it, so it draws circles and parts of circles (arcs). A divider has two sharp points and no pencil.",[71],"geometry box",{"id":73,"section":11,"level":49,"prompt":74,"check":75,"hints":86,"solution":88,"skills":89},"constructing-angles.q002","What is a **divider** mainly used for?",{"kind":52,"options":76,"correct":85},[77,79,81,83],{"id":55,"label":78},"Drawing straight lines",{"id":58,"label":80},"Measuring angles in degrees",{"id":61,"label":82},"Comparing and copying lengths",{"id":64,"label":84},"Drawing right angles",[61],[87],"A divider has no pencil.","A divider has two sharp metal points. Open it to fit a length, lift it without changing the gap, and prick the same length somewhere else. It **compares and copies lengths**; it cannot draw.",[71],{"id":91,"section":11,"level":49,"prompt":92,"check":93,"hints":104,"solution":105,"skills":106},"constructing-angles.q003","What are the three angles of the **45° set square**?",{"kind":52,"options":94,"correct":103},[95,97,99,101],{"id":55,"label":96},"30°, 60°, 90°",{"id":58,"label":98},"45°, 45°, 90°",{"id":61,"label":100},"45°, 60°, 75°",{"id":64,"label":102},"60°, 60°, 60°",[58],[],"The 45° set square is a right-angled triangle with two equal angles: 45° + 45° + 90° = 180°. The other set square has 30°, 60° and 90°.",[107],"set squares",{"id":109,"section":11,"level":110,"prompt":111,"check":112,"hints":117,"solution":120,"skills":121},"constructing-angles.q004","core","On Kabir's old ruler the 0 mark is **3 mm** in from the end. He lines up the **end** of the ruler with the start of a line and reads **8.3 cm** at the other end of the line. What is the true length of the line, in cm?",{"kind":113,"answer":114,"tolerance":115,"unit":116},"number",8.6,0,"cm",[118,119],"Convert to millimetres first.","The line starts 3 mm before the ruler's 0.","The reading 8.3 cm = 83 mm is the distance from the **0 mark** to the end of the line. But the line starts at the end of the ruler, 3 mm **before** the 0 mark, so that first 3 mm was never counted. The true length is 83 mm + 3 mm = **86 mm = 8.6 cm**. Always start from the 0 mark, not the end.",[122,43],"ruler",{"id":124,"section":11,"level":110,"prompt":125,"check":126,"hints":135,"solution":136,"skills":137},"constructing-angles.q005","True or false: in a ruler-and-compass **construction**, you may use the ruler's centimetre marks to measure lengths.",{"kind":52,"options":127,"correct":134},[128,131],{"id":129,"label":130},"t","True",{"id":132,"label":133},"f","False",[132],[],"**False.** In a construction the ruler is used only as a **straightedge**, to draw a straight line through two points. Lengths are copied with the compass. (A question may still give you a starting length, such as \"draw AB = 6 cm\", which you measure once before the construction begins.)",[138],"construction rules",{"id":140,"section":11,"level":110,"prompt":141,"check":142,"hints":153,"solution":155,"skills":156},"constructing-angles.q006","You need to draw a circle of radius **3.5 cm** and then a **123°** angle at its centre. Which two tools do you need, in order?",{"kind":52,"options":143,"correct":152},[144,146,148,150],{"id":55,"label":145},"Compass, then protractor",{"id":58,"label":147},"Divider, then set square",{"id":61,"label":149},"Protractor, then compass",{"id":64,"label":151},"Set square, then compass",[55],[154],"Is 123° made from 30°, 45°, 60° or 90°?","The circle needs a **compass** (open it to 3.5 cm against a ruler). 123° is not a set-square angle and not a standard construction, so it needs the **protractor**.",[71],{"id":158,"section":11,"level":159,"prompt":160,"check":161,"hints":172,"solution":174,"skills":175},"constructing-angles.q007","stretch","Why should the hinge of a compass be tight before you construct a 60° angle?",{"kind":52,"options":162,"correct":171},[163,165,167,169],{"id":55,"label":164},"So the width cannot change between the two arcs, keeping all three sides of the triangle equal",{"id":58,"label":166},"So the pencil stays sharp",{"id":61,"label":168},"So the arc is darker",{"id":64,"label":170},"It does not matter for 60°",[55],[173],"What makes the triangle equilateral?","The 60° construction works because OP = OQ = PQ (all equal to the compass width), making an equilateral triangle. If a loose hinge lets the width change between arcs, PQ is no longer equal to OP and the angle is no longer 60°.",[43,176],"60° construction",{"id":178,"section":15,"level":49,"prompt":179,"check":180,"hints":183,"solution":185,"skills":186},"constructing-angles.q008","The second arm of an **acute** angle crosses the protractor where the scales show **50** and **130**. What is the angle?",{"kind":113,"answer":181,"tolerance":115,"unit":182},50,"°",[184],"Acute angles are less than 90°.","Acute means less than 90°, so of the two numbers only **50°** fits. The other scale shows 130 because the scales always add to 180: 50 + 130 = 180.",[187],"protractor scales",{"id":189,"section":15,"level":49,"prompt":190,"check":191,"hints":193,"solution":195,"skills":196},"constructing-angles.q009","The correct scale reads **35** where the second arm crosses. What does the **other** scale read at that point?",{"kind":113,"answer":192,"tolerance":115},145,[194],"The two numbers add up to 180.","The two scales run in opposite directions and always add to 180 at the same point: 180 − 35 = **145**.",[187],{"id":198,"section":15,"level":110,"prompt":199,"check":200,"hints":202,"solution":203,"skills":204},"constructing-angles.q010","An angle is clearly wider than a right angle. Its second arm crosses where the scales show **70** and **110**. What is the angle?",{"kind":113,"answer":201,"tolerance":115,"unit":182},110,[],"Wider than a right angle means obtuse, more than 90°. Only **110°** fits. 70 is on the other scale (70 + 110 = 180).",[187,205],"estimation",{"id":207,"section":15,"level":110,"prompt":208,"check":209,"hints":220,"solution":222,"skills":223},"constructing-angles.q011","You line up an arm with the base line so that it points to the **left** of the centre point. Which of the protractor's two scales do you read?",{"kind":52,"options":210,"correct":219},[211,213,215,217],{"id":55,"label":212},"The scale whose 0 is at the right-hand end",{"id":58,"label":214},"The scale whose 0 is at the left-hand end",{"id":61,"label":216},"Whichever number is smaller",{"id":64,"label":218},"Whichever number is bigger",[58],[221],"Where is the 0 of each scale?","Always read the scale whose **0 lies on the lined-up arm**. The arm points left, so read the scale whose 0 is at the **left-hand end** of the base line. (Do not go by “inner” or “outer”: which row is printed inside differs from protractor to protractor.) “Smaller” or “bigger” only works after you have estimated whether the angle is acute or obtuse.",[187],{"id":225,"section":15,"level":110,"prompt":226,"check":227,"hints":229,"solution":231,"skills":232},"constructing-angles.q012","Ravi measures the angle at the tip of a slice of pizza and writes **150°**. The tip is clearly sharp and narrow. What is the most likely true size of the angle?",{"kind":113,"answer":228,"tolerance":115,"unit":182},30,[230],"What does the other scale show?","A sharp, narrow tip is acute, so 150° cannot be right. Ravi read the wrong scale; the correct reading is 180 − 150 = **30°**.",[233,187],"spot the mistake",{"id":235,"section":15,"level":110,"prompt":236,"check":237,"hints":239,"solution":241,"skills":242},"constructing-angles.q013","At which reading do **both** scales of a protractor show the same number?",{"kind":113,"answer":238,"tolerance":115},90,[240],"Solve x = 180 − x.","The two readings at a point are x and 180 − x. They are equal when x = 180 − x, so 2x = 180 and x = **90**. That is why a right angle can never be misread.",[187,243],"reasoning",{"id":245,"section":15,"level":159,"prompt":246,"check":247,"hints":248,"solution":250,"skills":251},"constructing-angles.q014","The two scales at the crossing point differ by **40**. The angle is obtuse. What is it?",{"kind":113,"answer":201,"tolerance":115,"unit":182},[249],"Write the two readings as x and 180 − x.","Call the angle x, so the other scale reads 180 − x. The difference is x − (180 − x) = 2x − 180 = 40, so 2x = 220 and x = **110°**. Check: the other scale reads 70, and 110 − 70 = 40. ✓",[187,252],"algebra",{"id":254,"section":15,"level":255,"prompt":256,"check":257,"hints":259,"solution":261,"skills":262},"constructing-angles.q015","challenge","The two readings at the crossing point differ by **100**, and the angle is acute. What is the angle?",{"kind":113,"answer":258,"tolerance":115,"unit":182},40,[260],"The bigger reading is 180 − x.","Let the angle be x (acute, so it is the smaller reading). The other reading is 180 − x. Then (180 − x) − x = 100, so 180 − 2x = 100, 2x = 80, and x = **40°**. Check: 40 and 140 differ by 100. ✓",[187,252],{"id":264,"section":19,"level":49,"prompt":265,"check":266,"hints":277,"solution":278,"skills":279},"constructing-angles.q016","Where must you place the **centre point** of the protractor when measuring an angle?",{"kind":52,"options":267,"correct":276},[268,270,272,274],{"id":55,"label":269},"On the vertex",{"id":58,"label":271},"At the end of the longer arm",{"id":61,"label":273},"Halfway along an arm",{"id":64,"label":275},"Anywhere inside the angle",[55],[],"The centre point goes exactly on the **vertex**, the point where the two arms meet. Then the base line is lined up along one arm.",[19],{"id":281,"section":19,"level":49,"prompt":282,"check":283,"hints":285,"solution":287,"skills":288},"constructing-angles.q017","At **2 o'clock**, what angle do the hands of a clock make?",{"kind":113,"answer":284,"tolerance":115,"unit":182},60,[286],"Each hour gap is 360 ÷ 12 degrees.","A clock face is 360° split into 12 equal hour gaps, so each gap is 360 ÷ 12 = 30°. At 2 o'clock the hands are 2 gaps apart: 2 × 30° = **60°**.",[289],"benchmarks",{"id":291,"section":19,"level":110,"prompt":292,"check":293,"hints":304,"solution":305,"skills":306},"constructing-angles.q018","An angle is drawn with arms only 1 cm long, so they do not reach the protractor's scale. What should you do?",{"kind":52,"options":294,"correct":303},[295,297,299,301],{"id":55,"label":296},"Extend both arms lightly with a ruler, then measure",{"id":58,"label":298},"Estimate and write that down",{"id":61,"label":300},"Measure the reflex angle instead",{"id":64,"label":302},"Use a bigger protractor only",[55],[],"An angle's size is the turn between its arms, not their length, so **extending the arms** with a ruler does not change it. Then the extended arm reaches the scale and can be read accurately.",[19],{"id":308,"section":19,"level":110,"prompt":309,"check":310,"hints":312,"solution":314,"skills":315},"constructing-angles.q019","Two angles of a triangle measure **58°** and **64°**. Without measuring, what should the third angle be?",{"kind":113,"answer":311,"tolerance":115,"unit":182},58,[313],"The three angles of a triangle add to 180°.","The angles of a triangle add up to 180°. 180° − 58° − 64° = **58°**. You can use this to check your protractor readings.",[19,316],"checking",{"id":318,"section":19,"level":110,"prompt":319,"check":320,"hints":322,"solution":324,"skills":325},"constructing-angles.q020","Three angles of a quadrilateral field (a kabaddi court drawn badly) measure **88°**, **95°** and **102°**. What must the fourth angle be?",{"kind":113,"answer":321,"tolerance":115,"unit":182},75,[323],"A quadrilateral splits into two triangles.","The angles of a quadrilateral add up to 360°. 360° − 88° − 95° − 102° = **75°**.",[19,316],{"id":327,"section":19,"level":159,"prompt":328,"check":329,"hints":340,"solution":342,"skills":343},"constructing-angles.q021","Anu measures the three angles of a triangle she drew as **61°**, **72°** and **49°**. What does this tell her?",{"kind":52,"options":330,"correct":339},[331,333,335,337],{"id":55,"label":332},"Her readings add to 182°, so at least one is off; with care each should be within about 1°",{"id":58,"label":334},"Her triangle is impossible",{"id":61,"label":336},"Everything is exactly right",{"id":64,"label":338},"She must have measured reflex angles",[55],[341],"Add the three readings.","61 + 72 + 49 = 182, but the angles of any triangle add to exactly 180°. So her measurements carry a total error of 2°, for example about 1° on two of them. That is normal for hand measurement; it tells her to re-check the readings, not that the triangle is impossible.",[43,316],{"id":345,"section":19,"level":159,"prompt":346,"check":347,"hints":351,"solution":353,"skills":354},"constructing-angles.q022","Put these measuring steps in the correct order and type the letters, for example ABCDE.\n\nA. Read where the second arm crosses the scale.\nB. Estimate the size of the angle.\nC. Line up the base line along one arm.\nD. Place the centre point on the vertex.\nE. Choose the scale with 0 on the lined-up arm.",{"kind":348,"accept":349},"text",[350],"BDCEA",[352],"The estimate comes before you touch the protractor.","Estimate first (**B**), place the centre on the vertex (**D**), line up the base line along one arm (**C**), choose the scale with 0 on that arm (**E**), then read the crossing (**A**): **BDCEA**. Finally compare the reading with your estimate.",[19,355],"steps",{"id":357,"section":23,"level":49,"prompt":358,"check":359,"hints":370,"solution":371,"skills":372},"constructing-angles.q023","You want to draw a **60°** angle. Your first arm points to the **right** from the vertex. Which scale do you count along?",{"kind":52,"options":360,"correct":369},[361,363,365,367],{"id":55,"label":362},"The scale that has 0 on the right",{"id":58,"label":364},"The scale that has 0 on the left",{"id":61,"label":366},"Either one",{"id":64,"label":368},"The scale that shows 120",[55],[],"Count from the 0 that sits on the arm you have drawn. The arm points right, so use the scale with **0 on the right**. The other scale would give 180° − 60° = 120°.",[373,187],"drawing angles",{"id":375,"section":23,"level":110,"prompt":376,"check":377,"hints":379,"solution":381,"skills":382},"constructing-angles.q024","Neha meant to draw **40°** but counted along the wrong scale. What angle did she actually draw?",{"kind":113,"answer":378,"tolerance":115,"unit":182},140,[380],"The two scales add to 180.","The 40 on the wrong scale sits where the correct scale reads 180 − 40 = **140**, so she drew a 140° angle. An estimate (\"40° is less than half a right angle\") would have warned her that her obtuse drawing was wrong.",[373,233],{"id":384,"section":23,"level":110,"prompt":385,"check":386,"hints":389,"solution":390,"skills":391},"constructing-angles.q025","Put the steps for drawing ∠ABC = 70° in order and type the letters, for example ABCD.\n\nA. Join B to the dot with a ruler.\nB. Draw ray BC.\nC. Mark a dot at 70 on the scale with 0 on BC.\nD. Place the centre point on B with the base line along BC.",{"kind":348,"accept":387},[388],"BDCA",[],"Draw the first arm (**B**), place the protractor (**D**), mark the dot on the correct scale (**C**), then join (**A**): **BDCA**. Finally check that the angle looks a little less than a right angle.",[373,355],{"id":393,"section":23,"level":110,"prompt":394,"check":395,"hints":406,"solution":408,"skills":409},"constructing-angles.q026","To draw an angle of **125°**, which is quicker: a protractor or set squares?",{"kind":52,"options":396,"correct":405},[397,399,401,403],{"id":55,"label":398},"A protractor: 125° is not a sum of set-square angles",{"id":58,"label":400},"Set squares: 90° + 35°",{"id":61,"label":402},"Set squares: 60° + 65°",{"id":64,"label":404},"Neither can do it",[55],[407],"Is 125 a multiple of 15?","Set squares carry only 30°, 45°, 60° and 90°. Sums and differences of these are all multiples of 15° (like 105° or 135°), and 125 is not a multiple of 15. So use the **protractor**.",[373,107],{"id":411,"section":23,"level":159,"prompt":412,"check":413,"hints":414,"solution":416,"skills":417},"constructing-angles.q027","With a half-circle protractor you want to draw a reflex angle of **250°**. What ordinary angle can you draw first, so that the angle round the outside is 250°?",{"kind":113,"answer":201,"tolerance":115,"unit":182},[415],"The two angles at the vertex add to 360°.","The ordinary angle and the reflex angle between the same arms make a full turn: 360° − 250° = **110°**. Draw 110°, then mark the reflex angle with an arc going the long way round.",[418,373],"reflex angles",{"id":420,"section":23,"level":159,"prompt":421,"check":422,"hints":424,"solution":425,"skills":426},"constructing-angles.q028","Another way to draw **200°**: extend the first arm backwards to make a straight line (180°), then turn a bit more. How many more degrees must you measure beyond the straight line?",{"kind":113,"answer":423,"tolerance":115,"unit":182},20,[],"200° = 180° + 20°. So measure **20°** beyond the straight line, on the side you are turning towards, and draw the second arm there.",[418,373],{"id":428,"section":27,"level":49,"prompt":429,"check":430,"hints":435,"solution":436,"skills":437},"constructing-angles.q029","True or false: an angle of **200°** is a reflex angle.",{"kind":52,"options":431,"correct":434},[432,433],{"id":129,"label":130},{"id":132,"label":133},[129],[],"**True.** A reflex angle is more than 180° and less than 360°, and 200° is in that range.",[438],"angle types",{"id":440,"section":27,"level":110,"prompt":441,"check":442,"hints":444,"solution":445,"skills":446},"constructing-angles.q030","The ordinary angle between two arms is **75°**. What is the reflex angle between them?",{"kind":113,"answer":443,"tolerance":115,"unit":182},285,[],"The two angles make a complete turn: 360° − 75° = **285°**.",[418],{"id":448,"section":27,"level":110,"prompt":449,"check":450,"hints":451,"solution":452,"skills":453},"constructing-angles.q031","A reflex angle measures **330°**. What is the ordinary angle between the same two arms?",{"kind":113,"answer":228,"tolerance":115,"unit":182},[],"360° − 330° = **30°**.",[418],{"id":455,"section":27,"level":110,"prompt":456,"check":457,"hints":459,"solution":461,"skills":462},"constructing-angles.q032","At **4 o'clock**, what is the **reflex** angle between the hands of a clock?",{"kind":113,"answer":458,"tolerance":115,"unit":182},240,[460],"First find the ordinary angle: 30° per hour gap.","The ordinary angle is 4 hour gaps × 30° = 120°. The reflex angle is 360° − 120° = **240°**.",[418,463],"clock",{"id":465,"section":27,"level":159,"prompt":466,"check":467,"hints":469,"solution":470,"skills":471},"constructing-angles.q033","To measure a reflex angle, Sam extends one arm backwards through the vertex and measures the extra piece beyond the straight line as **55°**. How big is the reflex angle?",{"kind":113,"answer":468,"tolerance":115,"unit":182},235,[],"The straight line gives 180°, and the extra piece adds 55°: 180° + 55° = **235°**. Check with the other method: the ordinary angle is 360° − 235° = 125°.",[418],{"id":473,"section":27,"level":159,"prompt":474,"check":475,"hints":477,"solution":478,"skills":479},"constructing-angles.q034","The minute hand of a clock turns clockwise from 12 to 9. Through how many degrees has it turned?",{"kind":113,"answer":476,"tolerance":115,"unit":182},270,[],"From 12 to 9 is 45 minutes, and the minute hand turns 6° per minute (360° ÷ 60): 45 × 6° = **270°**, a reflex angle, three quarters of a turn.",[418,463],{"id":481,"section":27,"level":255,"prompt":482,"check":483,"hints":485,"solution":487,"skills":488},"constructing-angles.q035","At a vertex, the reflex angle is **5 times** the ordinary angle. Find the reflex angle.",{"kind":113,"answer":484,"tolerance":115,"unit":182},300,[486],"The two angles add up to 360°.","Let the ordinary angle be x. Then x + 5x = 360°, so 6x = 360° and x = 60°. The reflex angle is 5 × 60° = **300°**. Check: 60 + 300 = 360. ✓",[418,252],{"id":490,"section":31,"level":49,"prompt":491,"check":492,"hints":503,"solution":504,"skills":505},"constructing-angles.q036","In the 60° construction, after drawing ray OA and an arc with centre O that cuts OA at P, what do you do next?",{"kind":52,"options":493,"correct":502},[494,496,498,500],{"id":55,"label":495},"With the same radius and centre P, cut the arc at Q",{"id":58,"label":497},"Open the compass wider and draw from A",{"id":61,"label":499},"Measure 60° with the protractor",{"id":64,"label":501},"Draw a line from P upwards",[55],[],"Keep the **same radius**, put the compass point on **P**, and cut the first arc at Q. Then ray OQ makes 60° with OA.",[176,355],{"id":507,"section":31,"level":49,"prompt":508,"check":509,"hints":514,"solution":515,"skills":516},"constructing-angles.q037","True or false: in the 60° construction, the compass width must stay the same for both arcs.",{"kind":52,"options":510,"correct":513},[511,512],{"id":129,"label":130},{"id":132,"label":133},[129],[],"**True.** The angle is 60° because OP = OQ = PQ, which makes triangle OPQ equilateral. That only happens if both arcs use the same radius.",[176],{"id":518,"section":31,"level":110,"prompt":519,"check":520,"hints":531,"solution":532,"skills":533},"constructing-angles.q038","Why is the angle made by the 60° construction exactly 60°?",{"kind":52,"options":521,"correct":530},[522,524,526,528],{"id":55,"label":523},"Triangle OPQ has three equal sides, so each of its angles is 180° ÷ 3 = 60°",{"id":58,"label":525},"Because the radius is 6 cm",{"id":61,"label":527},"Because the arcs are curved",{"id":64,"label":529},"Because Q is above P",[55],[],"OP and OQ are radii of the first arc, and PQ was drawn with the same radius, so all three sides are equal. An equilateral triangle has three equal angles adding to 180°, so each is **60°**.",[176,243],{"id":535,"section":31,"level":110,"prompt":536,"check":537,"hints":539,"solution":540,"skills":541},"constructing-angles.q039","In the 120° construction, how many times do you step the same radius along the arc, starting from P?",{"kind":113,"answer":538,"tolerance":115},2,[],"Each step of the radius along the arc adds 60°. Two steps (P to Q, then Q to R) give 60° + 60° = 120°, so step **2** times.",[542],"120° construction",{"id":544,"section":31,"level":110,"prompt":545,"check":546,"hints":557,"solution":559,"skills":560},"constructing-angles.q040","Segment AB is **9 cm** long. Which compass radius can you use to draw its perpendicular bisector?",{"kind":52,"options":547,"correct":556},[548,550,552,554],{"id":55,"label":549},"3 cm",{"id":58,"label":551},"4 cm",{"id":61,"label":553},"4.5 cm",{"id":64,"label":555},"5.5 cm",[64],[558],"Half of 9 is 4.5.","The radius must be **more than half of AB**, which is 4.5 cm; otherwise the arcs from A and B do not meet. Of the choices only **5.5 cm** is more than 4.5 cm. (At exactly 4.5 cm the arcs only touch at the midpoint, giving no second point to draw a line through.)",[561],"perpendicular bisector",{"id":563,"section":31,"level":110,"prompt":564,"check":565,"hints":568,"solution":569,"skills":570},"constructing-angles.q041","Put the steps for bisecting ∠AOB in order and type the letters.\n\nA. Draw ray OT.\nB. With centre O, draw an arc cutting OA at P and OB at Q.\nC. With centres P and Q and equal radii, draw arcs meeting at T.",{"kind":348,"accept":566},[567],"BCA",[],"First the arc from the vertex (**B**), then the equal arcs from P and Q (**C**), then the ray from the vertex through their crossing (**A**): **BCA**.",[571,355],"angle bisector",{"id":573,"section":31,"level":110,"prompt":574,"check":575,"hints":586,"solution":587,"skills":588},"constructing-angles.q042","Which method constructs a **90°** angle at O on ray OA?",{"kind":52,"options":576,"correct":585},[577,579,581,583],{"id":55,"label":578},"Bisect the angle between the 60° and 120° marks",{"id":58,"label":580},"Bisect the 60° angle",{"id":61,"label":582},"Step the radius three times",{"id":64,"label":584},"Bisect the 120° angle",[55],[],"The ray halfway between the 60° and 120° marks is at (60° + 120°) ÷ 2 = **90°**. Bisecting 60° gives 30°; bisecting 120° gives 60°; three steps give 180°.",[589],"90° construction",{"id":591,"section":31,"level":159,"prompt":592,"check":593,"hints":595,"solution":596,"skills":597},"constructing-angles.q043","An angle of **84°** is bisected with ruler and compass. What is each half?",{"kind":113,"answer":594,"tolerance":115,"unit":182},42,[],"Bisecting cuts the angle into two equal parts: 84° ÷ 2 = **42°**. The bisector construction works for any angle, not only special ones.",[571],{"id":599,"section":31,"level":159,"prompt":600,"check":601,"hints":603,"solution":604,"skills":605},"constructing-angles.q044","Segment AB is **11 cm** long. Its perpendicular bisector meets AB at M. How long is AM?",{"kind":113,"answer":602,"tolerance":115,"unit":116},5.5,[],"The perpendicular bisector passes through the midpoint, so AM = 11 ÷ 2 = **5.5 cm**.",[561],{"id":607,"section":31,"level":159,"prompt":608,"check":609,"hints":620,"solution":621,"skills":622},"constructing-angles.q045","To **copy** ∠AOB onto a ray O′A′, which two lengths must you carry across with the compass?",{"kind":52,"options":610,"correct":619},[611,613,615,617],{"id":55,"label":612},"The arc radius from the vertex, and the distance PQ between the points where that arc cuts the arms",{"id":58,"label":614},"The lengths of the two arms",{"id":61,"label":616},"The distance from O to B only",{"id":64,"label":618},"No lengths: you use a protractor",[55],[],"Draw an arc of some radius from O, cutting the arms at P and Q; draw the **same radius** arc from O′, cutting O′A′ at P′. Then set the compass to the **chord PQ** and cut the new arc from P′ at Q′. Triangles OPQ and O′P′Q′ have equal sides, so ∠A′O′Q′ = ∠AOB.",[623],"copying an angle",{"id":625,"section":31,"level":255,"prompt":626,"check":627,"hints":638,"solution":640,"skills":641},"constructing-angles.q046","In the 60° construction, Tara draws the first arc with radius 5 cm but her compass slips open to **6 cm** for the second arc (from P). Is her angle bigger or smaller than 60°?",{"kind":52,"options":628,"correct":637},[629,631,633,635],{"id":55,"label":630},"Bigger than 60°",{"id":58,"label":632},"Smaller than 60°",{"id":61,"label":634},"Exactly 60°",{"id":64,"label":636},"It depends on the length of OA",[55],[639],"In a triangle, a longer opposite side means a bigger angle.","Now OP = OQ = 5 cm but PQ = 6 cm. The side opposite the angle at O is longer than the other two, so the angle at O is **bigger** than in the equilateral case. (A careful drawing or calculation gives about 73.7°.)",[176,243],{"id":643,"section":31,"level":255,"prompt":644,"check":645,"hints":656,"solution":657,"skills":658},"constructing-angles.q047","In the angle-bisector construction, why must the arcs from P and Q have a radius **more than half of PQ**?",{"kind":52,"options":646,"correct":655},[647,649,651,653],{"id":55,"label":648},"Otherwise the two arcs do not meet, so there is no point T",{"id":58,"label":650},"Otherwise the angle doubles",{"id":61,"label":652},"To make the triangle equilateral",{"id":64,"label":654},"There is no reason; any radius works",[55],[],"P and Q are a distance PQ apart. Two circles of radius r centred at P and Q meet only if 2r is at least PQ. If r is less than half of PQ, the arcs never reach each other and T cannot be found. (At exactly half they touch at one point on PQ itself, which is not useful.)",[571,243],{"id":660,"section":35,"level":49,"prompt":661,"check":662,"hints":663,"solution":664,"skills":665},"constructing-angles.q048","You construct 60° and then bisect it. What angle do you get?",{"kind":113,"answer":228,"tolerance":115,"unit":182},[],"Bisecting halves the angle: 60° ÷ 2 = **30°**.",[666],"bisecting",{"id":668,"section":35,"level":110,"prompt":669,"check":670,"hints":681,"solution":682,"skills":683},"constructing-angles.q049","Which construction gives **45°**?",{"kind":52,"options":671,"correct":680},[672,674,676,678],{"id":55,"label":673},"Bisect a constructed 90°",{"id":58,"label":675},"Bisect 60°",{"id":61,"label":677},"Bisect 120°",{"id":64,"label":679},"Step the radius once",[55],[],"90° ÷ 2 = **45°**, so construct a right angle and bisect it.",[666],{"id":685,"section":35,"level":110,"prompt":686,"check":687,"hints":698,"solution":699,"skills":700},"constructing-angles.q050","How can you construct **15°**?",{"kind":52,"options":688,"correct":697},[689,691,693,695],{"id":55,"label":690},"Construct 60°, bisect to 30°, bisect again",{"id":58,"label":692},"Bisect 45°",{"id":61,"label":694},"Divide 45° into three",{"id":64,"label":696},"Step the radius a quarter of the way",[55],[],"60° → 30° → **15°** by bisecting twice. (Bisecting 45° gives 22.5°, not 15°. Dividing into three equal parts is not a ruler-and-compass move in general.)",[666,701],"combinations",{"id":703,"section":35,"level":110,"prompt":704,"check":705,"hints":716,"solution":717,"skills":718},"constructing-angles.q051","Which combination gives **75°**?",{"kind":52,"options":706,"correct":715},[707,709,711,713],{"id":55,"label":708},"60° + 15°: bisect between the 60° and 90° marks",{"id":58,"label":710},"45° + 45°",{"id":61,"label":712},"90° − 30° − 30°",{"id":64,"label":714},"120° ÷ 2",[55],[],"75° = 60° + 15°. The ray halfway between the 60° and 90° marks is at (60° + 90°) ÷ 2 = **75°**.",[701],{"id":720,"section":35,"level":110,"prompt":721,"check":722,"hints":724,"solution":725,"skills":726},"constructing-angles.q052","Bisecting the angle between the **90°** and **120°** marks gives which angle?",{"kind":113,"answer":723,"tolerance":115,"unit":182},105,[],"Halfway between 90° and 120° is (90 + 120) ÷ 2 = **105°**, which is also 90° + 15°.",[701],{"id":728,"section":35,"level":110,"prompt":729,"check":730,"hints":740,"solution":741,"skills":742},"constructing-angles.q053","Which method gives **135°** at O on a straight line?",{"kind":52,"options":731,"correct":739},[732,734,735,737],{"id":55,"label":733},"Construct 90°, then bisect the other right angle on the straight line: 90° + 45°",{"id":58,"label":677},{"id":61,"label":736},"Step the radius twice",{"id":64,"label":738},"Bisect 60° three times",[55],[],"On a straight line, the perpendicular at O makes two right angles. Bisect the second one to get 45° more: 90° + 45° = **135°**, which is also 180° − 45°.",[701],{"id":744,"section":35,"level":110,"prompt":745,"check":746,"hints":748,"solution":749,"skills":750},"constructing-angles.q054","AOB is a straight line. You construct a ray OC so that ∠BOC = 30°. What is ∠AOC?",{"kind":113,"answer":747,"tolerance":115,"unit":182},150,[],"∠AOC and ∠BOC together make the straight angle AOB, so ∠AOC = 180° − 30° = **150°**. This is the standard way to construct 150°.",[701,751],"straight angle",{"id":753,"section":35,"level":159,"prompt":754,"check":755,"hints":764,"solution":765,"skills":766},"constructing-angles.q055","How can you construct **22.5°**?",{"kind":52,"options":756,"correct":763},[757,758,760,761],{"id":55,"label":692},{"id":58,"label":759},"Bisect 30°",{"id":61,"label":738},{"id":64,"label":762},"It is impossible",[55],[],"90° → 45° → **22.5°** by bisecting twice. Half-degree angles are no problem for a construction.",[666],{"id":768,"section":35,"level":159,"prompt":769,"check":770,"hints":781,"solution":783,"skills":784},"constructing-angles.q056","Using only 60° constructions, stepping, bisecting and straight lines, which of these angles **cannot** be made?",{"kind":52,"options":771,"correct":780},[772,774,776,778],{"id":55,"label":773},"20°",{"id":58,"label":775},"15°",{"id":61,"label":777},"165°",{"id":64,"label":779},"7.5°",[55],[782],"Which one needs dividing by 3?","15° = 60° halved twice; 7.5° = 60° halved three times; 165° = 180° − 15°. But **20°** is a third of 60°, and cutting an angle into three equal parts is not possible in general with ruler and compass. In fact 20° cannot be constructed at all (proved by Pierre Wantzel in 1837).",[785],"constructible angles",{"id":787,"section":35,"level":159,"prompt":788,"check":789,"hints":800,"solution":801,"skills":802},"constructing-angles.q057","Which plan constructs **165°** with ruler and compass?",{"kind":52,"options":790,"correct":799},[791,793,795,797],{"id":55,"label":792},"Construct 15° at one end of a straight line; the angle on the other side is 180° − 15°",{"id":58,"label":794},"Step the radius three times and bisect",{"id":61,"label":796},"Add 120° and 60°",{"id":64,"label":798},"Bisect 150° twice",[55],[],"180° − 15° = **165°**. Draw a straight line through O, construct 15° (60° halved twice) on one side; the remaining angle on the straight line is 165°.",[701],{"id":804,"section":35,"level":159,"prompt":805,"check":806,"hints":808,"solution":809,"skills":810},"constructing-angles.q058","Starting from 60°, how many bisections does it take to reach **7.5°**?",{"kind":113,"answer":807,"tolerance":115},3,[],"60° → 30° → 15° → 7.5°: **3** bisections. Each bisection halves the angle, and 60 ÷ 2 ÷ 2 ÷ 2 = 7.5.",[666],{"id":812,"section":35,"level":255,"prompt":813,"check":814,"hints":816,"solution":818,"skills":819},"constructing-angles.q059","Starting from 60° and bisecting again and again, how many bisections are needed before the angle is **less than 1°**?",{"kind":113,"answer":815,"tolerance":115},6,[817],"Keep halving and count.","60 → 30 → 15 → 7.5 → 3.75 → 1.875 → 0.9375. After 5 bisections it is 1.875° (still at least 1°); after **6** it is 0.9375°, less than 1°. (In practice, drawing such tiny angles accurately is very hard.)",[666,243],{"id":821,"section":35,"level":255,"prompt":822,"check":823,"hints":834,"solution":835,"skills":836},"constructing-angles.q060","A regular 24-sided polygon has 24 equal angles at its centre. Can its centre angle be constructed with ruler and compass using what you know?",{"kind":52,"options":824,"correct":833},[825,827,829,831],{"id":55,"label":826},"Yes: 360° ÷ 24 = 15°, which is 60° bisected twice",{"id":58,"label":828},"No: 24 is too many sides",{"id":61,"label":830},"No: 360 ÷ 24 is not a whole number",{"id":64,"label":832},"Only with a protractor",[55],[],"The centre angle is 360° ÷ 24 = **15°**, and 15° = 60° ÷ 2 ÷ 2, so it can be constructed. Stepping 15° round the circle 24 times gives the polygon's corners.",[785,837],"polygons",{"id":839,"section":35,"level":255,"prompt":840,"check":841,"hints":852,"solution":853,"skills":854},"constructing-angles.q061","Why can't a 20° angle be constructed by bisecting, even though 20° is a whole number?",{"kind":52,"options":842,"correct":851},[843,845,847,849],{"id":55,"label":844},"Bisecting only halves angles; 20° is a third of 60°, and trisecting 60° with ruler and compass is impossible",{"id":58,"label":846},"Because 20 is not a multiple of 10",{"id":61,"label":848},"Because 20° is acute",{"id":64,"label":850},"It can: bisect 40°",[55],[],"Starting from 60° or 90°, bisecting and adding or subtracting always give multiples of 15° and their halves, quarters and so on. 20° would need 60° ÷ 3. Mathematicians proved in 1837 (Wantzel) that a 60° angle **cannot** be trisected with ruler and compass, so 20° is not constructible. (\"Bisect 40°\" does not help, because 40° is not constructible either.)",[785,855],"trisection",{"id":857,"section":39,"level":49,"prompt":858,"check":859,"hints":868,"solution":869,"skills":870},"constructing-angles.q062","Which angles are on the **30°–60°** set square?",{"kind":52,"options":860,"correct":867},[861,862,864,865],{"id":55,"label":96},{"id":58,"label":863},"30°, 60°, 60°",{"id":61,"label":98},{"id":64,"label":866},"30°, 30°, 120°",[55],[],"It is a right-angled triangle with the other two angles 30° and 60°: 30 + 60 + 90 = 180.",[107],{"id":872,"section":39,"level":49,"prompt":873,"check":874,"hints":875,"solution":876,"skills":877},"constructing-angles.q063","Placing the 45° corner of one set square next to the 30° corner of the other gives which angle?",{"kind":113,"answer":321,"tolerance":115,"unit":182},[],"Angles placed side by side add: 45° + 30° = **75°**.",[107,701],{"id":879,"section":39,"level":110,"prompt":880,"check":881,"hints":892,"solution":893,"skills":894},"constructing-angles.q064","Which two set-square corners placed side by side make **105°**?",{"kind":52,"options":882,"correct":891},[883,885,887,889],{"id":55,"label":884},"45° and 60°",{"id":58,"label":886},"30° and 45°",{"id":61,"label":888},"45° and 90°",{"id":64,"label":890},"30° and 90°",[55],[],"45° + 60° = **105°**.",[107,701],{"id":896,"section":39,"level":110,"prompt":897,"check":898,"hints":909,"solution":910,"skills":911},"constructing-angles.q065","How can you use set squares to draw **15°**?",{"kind":52,"options":899,"correct":908},[900,902,904,906],{"id":55,"label":901},"Draw 45°, then place the 30° corner inside it on the same arm; the gap left is 15°",{"id":58,"label":903},"Put 45° next to 60°",{"id":61,"label":905},"Halve the 30° corner by eye",{"id":64,"label":907},"It cannot be done with set squares",[55],[],"Put both corners at the same vertex along the same arm. The 45° angle is 30° + the gap, so the gap is 45° − 30° = **15°**. (Also 60° − 45° = 15°.)",[107,701],{"id":913,"section":39,"level":159,"prompt":914,"check":915,"hints":917,"solution":919,"skills":920},"constructing-angles.q066","You place **one** corner of the 45° set square (45° or 90°) next to **one** corner of the 30°–60° set square (30°, 60° or 90°), so the angles add. How many **different** angles less than 180° can you make?",{"kind":113,"answer":916,"tolerance":115},5,[918],"List all 2 × 3 = 6 sums first.","The sums are 45 + 30 = 75, 45 + 60 = 105, 45 + 90 = 135, 90 + 30 = 120, 90 + 60 = 150 and 90 + 90 = 180. Leaving out 180°, and noting 135° appears only once, the different angles are 75°, 105°, 120°, 135° and 150°: **5** angles.",[107,701],{"id":922,"section":39,"level":159,"prompt":923,"check":924,"hints":934,"solution":936,"skills":937},"constructing-angles.q067","Which of these angles can **not** be made from set-square corners by adding or subtracting?",{"kind":52,"options":925,"correct":933},[926,928,930,931],{"id":55,"label":927},"100°",{"id":58,"label":929},"150°",{"id":61,"label":775},{"id":64,"label":932},"120°",[55],[935],"Look for a pattern: are all set-square angles multiples of some number?","Every set-square angle is a multiple of 15° (30, 45, 60, 90), so every sum or difference is a multiple of 15° too. 150, 15 and 120 are multiples of 15; **100** is not (100 ÷ 15 is not a whole number).",[107,243],{"id":939,"section":39,"level":255,"prompt":940,"check":941,"hints":951,"solution":952,"skills":953},"constructing-angles.q068","A carpenter needs a **165°** angle and has only set squares and a straight edge. Which plan works?",{"kind":52,"options":942,"correct":950},[943,945,947,949],{"id":55,"label":944},"Draw a straight line; make 15° (45° − 30°) at one end; the remaining angle on the line is 165°",{"id":58,"label":946},"Add 90° + 45° + 45°",{"id":61,"label":948},"Add 120° + 45° using one corner",{"id":64,"label":762},[55],[],"180° − 15° = **165°**. The 15° comes from 45° − 30°. (Option c fails because neither set square has a 120° corner; 90° + 45° + 45° = 180°, not 165°.)",[107,701],{"id":955,"section":43,"level":49,"prompt":956,"check":957,"hints":968,"solution":969,"skills":970},"constructing-angles.q069","The target was **60°**. Your construction measures **59°**. With a tolerance of ± 1°, is it accurate?",{"kind":52,"options":958,"correct":967},[959,961,963,965],{"id":55,"label":960},"Yes, it is within 1°",{"id":58,"label":962},"No, it must be exactly 60°",{"id":61,"label":964},"No, 59 is acute",{"id":64,"label":966},"Only if the arms are long",[55],[],"± 1° means anything from 59° to 61° counts as accurate. 59° is inside that range, so **yes**.",[43],{"id":972,"section":43,"level":110,"prompt":973,"check":974,"hints":985,"solution":986,"skills":987},"constructing-angles.q070","The target is **45°**, with tolerance ± 1°. Which measured result is **not** acceptable?",{"kind":52,"options":975,"correct":984},[976,978,980,982],{"id":55,"label":977},"44°",{"id":58,"label":979},"45°",{"id":61,"label":981},"46°",{"id":64,"label":983},"47°",[64],[],"Acceptable results are 44° to 46°. **47°** is 2° off, outside the tolerance.",[43],{"id":989,"section":43,"level":159,"prompt":990,"check":991,"hints":995,"solution":998,"skills":999},"constructing-angles.q071","A protractor has a radius of **5 cm**. How far apart, in millimetres, are two neighbouring 1° marks along its edge? Give your answer to one decimal place.",{"kind":113,"answer":992,"tolerance":993,"unit":994},0.9,0.05,"mm",[996,997],"Circumference = 2 × π × radius.","Share it among 360 degrees.","The edge is part of a circle of radius 50 mm. A full circle's edge is 2 × π × 50 ≈ 314.2 mm, shared among 360 degrees, so each degree takes 314.2 ÷ 360 ≈ **0.9 mm** (more exactly 0.87 mm). That is why sharp pencils matter.",[43,1000],"circles",{"id":1002,"section":43,"level":159,"prompt":1003,"check":1004,"hints":1015,"solution":1016,"skills":1017},"constructing-angles.q072","Why do longer arms help you measure an angle more accurately?",{"kind":52,"options":1005,"correct":1014},[1006,1008,1010,1012],{"id":55,"label":1007},"The same small wobble of a line changes the direction less when the line is long",{"id":58,"label":1009},"Longer arms make the angle bigger",{"id":61,"label":1011},"The protractor reads longer lines on a different scale",{"id":64,"label":1013},"They do not help",[55],[],"If the end of a line is misplaced by, say, 1 mm, the direction changes by a large angle when the line is short and by a small angle when it is long. Longer arms also cross the scale clearly. The angle itself is the same whatever the arm length.",[43,243],{"id":1019,"section":43,"level":255,"prompt":1020,"check":1021,"hints":1024,"solution":1026,"skills":1027},"constructing-angles.q073","The end of a **5 cm** arm is drawn **1 mm** out of place, sideways. Roughly how many degrees does the arm's direction change? Give your answer to one decimal place.",{"kind":113,"answer":1022,"tolerance":1023,"unit":182},1.1,0.1,[1025],"How far does the end of a 5 cm arm move for each 1° of turn?","Use the idea from the previous question: 1° of turn moves the end of a 50 mm arm by about 50 × π ÷ 180 ≈ 0.87 mm. A 1 mm slip is therefore about 1 ÷ 0.87 ≈ **1.1°**. (A more exact calculation gives 1.15°.) So on short arms, a 1 mm slip already uses up the ± 1° tolerance.",[43,243],{"id":1029,"section":43,"level":255,"prompt":1030,"check":1031,"hints":1042,"solution":1043,"skills":1044},"constructing-angles.q074","Priya constructs a 90° angle and checks it: her protractor reads **93°**. She also checks it by fitting the corner of a set square, and there is a thin gap. Which is the **best** next step?",{"kind":52,"options":1032,"correct":1041},[1033,1035,1037,1039],{"id":55,"label":1034},"Look at her construction arcs to find where the compass slipped, then redraw with a tight compass and a sharp pencil",{"id":58,"label":1036},"Write 90° anyway because it was constructed",{"id":61,"label":1038},"Change the protractor reading to 90°",{"id":64,"label":1040},"Decide that 90° cannot be constructed",[55],[],"A careful construction should be within ± 1°. Two independent checks both say her angle is off by about 3°, so something went wrong in the drawing, usually a compass whose width changed or a point that slipped. The kept construction arcs show where. A construction is exact **in principle**, but a real drawing is only as good as its execution.",[43,316],[1046,1047,1048,1049,1050,1051,1052,1053,1054,1055],"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","needs_review",{"generatedBy":1058,"notes":1059},"claude-code","Draft generated with Python; every numeric answer was computed and asserted. 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