[{"data":1,"prerenderedAt":989},["ShallowReactive",2],{"questions:angles":3},{"bank":4,"contentHash":977,"dependencyHashes":978,"releaseId":988},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":964,"reviewStatus":973,"authoring":974},1,"angles","Angles question bank","Practise angles from naming and types through clock angles, complementary and supplementary pairs, linear pairs, vertically opposite angles and angles around a point, up to transversals, triangles and olympiad-style puzzles. Every question has a worked solution: try it first, use a hint if you are stuck, then read the solution to check your reasoning. Give a reason for every step, and never assume an angle from how a diagram looks.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"what-is","What is an angle and naming angles","Vertex, arms, interior and exterior, angle as turn, and naming with ∠ABC.",{"id":15,"title":16,"description":17},"types","Types of angles","Zero, acute, right, obtuse, straight, reflex and complete angles, and fractions of a turn.",{"id":19,"title":20,"description":21},"clock-direction","Clock and direction angles","Angles between clock hands, the clock formula, compass directions and turns.",{"id":23,"title":24,"description":25},"comp-supp","Complementary and supplementary angles","Pairs adding to 90° and 180°, finding complements and supplements, and algebra puzzles.",{"id":27,"title":28,"description":29},"linear-adjacent","Linear pairs and adjacent angles","Adjacent angles, linear pairs and angles along a straight line.",{"id":31,"title":32,"description":33},"vert-opp","Vertically opposite angles","Why they are equal, and finding angles where lines cross.",{"id":35,"title":36,"description":37},"at-point","Angles around a point","Angles adding to 360°, pie charts and tiling.",{"id":39,"title":40,"description":41},"missing","Missing-angle problems","Multi-step problems that combine angle facts, with reasons.",{"id":43,"title":44,"description":45},"transversal-triangle","Transversals, triangles and polygons","Parallel lines and transversals, triangle and polygon angle sums, exterior angles (stretch and challenge).",[47,61,86,106,119,129,147,164,183,198,208,217,242,252,271,285,304,318,328,350,358,367,376,390,399,409,418,428,439,448,458,468,483,500,508,519,528,545,554,564,573,582,594,613,621,632,641,650,659,667,675,693,701,710,727,735,744,752,759,767,776,784,795,804,812,820,830,838,847,865,873,881,889,903,911,919,927,936,946,955],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":56,"solution":58,"skills":59},"angles.q001","foundation","An angle is formed by two rays that share a common end point. What is that common end point called?",{"kind":52,"accept":53},"text",[54,55],"vertex","the vertex",[57],"It is the corner point.","The common end point of the two rays (arms) is the **vertex** of the angle. For example, in ∠ABC the vertex is B.",[60],"vocabulary",{"id":62,"section":11,"level":49,"prompt":63,"check":64,"hints":80,"solution":82,"skills":83},"angles.q002","Angle P is drawn with arms 2 cm long and angle Q with arms 9 cm long. Both have the same opening. Which statement is true?",{"kind":65,"options":66,"correct":79},"choice",[67,70,73,76],{"id":68,"label":69},"a","∠Q is bigger because its arms are longer",{"id":71,"label":72},"b","∠P is bigger",{"id":74,"label":75},"c","∠P = ∠Q",{"id":77,"label":78},"d","They cannot be compared",[74],[81],"What does an angle measure: length or turn?","The size of an angle is the **amount of turn** between its arms. Arm length does not matter, so the angles are equal: **∠P = ∠Q**.",[84,85],"angle as turn","misconceptions",{"id":87,"section":11,"level":49,"prompt":88,"check":89,"hints":101,"solution":103,"skills":104},"angles.q003","In the angle ∠XYZ, which letter names the vertex?",{"kind":65,"options":90,"correct":100},[91,94,97],{"id":92,"label":93},"x","X",{"id":95,"label":96},"y","Y",{"id":98,"label":99},"z","Z",[95],[102],"Middle letter = corner.","With three letters, the vertex is always the **middle** letter. In ∠XYZ the vertex is **Y**, and the arms are YX and YZ.",[105],"naming angles",{"id":107,"section":11,"level":108,"prompt":109,"check":110,"hints":114,"solution":116,"skills":117},"angles.q004","core","Three rays OA, OB and OC start from O, with OB between OA and OC. How many different angles (less than 180°) are formed at O?",{"kind":111,"answer":112,"tolerance":113},"number",3,0,[115],"Count the pairs of rays: OA–OB, OB–OC, OA–OC.","Any two of the three rays make an angle: ∠AOB, ∠BOC and ∠AOC. That is **3** angles. This is why we cannot just write ∠O here: it would not say which angle we mean.",[105,118],"counting",{"id":120,"section":11,"level":108,"prompt":121,"check":122,"hints":124,"solution":126,"skills":127},"angles.q005","Four rays start from the same point O, all on one side of a line through O, and no two point the same way. How many angles between pairs of rays are there?",{"kind":111,"answer":123,"tolerance":113},6,[125],"List the pairs of rays systematically.","Each pair of rays makes one angle. With 4 rays the pairs are: 1–2, 1–3, 1–4, 2–3, 2–4, 3–4, which is **6** angles. (In general, n rays make n × (n − 1) ÷ 2 angles: 4 × 3 ÷ 2 = 6.)",[118,128],"reasoning",{"id":130,"section":11,"level":108,"prompt":131,"check":132,"hints":143,"solution":145,"skills":146},"angles.q006","In triangle PQR, which of these names the angle at vertex R? Choose all that apply.",{"kind":65,"options":133,"correct":142},[134,136,138,140],{"id":68,"label":135},"∠PRQ",{"id":71,"label":137},"∠QRP",{"id":74,"label":139},"∠RPQ",{"id":77,"label":141},"∠PQR",[68,71],[144],"Look for R in the middle.","The angle at R has R in the middle: **∠PRQ** or **∠QRP** (the arms can be read in either order). ∠RPQ is at P and ∠PQR is at Q.",[105],{"id":148,"section":11,"level":49,"prompt":149,"check":150,"hints":161,"solution":162,"skills":163},"angles.q007","Point M lies between the two arms of ∠ABC, not on either arm. Where is M?",{"kind":65,"options":151,"correct":160},[152,154,156,158],{"id":68,"label":153},"In the interior of the angle",{"id":71,"label":155},"In the exterior of the angle",{"id":74,"label":157},"On the angle",{"id":77,"label":159},"At the vertex",[68],[],"Points between the arms are in the **interior** of the angle. Points on the arms (including the vertex) are *on* the angle, and all other points are in the exterior.",[60],{"id":165,"section":15,"level":49,"prompt":166,"check":167,"hints":178,"solution":180,"skills":181},"angles.q008","What type of angle measures **135°**?",{"kind":65,"options":168,"correct":177},[169,171,173,175],{"id":68,"label":170},"Acute",{"id":71,"label":172},"Right",{"id":74,"label":174},"Obtuse",{"id":77,"label":176},"Reflex",[74],[179],"Compare with 90° and 180°.","135° is more than 90° and less than 180°, so it is **obtuse**.",[182],"angle types",{"id":184,"section":15,"level":49,"prompt":185,"check":186,"hints":195,"solution":196,"skills":197},"angles.q009","What type of angle measures **210°**?",{"kind":65,"options":187,"correct":194},[188,189,191,192],{"id":68,"label":174},{"id":71,"label":190},"Straight",{"id":74,"label":176},{"id":77,"label":193},"Complete",[74],[],"210° is more than 180° and less than 360°, so it is a **reflex** angle.",[182],{"id":199,"section":15,"level":49,"prompt":200,"check":201,"hints":204,"solution":206,"skills":207},"angles.q010","How many degrees are there in a **straight angle**?",{"kind":111,"answer":202,"tolerance":113,"unit":203},180,"°",[205],"It is half of a full turn.","A straight angle is a half turn: the arms point in opposite directions. Half of 360° is **180°**.",[182],{"id":209,"section":15,"level":49,"prompt":210,"check":211,"hints":213,"solution":214,"skills":215},"angles.q011","How many right angles make one complete turn?",{"kind":111,"answer":212,"tolerance":113},4,[],"A complete turn is 360° and a right angle is 90°, so 360 ÷ 90 = **4** right angles.",[182,216],"division",{"id":218,"section":15,"level":108,"prompt":219,"check":220,"hints":237,"solution":239,"skills":240},"angles.q012","Which of these angles are **acute**? Choose all that apply: 0°, 1°, 45°, 89°, 90°, 91°.",{"kind":65,"options":221,"correct":236},[222,224,226,228,230,233],{"id":68,"label":223},"0°",{"id":71,"label":225},"1°",{"id":74,"label":227},"45°",{"id":77,"label":229},"89°",{"id":231,"label":232},"e","90°",{"id":234,"label":235},"f","91°",[71,74,77],[238],"The boundaries 0° and 90° are not acute.","Acute angles are **more than 0° and less than 90°**: 1°, 45° and 89°. 0° is a zero angle, 90° is a right angle and 91° is obtuse.",[182,241],"boundaries",{"id":243,"section":15,"level":108,"prompt":244,"check":245,"hints":247,"solution":249,"skills":250},"angles.q013","The smaller opening between two arms is **48°**. What is the reflex angle between the same two arms?",{"kind":111,"answer":246,"tolerance":113,"unit":203},312,[248],"Small angle + reflex angle = 360°.","The two openings together make a full turn, so the reflex angle = 360° − 48° = **312°**.",[251],"reflex angles",{"id":253,"section":15,"level":108,"prompt":254,"check":255,"hints":266,"solution":268,"skills":269},"angles.q014","A fan has **5** equally spaced blades. What is the angle between neighbouring blades, and what type is it?",{"kind":65,"options":256,"correct":265},[257,259,261,263],{"id":68,"label":258},"60°, acute",{"id":71,"label":260},"72°, acute",{"id":74,"label":262},"72°, obtuse",{"id":77,"label":264},"90°, right",[71],[267],"Share 360° equally.","The five equal gaps make a full turn: 360° ÷ 5 = **72°**, which is less than 90°, so it is **acute**.",[270,182],"fractions of a turn",{"id":272,"section":15,"level":108,"prompt":273,"check":274,"hints":282,"solution":283,"skills":284},"angles.q015","True or false: a straight angle is equal to two right angles.",{"kind":65,"options":275,"correct":281},[276,279],{"id":277,"label":278},"t","True",{"id":234,"label":280},"False",[277],[],"**True.** A straight angle is 180° and 2 × 90° = 180°. Folding a straight line in half makes two right angles.",[182],{"id":286,"section":15,"level":287,"prompt":288,"check":289,"hints":300,"solution":301,"skills":302},"angles.q016","stretch","An angle is **3 times** as big as a right angle. What type is it, and how big is it?",{"kind":65,"options":290,"correct":299},[291,293,295,297],{"id":68,"label":292},"270°, reflex",{"id":71,"label":294},"270°, obtuse",{"id":74,"label":296},"180°, straight",{"id":77,"label":298},"300°, reflex",[68],[],"3 × 90° = **270°**. This is more than 180° and less than 360°, so it is **reflex**. It is the angle turned going from north to west, clockwise.",[182,303],"multiplication",{"id":305,"section":15,"level":287,"prompt":306,"check":307,"hints":314,"solution":316,"skills":317},"angles.q017","Two angles of the same type are added and the result is a **reflex** angle. Which type could they both be? Choose all that apply.",{"kind":65,"options":308,"correct":313},[309,310,311,312],{"id":68,"label":170},{"id":71,"label":172},{"id":74,"label":174},{"id":77,"label":190},[74],[315],"Find the smallest and largest possible sums for each type.","Two acute angles add to less than 180°. Two right angles make exactly 180° (straight). Two straight angles make 360° (complete). Two **obtuse** angles are each between 90° and 180°, so their sum is between 180° and 360°: always reflex.",[128,182],{"id":319,"section":19,"level":49,"prompt":320,"check":321,"hints":323,"solution":325,"skills":326},"angles.q018","What is the smaller angle between the hands of a clock at **2:00**?",{"kind":111,"answer":322,"tolerance":113,"unit":203},60,[324],"Each hour-gap is 30°.","From 12 to 2 there are 2 gaps, and each gap is 360° ÷ 12 = 30°. So the angle is 2 × 30° = **60°**.",[327],"clock angles",{"id":329,"section":19,"level":49,"prompt":330,"check":331,"hints":345,"solution":346,"skills":347},"angles.q019","Riya faces **north** and turns a quarter turn **clockwise**. Which way does she face now?",{"kind":65,"options":332,"correct":344},[333,336,338,341],{"id":334,"label":335},"n","North",{"id":231,"label":337},"East",{"id":339,"label":340},"s","South",{"id":342,"label":343},"w","West",[231],[],"Clockwise goes N → E → S → W. A quarter turn (90°) clockwise from north brings her to **east**.",[348,349],"directions","turns",{"id":351,"section":19,"level":49,"prompt":352,"check":353,"hints":354,"solution":356,"skills":357},"angles.q020","Through how many degrees does the **minute hand** turn from 4:10 to 4:40?",{"kind":111,"answer":202,"tolerance":113,"unit":203},[355],"The minute hand turns 6° every minute.","30 minutes is half an hour, so the minute hand turns half a full turn: 30 × 6° = **180°**.",[327],{"id":359,"section":19,"level":108,"prompt":360,"check":361,"hints":363,"solution":365,"skills":366},"angles.q021","What is the smaller angle between the clock hands at **3:30**?",{"kind":111,"answer":362,"tolerance":113,"unit":203},75,[364],"The hour hand is half-way between 3 and 4.","Minute hand at 6: 30 × 6° = 180° from 12. Hour hand: 3 × 30° = 90°, plus 30 × 0.5° = 15° for the 30 minutes, so 105°. The angle is 180° − 105° = **75°**, not 90°.",[327],{"id":368,"section":19,"level":108,"prompt":369,"check":370,"hints":372,"solution":374,"skills":375},"angles.q022","What is the **reflex** angle between the clock hands at **8:00**?",{"kind":111,"answer":371,"tolerance":113,"unit":203},240,[373],"Find the smaller angle first.","At 8:00 the smaller angle is 4 gaps (8 to 12) × 30° = 120°. The reflex angle is 360° − 120° = **240°**.",[327,251],{"id":377,"section":19,"level":108,"prompt":378,"check":379,"hints":386,"solution":388,"skills":389},"angles.q023","Arjun faces **west** and turns **270° clockwise**. Which way does he face now?",{"kind":65,"options":380,"correct":385},[381,382,383,384],{"id":334,"label":335},{"id":231,"label":337},{"id":339,"label":340},{"id":342,"label":343},[339],[387],"270° clockwise = 90° anticlockwise.","270° clockwise is three quarter turns: W → N → E → **S**. (Equivalently, 270° clockwise is the same as 90° anticlockwise, and anticlockwise from west is south.)",[348,349],{"id":391,"section":19,"level":108,"prompt":392,"check":393,"hints":395,"solution":397,"skills":398},"angles.q024","What is the smallest angle between the directions **north-east** and **south**?",{"kind":111,"answer":394,"tolerance":113,"unit":203},135,[396],"Measure both directions clockwise from north.","Clockwise from north: NE is at 45° and S is at 180°. The angle between them is 180° − 45° = **135°** (the other way round is 225°, which is larger).",[348],{"id":400,"section":19,"level":287,"prompt":401,"check":402,"hints":404,"solution":406,"skills":407},"angles.q025","Use the clock formula |30h − 5.5m| to find the smaller angle between the hands at **7:20**.",{"kind":111,"answer":403,"tolerance":113,"unit":203},100,[405],"30 × 7 = 210, 5.5 × 20 = 110.","h = 7, m = 20: 30 × 7 = 210 and 5.5 × 20 = 110. |210 − 110| = **100°**, which is less than 180°, so it is the smaller angle.",[327,408],"formula",{"id":410,"section":19,"level":287,"prompt":411,"check":412,"hints":414,"solution":416,"skills":417},"angles.q026","What is the smaller angle between the clock hands at **9:45**?",{"kind":111,"answer":413,"tolerance":113,"unit":203},22.5,[415],"The hour hand has moved three-quarters of the way from 9 to 10.","Hour hand: 9 × 30° + 45 × 0.5° = 270° + 22.5° = 292.5°. Minute hand: 45 × 6° = 270°. Difference: **22.5°**. Formula check: |270 − 247.5| = 22.5°.",[327],{"id":419,"section":19,"level":287,"prompt":420,"check":421,"hints":424,"solution":426,"skills":427},"angles.q027","The hour hand of a clock turns through **75°**. How many minutes have passed?",{"kind":111,"answer":422,"tolerance":113,"unit":423},150,"min",[425],"30° per hour, or 0.5° per minute.","The hour hand turns 0.5° per minute, so 75° takes 75 ÷ 0.5 = **150 minutes** (2½ hours).",[327,216],{"id":429,"section":19,"level":430,"prompt":431,"check":432,"hints":434,"solution":437,"skills":438},"angles.q028","challenge","At what time, to the nearest minute, between **5:00 and 6:00** do the hour and minute hands overlap? Give the number of minutes past 5.",{"kind":111,"answer":433,"tolerance":113,"unit":423},27,[435,436],"How far ahead is the hour hand at 5:00?","The minute hand gains 5.5° per minute.","At 5:00 the minute hand is 150° behind the hour hand. It gains 6° − 0.5° = 5.5° per minute, so it catches up after 150 ÷ 5.5 = 300⁄11 = **27 3⁄11 minutes**, about **5:27** (5:27:16).",[327,128],{"id":440,"section":19,"level":430,"prompt":441,"check":442,"hints":444,"solution":446,"skills":447},"angles.q029","How many times in a full day (24 hours) are the hands of a clock at **right angles**?",{"kind":111,"answer":443,"tolerance":113},44,[445],"How many times does the minute hand overtake the hour hand in 12 hours?","The minute hand laps the hour hand 11 times every 12 hours (it goes round 12 times, the hour hand once). In each lap the hands are at right angles twice. So 11 × 2 = 22 times in 12 hours, and **44** times in 24 hours.",[327,128],{"id":449,"section":23,"level":49,"prompt":450,"check":451,"hints":453,"solution":455,"skills":456},"angles.q030","What is the **complement** of **40°**?",{"kind":111,"answer":452,"tolerance":113,"unit":203},50,[454],"Complement = 90° − angle.","Complementary angles add to 90°, so the complement is 90° − 40° = **50°**.",[457],"complementary",{"id":459,"section":23,"level":49,"prompt":460,"check":461,"hints":463,"solution":465,"skills":466},"angles.q031","What is the **supplement** of **40°**?",{"kind":111,"answer":462,"tolerance":113,"unit":203},140,[464],"Supplement = 180° − angle.","Supplementary angles add to 180°, so the supplement is 180° − 40° = **140°**.",[467],"supplementary",{"id":469,"section":23,"level":49,"prompt":470,"check":471,"hints":480,"solution":481,"skills":482},"angles.q032","Are **65°** and **25°** complementary, supplementary or neither?",{"kind":65,"options":472,"correct":479},[473,475,477],{"id":74,"label":474},"Complementary",{"id":339,"label":476},"Supplementary",{"id":334,"label":478},"Neither",[74],[],"65° + 25° = **90°**, so they are **complementary**.",[457],{"id":484,"section":23,"level":108,"prompt":485,"check":486,"hints":497,"solution":498,"skills":499},"angles.q033","Which pair of angles is **supplementary**?",{"kind":65,"options":487,"correct":496},[488,490,492,494],{"id":68,"label":489},"80° and 20°",{"id":71,"label":491},"95° and 85°",{"id":74,"label":493},"120° and 70°",{"id":77,"label":495},"45° and 45°",[71],[],"Add each pair: 80 + 20 = 100, **95 + 85 = 180**, 120 + 70 = 190, 45 + 45 = 90. Only 95° and 85° add to 180°.",[467],{"id":501,"section":23,"level":108,"prompt":502,"check":503,"hints":505,"solution":506,"skills":507},"angles.q034","Two complementary angles are **equal**. How big is each?",{"kind":111,"answer":504,"tolerance":113,"unit":203},45,[],"Two equal angles adding to 90°: each is 90° ÷ 2 = **45°**.",[457],{"id":509,"section":23,"level":108,"prompt":510,"check":511,"hints":513,"solution":516,"skills":517},"angles.q035","An angle is **4 times** its complement. Find the angle.",{"kind":111,"answer":512,"tolerance":113,"unit":203},72,[514,515],"Let the complement be c.","c + 4c = 90.","Let the complement be c; the angle is 4c. They add to 90°: c + 4c = 5c = 90°, so c = 18° and the angle is 4 × 18° = **72°**. Check: 72 + 18 = 90 ✓.",[457,518],"algebra",{"id":520,"section":23,"level":108,"prompt":521,"check":522,"hints":524,"solution":526,"skills":527},"angles.q036","An angle is **30° more** than its supplement. Find the angle.",{"kind":111,"answer":523,"tolerance":113,"unit":203},105,[525],"Let the supplement be s; the angle is s + 30.","Let the supplement be s; the angle is s + 30°. Together: s + s + 30° = 180°, so 2s = 150° and s = 75°. The angle is **105°**. Check: 105 − 75 = 30 ✓ and 105 + 75 = 180 ✓.",[467,518],{"id":529,"section":23,"level":287,"prompt":530,"check":531,"hints":542,"solution":543,"skills":544},"angles.q037","Spot the mistake. Kiran says: “The supplement of 110° is 70°, and the complement of 110° is −20°.” What is wrong?",{"kind":65,"options":532,"correct":541},[533,535,537,539],{"id":68,"label":534},"Nothing is wrong",{"id":71,"label":536},"The supplement should be 250°",{"id":74,"label":538},"110° has no complement, because a complement must be a positive angle less than 90°",{"id":77,"label":540},"The complement is 20°",[74],[],"The supplement is correct: 180° − 110° = 70°. But complements only exist for **acute** angles. 90° − 110° is negative, which is not an angle in this sense, so **110° has no complement**.",[457,85],{"id":546,"section":23,"level":287,"prompt":547,"check":548,"hints":549,"solution":552,"skills":553},"angles.q038","The supplement of an angle is **three times** its complement. Find the angle.",{"kind":111,"answer":504,"tolerance":113,"unit":203},[550,551],"Write both in terms of x.","180 − x = 3(90 − x).","Let the angle be x. Supplement = 180 − x, complement = 90 − x. So 180 − x = 3(90 − x) = 270 − 3x. Then 2x = 90 and x = **45°**. Check: supplement 135°, complement 45°, and 135 = 3 × 45 ✓.",[457,467,518],{"id":555,"section":23,"level":287,"prompt":556,"check":557,"hints":559,"solution":561,"skills":562},"angles.q039","Two supplementary angles are in the ratio **7 : 11**. Find the **smaller** angle.",{"kind":111,"answer":558,"tolerance":113,"unit":203},70,[560],"Divide 180° into 18 equal parts.","7 + 11 = 18 parts make 180°, so one part = 10°. The angles are 70° and 110°. The smaller is **70°**.",[467,563],"ratio",{"id":565,"section":23,"level":430,"prompt":566,"check":567,"hints":569,"solution":571,"skills":572},"angles.q040","The difference between an angle's supplement and **twice** its complement is **40°**. Find the angle.",{"kind":111,"answer":568,"tolerance":113,"unit":203},40,[570],"Simplify (180 − x) − 2(90 − x) before solving.","Supplement − 2 × complement = (180 − x) − 2(90 − x) = 180 − x − 180 + 2x = x. So the angle itself is **40°**. Check: supplement 140°, complement 50°, 140 − 100 = 40 ✓.",[518,128],{"id":574,"section":27,"level":49,"prompt":575,"check":576,"hints":578,"solution":579,"skills":580},"angles.q041","Two angles form a **linear pair**. One is **70°**. What is the other?",{"kind":111,"answer":577,"tolerance":113,"unit":203},110,[],"A linear pair lies along a straight line, so the angles add to 180°: 180° − 70° = **110°**.",[581],"linear pair",{"id":583,"section":27,"level":49,"prompt":584,"check":585,"hints":590,"solution":591,"skills":592},"angles.q042","True or false: every pair of **adjacent** angles adds up to 180°.",{"kind":65,"options":586,"correct":589},[587,588],{"id":277,"label":278},{"id":234,"label":280},[234],[],"**False.** Adjacent angles only need a common vertex, a common arm and no overlap. They can add to anything, such as 20° + 30° = 50°. Only a **linear pair** (outer arms on a straight line) must add to 180°.",[593,85],"adjacent angles",{"id":595,"section":27,"level":108,"prompt":596,"check":597,"hints":610,"solution":611,"skills":612},"angles.q043","Which conditions must two angles meet to be **adjacent**? Choose all that apply.",{"kind":65,"options":598,"correct":609},[599,601,603,605,607],{"id":68,"label":600},"A common vertex",{"id":71,"label":602},"A common arm",{"id":74,"label":604},"Their interiors do not overlap",{"id":77,"label":606},"They add up to 180°",{"id":231,"label":608},"They are equal",[68,71,74],[],"Adjacent angles have a **common vertex**, a **common arm**, and **non-overlapping interiors**. Their sizes can be anything; adding to 180° is extra (that makes a linear pair).",[593],{"id":614,"section":27,"level":108,"prompt":615,"check":616,"hints":617,"solution":619,"skills":620},"angles.q044","The angles of a linear pair are **(2x + 10)°** and **(3x − 30)°**. Find x.",{"kind":111,"answer":568,"tolerance":113},[618],"Add the two expressions and set the total to 180.","Linear pair: (2x + 10) + (3x − 30) = 180. So 5x − 20 = 180, 5x = 200, **x = 40**. The angles are 90° and 90°.",[581,518],{"id":622,"section":27,"level":108,"prompt":623,"check":624,"hints":629,"solution":630,"skills":631},"angles.q045","True or false: if two angles are supplementary, they must form a linear pair.",{"kind":65,"options":625,"correct":628},[626,627],{"id":277,"label":278},{"id":234,"label":280},[234],[],"**False.** A linear pair must also be **adjacent** with outer arms on a line. 120° at one corner of a room and 60° somewhere else are supplementary but not a linear pair. (The reverse is true: every linear pair is supplementary.)",[581,128],{"id":633,"section":27,"level":287,"prompt":634,"check":635,"hints":637,"solution":639,"skills":640},"angles.q046","Ray OC stands on line AB. ∠AOC is **four times** ∠COB. Find ∠AOC.",{"kind":111,"answer":636,"tolerance":113,"unit":203},144,[638],"5 equal parts make 180°.","∠AOC + ∠COB = 180° (linear pair). If ∠COB = x, then 4x + x = 180°, so x = 36° and ∠AOC = 4 × 36° = **144°**.",[581,563],{"id":642,"section":27,"level":287,"prompt":643,"check":644,"hints":645,"solution":647,"skills":648},"angles.q047","Three rays OC, OD and OE stand on the same side of straight line AB at point O, in the order A, C, D, E, B. ∠AOC = 50°, ∠COD = 45° and ∠DOE = 35°. Find ∠EOB.",{"kind":111,"answer":452,"tolerance":113,"unit":203},[646],"All the angles along one side of a straight line add to 180°.","The four angles ∠AOC, ∠COD, ∠DOE and ∠EOB lie side by side along the straight line, so they add to 180°: ∠EOB = 180° − 50° − 45° − 35° = **50°**.",[649],"angles on a line",{"id":651,"section":27,"level":430,"prompt":652,"check":653,"hints":655,"solution":657,"skills":658},"angles.q048","In a linear pair, one angle is **36° less than twice** the other. Find the **larger** angle.",{"kind":111,"answer":654,"tolerance":113,"unit":203},108,[656],"Write the pair as x and 2x − 36.","Let the other angle be x; this one is 2x − 36. Together: x + 2x − 36 = 180, so 3x = 216 and x = 72°. The angles are 72° and 2 × 72 − 36 = 108°. The larger is **108°**.",[581,518],{"id":660,"section":31,"level":49,"prompt":661,"check":662,"hints":663,"solution":664,"skills":665},"angles.q049","Two straight lines cross. One of the angles is **50°**. What is the angle **vertically opposite** to it?",{"kind":111,"answer":452,"tolerance":113,"unit":203},[],"Vertically opposite angles are equal, so it is **50°**.",[666],"vertically opposite",{"id":668,"section":31,"level":49,"prompt":669,"check":670,"hints":672,"solution":673,"skills":674},"angles.q050","Two straight lines cross. One of the angles is **50°**. What is the size of each of the two angles **next to** it?",{"kind":111,"answer":671,"tolerance":113,"unit":203},130,[],"Each neighbour forms a linear pair with the 50° angle: 180° − 50° = **130°**.",[581],{"id":676,"section":31,"level":108,"prompt":677,"check":678,"hints":689,"solution":690,"skills":691},"angles.q051","Two lines intersect so that one angle is **90°**. What are the other three angles?",{"kind":65,"options":679,"correct":688},[680,682,684,686],{"id":68,"label":681},"90°, 90°, 90°",{"id":71,"label":683},"90°, 180°, 90°",{"id":74,"label":685},"45°, 90°, 45°",{"id":77,"label":687},"It depends on the lines",[68],[],"The opposite angle equals 90° (vertically opposite), and each neighbour is 180° − 90° = 90° (linear pair). So **all four are 90°**: the lines are perpendicular.",[666,692],"perpendicular",{"id":694,"section":31,"level":108,"prompt":695,"check":696,"hints":697,"solution":699,"skills":700},"angles.q052","Two vertically opposite angles measure **(3x + 12)°** and **(5x − 20)°**. Find each angle.",{"kind":111,"answer":322,"tolerance":113,"unit":203},[698],"Set the two expressions equal.","Vertically opposite angles are equal: 3x + 12 = 5x − 20, so 2x = 32 and x = 16. Each angle is 3 × 16 + 12 = **60°**.",[666,518],{"id":702,"section":31,"level":108,"prompt":703,"check":704,"hints":706,"solution":707,"skills":708},"angles.q053","Scissors are opened so that the blades make an angle of **28°**. What is the angle between the handles?",{"kind":111,"answer":705,"tolerance":113,"unit":203},28,[],"Each blade and its handle form one straight piece, so the blades and handles make two crossing lines. The angle between the handles is **vertically opposite** the blade angle, so it is also **28°**.",[666,709],"real life",{"id":711,"section":31,"level":287,"prompt":712,"check":713,"hints":723,"solution":724,"skills":725},"angles.q054","Which reason explains why vertically opposite angles are equal?",{"kind":65,"options":714,"correct":722},[715,717,719,721],{"id":68,"label":716},"They look the same size",{"id":71,"label":718},"Each forms a linear pair with the same neighbouring angle, so each is 180° minus that angle",{"id":74,"label":720},"They are corresponding angles",{"id":77,"label":606},[71],[],"If ∠1 and ∠3 are opposite and ∠2 is next to both, then ∠1 + ∠2 = 180° and ∠3 + ∠2 = 180° (linear pairs). So **∠1 = ∠3 = 180° − ∠2**. This is Euclid's Proposition I.15.",[726,666],"proof",{"id":728,"section":31,"level":287,"prompt":729,"check":730,"hints":731,"solution":733,"skills":734},"angles.q055","Two lines cross. Two **neighbouring** angles are **(x + 20)°** and **(2x + 10)°**. Find the larger angle.",{"kind":111,"answer":577,"tolerance":113,"unit":203},[732],"Neighbours at a crossing add to 180°.","Neighbouring angles at a crossing form a linear pair: (x + 20) + (2x + 10) = 180, so 3x = 150 and x = 50. The angles are 70° and 110°; the larger is **110°**.",[581,518],{"id":736,"section":31,"level":430,"prompt":737,"check":738,"hints":740,"solution":742,"skills":743},"angles.q056","Three lines pass through one point, making six angles. Going round, they are **a, 2a, 3a, a, 2a, 3a**. Find a.",{"kind":111,"answer":739,"tolerance":113,"unit":203},30,[741],"Three neighbouring angles make a straight line.","Three neighbouring angles lie along a straight line, so a + 2a + 3a = 180°, giving 6a = 180° and **a = 30°**. The angles are 30°, 60°, 90° repeated, and the six add to 360° ✓.",[649,518],{"id":745,"section":35,"level":49,"prompt":746,"check":747,"hints":748,"solution":749,"skills":750},"angles.q057","Three angles around a point are **100°**, **120°** and **x**. Find x.",{"kind":111,"answer":462,"tolerance":113,"unit":203},[],"Angles around a point add to 360°: x = 360° − 100° − 120° = **140°**.",[751],"angles at a point",{"id":753,"section":35,"level":49,"prompt":754,"check":755,"hints":756,"solution":757,"skills":758},"angles.q058","A round cake is cut into **12** equal slices from the centre. What angle is at the tip of each slice?",{"kind":111,"answer":739,"tolerance":113,"unit":203},[],"The slices fill the full turn around the centre: 360° ÷ 12 = **30°**.",[751,216],{"id":760,"section":35,"level":108,"prompt":761,"check":762,"hints":763,"solution":765,"skills":766},"angles.q059","Four angles around a point are **x, 2x, 3x** and **4x**. Find the largest angle.",{"kind":111,"answer":636,"tolerance":113,"unit":203},[764],"Add the multiples of x.","x + 2x + 3x + 4x = 10x = 360°, so x = 36°. The largest is 4 × 36° = **144°**.",[751,518],{"id":768,"section":35,"level":108,"prompt":769,"check":770,"hints":771,"solution":773,"skills":774},"angles.q060","In a pie chart of 30 students, **12** chose football. What is the angle of the football slice?",{"kind":111,"answer":636,"tolerance":113,"unit":203},[772],"360 ÷ 30 = ?","The whole circle, 360°, stands for 30 students, so each student is 12°. Football: 12 × 12° = **144°**.",[751,775],"pie charts",{"id":777,"section":35,"level":287,"prompt":778,"check":779,"hints":780,"solution":782,"skills":783},"angles.q061","Five angles around a point are **75°, 85°, 50°, y** and **2y**. Find y.",{"kind":111,"answer":452,"tolerance":113,"unit":203},[781],"The five angles add to 360°.","75 + 85 + 50 = 210, so y + 2y = 360 − 210 = 150. Then 3y = 150 and **y = 50°**. The angles are 75°, 85°, 50°, 50° and 100°, which add to 360° ✓.",[751,518],{"id":785,"section":35,"level":287,"prompt":786,"check":787,"hints":792,"solution":793,"skills":794},"angles.q062","True or false: four **obtuse** angles can fit exactly around a point.",{"kind":65,"options":788,"correct":791},[789,790],{"id":277,"label":278},{"id":234,"label":280},[234],[],"**False.** Each obtuse angle is more than 90°, so four of them add to more than 4 × 90° = 360°. They would overlap.",[751,128],{"id":796,"section":35,"level":430,"prompt":797,"check":798,"hints":799,"solution":801,"skills":802},"angles.q063","Floor tiles meet at a point: **two squares** and **one regular hexagon**. Could a third shape fill the gap exactly? What angle does it need?",{"kind":111,"answer":322,"tolerance":113,"unit":203},[800],"A regular hexagon's corner is 120°.","Two squares give 2 × 90° = 180°, and a regular hexagon's corner is 120°. That makes 300°, so the gap is 360° − 300° = **60°**, exactly the corner of an equilateral triangle. So an equilateral triangle fits.",[751,803],"tiling",{"id":805,"section":39,"level":108,"prompt":806,"check":807,"hints":808,"solution":810,"skills":811},"angles.q064","Lines AB and CD cross at O. ∠AOC = **(2x)°** and ∠BOD = **(x + 35)°**. Find ∠AOD.",{"kind":111,"answer":577,"tolerance":113,"unit":203},[809],"First find x using vertically opposite angles.","∠AOC and ∠BOD are vertically opposite, so 2x = x + 35 and x = 35. ∠AOC = 70°. ∠AOD forms a linear pair with ∠AOC: 180° − 70° = **110°**.",[666,581,518],{"id":813,"section":39,"level":108,"prompt":814,"check":815,"hints":817,"solution":818,"skills":819},"angles.q065","OE is perpendicular to line AB at O, and ray OF lies between OE and OB with ∠EOF = **34°**. Find ∠FOB.",{"kind":111,"answer":816,"tolerance":113,"unit":203},56,[],"∠EOB = 90° (OE ⊥ AB). OF lies inside it, so ∠FOB = 90° − 34° = **56°** (adjacent angles).",[457,593],{"id":821,"section":39,"level":287,"prompt":822,"check":823,"hints":825,"solution":827,"skills":828},"angles.q066","Lines PQ and RS cross at O. Ray OT bisects ∠POS, and ∠POR = **130°**. Find ∠TOS.",{"kind":111,"answer":824,"tolerance":113,"unit":203},25,[826],"Find ∠POS first.","∠POS forms a linear pair with ∠POR: 180° − 130° = 50°. OT bisects it, so ∠TOS = 50° ÷ 2 = **25°**.",[581,829],"bisector",{"id":831,"section":39,"level":287,"prompt":832,"check":833,"hints":835,"solution":836,"skills":837},"angles.q067","Two lines cross at O. A third ray OX lies inside one of the angles and splits it into **25°** and **40°**. What is the angle **vertically opposite** to the whole angle that OX splits?",{"kind":111,"answer":834,"tolerance":113,"unit":203},65,[],"The whole angle is 25° + 40° = 65° (adjacent angles). Its vertically opposite angle is equal: **65°**.",[593,666],{"id":839,"section":39,"level":287,"prompt":840,"check":841,"hints":843,"solution":845,"skills":846},"angles.q068","Spot the mistake. Meena solves: “∠1 and ∠2 are a linear pair; ∠1 = 64°, so ∠2 = 90° − 64° = 26°.” What is the correct value of ∠2?",{"kind":111,"answer":842,"tolerance":113,"unit":203},116,[844],"Linear pair: which total?","A linear pair adds to **180°**, not 90°. So ∠2 = 180° − 64° = **116°**. Meena used the complementary rule by mistake.",[581,85],{"id":848,"section":39,"level":430,"prompt":849,"check":850,"hints":861,"solution":863,"skills":864},"angles.q069","Lines AB and CD cross at O. OE bisects ∠AOC and OF bisects ∠BOD. If ∠AOC = **70°**, which statement about ∠EOF (measured through OC and OB) is true?",{"kind":65,"options":851,"correct":860},[852,854,856,858],{"id":68,"label":853},"∠EOF = 180°: E, O and F lie on one straight line",{"id":71,"label":855},"∠EOF = 70°",{"id":74,"label":857},"∠EOF = 110°",{"id":77,"label":859},"∠EOF = 140°",[68],[862],"Add up the angles from OE to OF going through OC and OB.","∠BOD = ∠AOC = 70° (vertically opposite). ∠EOC = 35° (bisector), ∠COB = 180° − 70° = 110° (linear pair), ∠BOF = 35° (bisector). Going from OE through OC and OB to OF: 35° + 110° + 35° = **180°**. So E, O, F are on one straight line: the bisectors of vertically opposite angles line up.",[829,128],{"id":866,"section":43,"level":108,"prompt":867,"check":868,"hints":869,"solution":870,"skills":871},"angles.q070","Two parallel lines are cut by a transversal. One angle is **65°**. What is its **corresponding** angle?",{"kind":111,"answer":834,"tolerance":113,"unit":203},[],"When lines are parallel, corresponding angles are **equal**: **65°**.",[872],"transversals",{"id":874,"section":43,"level":108,"prompt":875,"check":876,"hints":878,"solution":879,"skills":880},"angles.q071","Two parallel lines are cut by a transversal. One **co-interior** angle is **65°**. What is the other?",{"kind":111,"answer":877,"tolerance":113,"unit":203},115,[],"Co-interior angles on parallel lines add to 180°: 180° − 65° = **115°**.",[872],{"id":882,"section":43,"level":108,"prompt":883,"check":884,"hints":885,"solution":886,"skills":887},"angles.q072","Two angles of a triangle are **50°** and **60°**. Find the third angle.",{"kind":111,"answer":558,"tolerance":113,"unit":203},[],"The angles of a triangle add to 180°: 180° − 50° − 60° = **70°**.",[888],"triangle angle sum",{"id":890,"section":43,"level":287,"prompt":891,"check":892,"hints":899,"solution":900,"skills":901},"angles.q073","A transversal cuts two lines. Alternate interior angles are **72°** and **68°**. Are the lines parallel?",{"kind":65,"options":893,"correct":898},[894,896],{"id":95,"label":895},"Yes",{"id":334,"label":897},"No",[334],[],"For parallel lines, alternate interior angles must be **equal**. 72° ≠ 68°, so the lines are **not parallel**; they meet on the side where the co-interior angles add to less than 180°.",[872,902],"converse",{"id":904,"section":43,"level":287,"prompt":905,"check":906,"hints":907,"solution":908,"skills":909},"angles.q074","Three angles of a quadrilateral are **80°, 95°** and **110°**. Find the fourth.",{"kind":111,"answer":362,"tolerance":113,"unit":203},[],"A quadrilateral splits into two triangles, so its angles add to 360°: 360° − 80° − 95° − 110° = **75°**.",[910],"quadrilateral angle sum",{"id":912,"section":43,"level":287,"prompt":913,"check":914,"hints":915,"solution":917,"skills":918},"angles.q075","In a right-angled triangle, one acute angle is **(2x)°** and the other is **(x + 15)°**. Find x.",{"kind":111,"answer":824,"tolerance":113},[916],"The acute angles add to 90°.","The two acute angles of a right triangle are complementary: 2x + x + 15 = 90, so 3x = 75 and **x = 25**. The angles are 50° and 40°.",[888,457],{"id":920,"section":43,"level":287,"prompt":921,"check":922,"hints":923,"solution":924,"skills":925},"angles.q076","In triangle ABC, side BC is extended to D. ∠A = **48°** and ∠B = **62°**. Find the exterior angle ∠ACD.",{"kind":111,"answer":577,"tolerance":113,"unit":203},[],"Exterior angle = sum of the interior opposite angles: ∠ACD = 48° + 62° = **110°**. (Check: ∠ACB = 180° − 110° = 70°, and 48 + 62 + 70 = 180 ✓.)",[926],"exterior angle",{"id":928,"section":43,"level":430,"prompt":929,"check":930,"hints":931,"solution":933,"skills":934},"angles.q077","Parallel lines AB and CD have a bent path between them, bending at E (pointing between the lines). ∠BAE = **(x + 10)°**, ∠DCE = **(2x)°** and the bend ∠AEC = **100°**. Find x.",{"kind":111,"answer":739,"tolerance":113},[932],"Bend angle = sum of the two outer angles.","Draw a line through E parallel to AB. The bend splits into two alternate angles: ∠AEC = ∠BAE + ∠DCE. So 100 = x + 10 + 2x, giving 3x = 90 and **x = 30**. The outer angles are 40° and 60°.",[872,935],"construction",{"id":937,"section":43,"level":430,"prompt":938,"check":939,"hints":941,"solution":943,"skills":944},"angles.q078","Each interior angle of a regular polygon is **140°**. How many sides does it have?",{"kind":111,"answer":940,"tolerance":113},9,[942],"Use exterior angles.","Each exterior angle is 180° − 140° = 40°. The exterior angles add to 360°, so there are 360 ÷ 40 = **9** sides (a regular nonagon). Check: (9 − 2) × 180° ÷ 9 = 1260° ÷ 9 = 140° ✓.",[945,926],"polygons",{"id":947,"section":43,"level":430,"prompt":948,"check":949,"hints":950,"solution":952,"skills":953},"angles.q079","In triangle ABC, AB = AC and the exterior angle at C (with BC extended beyond C) is **115°**. Find ∠A.",{"kind":111,"answer":452,"tolerance":113,"unit":203},[951],"Find ∠ACB first, then use equal base angles.","∠ACB = 180° − 115° = 65° (linear pair). AB = AC, so ∠B = ∠ACB = 65° (isosceles triangle). ∠A = 180° − 65° − 65° = **50°**. Check with the exterior angle property: ∠A + ∠B = 50° + 65° = 115° ✓.",[926,954],"isosceles",{"id":956,"section":43,"level":430,"prompt":957,"check":958,"hints":959,"solution":961,"skills":962},"angles.q080","What is the sum of the five tip angles of any five-pointed star drawn with straight lines?",{"kind":111,"answer":202,"tolerance":113,"unit":203},[960],"Use the exterior angle property twice.","Take the small triangle at one tip. Each of its other two angles is an exterior angle of another small triangle, so it equals the sum of two other tip angles. The triangle's angle sum then gives: sum of all five tips = **180°**. A regular star has tips of 36° each.",[963,926],"angle chasing",[965,966,967,968,969,970,971,972],"angles-ncert-class7-lines-angles","angles-ncert-class7-triangle","angles-mathsisfun-angles","angles-mathsisfun-parallel","angles-khan-angle-basics","angles-wiki-clock-angle","angles-ncert-class6-lines-angles","angles-mathsisfun-complementary","needs_review",{"generatedBy":975,"notes":976},"claude-code","Draft; every numeric answer computed and asserted in the Python generator. Pending owner review.","4b64777cb328510e4d9621eec54e61ec8f27ba258a339a0d4e37a6f99d748028",{"logic:questions":979,"source:angles-khan-angle-basics":980,"source:angles-mathsisfun-angles":981,"source:angles-mathsisfun-complementary":982,"source:angles-mathsisfun-parallel":983,"source:angles-ncert-class6-lines-angles":984,"source:angles-ncert-class7-lines-angles":985,"source:angles-ncert-class7-triangle":986,"source:angles-wiki-clock-angle":987},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","68741673c0dc6c96760fde888327818c508788257b5306bad48ab148615541a4","2dab8c7c8736bbad61c4a7223c297c3b250b20399d5b5544c02434cab39a0c9c","7656284145acec3d6ead6ded762ed0eef2bb3137446c527350b7b999562580df","3cc0370817500ac4458d13f159f32213300db7e939f725c2dc0f989a699c8609","36960f0a34d1e3f73475a665e4b12c1140dfbbeac1c1b2772bf71a8d3b9522a5","548a49fd09ac1b7cec7c74b038e1a87c5d3add389d6f27376ac054da17d20972","286f44f488c463767261100625e631e8edb823f1f296976d251359a4bc9677ee","141f5516f8832cc0eea67b34a5726cd12ed7309036b178da74f753ca6e4ae706","preview-7e1cbbcc4f",1789899599553]