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Therefore, adjacent angles lying on the same straight line must have a total of 180°."},{"text":"360°.","correct":false,"explanation":"This confuses a straight line with a full turn around a point. A straight line is a half-turn, so its angles add to 180°."},{"text":"90°.","correct":false,"explanation":"This mistakes a straight angle for a right angle. A right angle is 90°, but angles on a straight line add to 180°."},{"text":"270°.","correct":false,"explanation":"This represents three-quarters of a full turn rather than a straight line. A straight line forms a half-turn, so its angles total 180°."}]},{"id":"4d871dfe-e0ae-ef06-90d8-82e3d8e71196","cardType":"cloze","points":null,"front":"Angles around a point add to [ ].","back":"Angles around a point add to 360°.","mcqOptions":[{"text":"360°","correct":true,"explanation":"Angles around a single point make one complete turn. A complete turn measures 360°, so all the angles around the point must total 360°."},{"text":"180°","correct":false,"explanation":"This confuses angles around a point with angles on a straight line. Around a point is a complete turn, so the angles add to 360°."},{"text":"90°","correct":false,"explanation":"This gives the size of one right angle, only a quarter-turn. Angles all the way around a point form a full turn of 360°."},{"text":"270°","correct":false,"explanation":"This accounts for only three-quarters of a turn. Angles around a point complete the entire turn, so their total is 360°."}]},{"id":"4da7298d-ddaa-51ed-c145-370654c455cc","cardType":"basic","points":1,"front":"What do angles in a triangle add to?","back":"180°.","mcqOptions":[{"text":"180°.","correct":true,"explanation":"The three interior angles of every triangle have a total of 180°, regardless of the triangle’s side lengths, angle sizes or orientation."},{"text":"360°.","correct":false,"explanation":"This confuses the triangle’s interior angle sum with a full turn. The three interior angles of any triangle always add to 180°."},{"text":"90°.","correct":false,"explanation":"A right triangle contains one 90° angle, but that is not the total. All three interior angles of a triangle add to 180°."},{"text":"270°.","correct":false,"explanation":"This is three right angles, not the interior angle sum of a triangle. Every triangle’s three interior angles have a total of 180°."}]},{"id":"d127580d-88a5-2f59-8f32-2d8a635db8a2","cardType":"cloze","points":null,"front":"In a regular polygon, all sides are [ ] and all interior angles are [ ].","back":"In a regular polygon, all sides are equal and all interior angles are equal.","mcqOptions":[{"text":"In a regular polygon, all sides are equal and all interior angles are equal.","correct":true,"explanation":"A polygon is regular only when it is both equilateral and equiangular. Therefore, every side has equal length and every interior angle has equal size."},{"text":"In a regular polygon, all sides are parallel and all interior angles are equal.","correct":false,"explanation":"Regular polygons have equal side lengths, not all parallel sides; adjacent sides must meet. Their interior angles are also equal, so both equality conditions are required."},{"text":"In a regular polygon, all sides are equal and all interior angles are different.","correct":false,"explanation":"Equal sides alone do not give the full definition of a regular polygon. A regular polygon must also have equal interior angles, rather than different ones."},{"text":"In a regular polygon, all sides are different and all interior angles are different.","correct":false,"explanation":"This reverses both defining properties of a regular polygon. For a polygon to be regular, all its sides and all its interior angles must be equal."}]},{"id":"7a3b8619-0ab6-e3f7-0623-f8ddcf47a79a","cardType":"basic","points":1,"front":"How many triangles can an n-sided polygon be split into from one vertex?","back":"n - 2 triangles.","mcqOptions":[{"text":"n - 2 triangles.","correct":true,"explanation":"Drawing diagonals from one vertex divides an n-sided polygon into n - 2 non-overlapping triangles. This result is used to derive the polygon interior angle sum."},{"text":"n - 1 triangles.","correct":false,"explanation":"This common off-by-one error counts one extra region. Splitting from one vertex creates n - 2 triangles because the chosen vertex belongs to the two boundary sides."},{"text":"n triangles.","correct":false,"explanation":"This incorrectly assumes one triangle for every side. Diagonals drawn from one vertex actually divide an n-sided polygon into n - 2 triangles."},{"text":"n - 3 triangles.","correct":false,"explanation":"This is the number of diagonals drawn from one vertex, not the number of triangles formed. Those diagonals create n - 2 triangles."}]},{"id":"803634f4-9c26-ce5a-22ec-60775489d14b","cardType":"cloze","points":null,"front":"For an n-sided polygon, the interior angle sum is [ ].","back":"For an n-sided polygon, the interior angle sum is (n - 2) × 180°.","mcqOptions":[{"text":"(n - 2) × 180°","correct":true,"explanation":"An n-sided polygon can be divided from one vertex into n - 2 triangles. Each triangle contributes 180°, giving an interior angle sum of (n - 2) × 180°."},{"text":"(n - 1) × 180°","correct":false,"explanation":"This counts one too many triangles in the polygon. The correct split produces n - 2 triangles, so the interior angle sum is (n - 2) × 180°."},{"text":"n × 180°","correct":false,"explanation":"This incorrectly assigns a whole triangle to every side. An n-sided polygon splits into n - 2 triangles, making the sum (n - 2) × 180°."},{"text":"(n - 2) × 360°","correct":false,"explanation":"This uses 360° for each triangle, confusing a triangle’s angle sum with a full turn. Each triangle contributes 180°, so the formula is (n - 2) × 180°."}]},{"id":"f4f1b722-4ede-c203-ba5d-e1eb46a22cd2","cardType":"basic","points":1,"front":"What is the interior angle sum of a pentagon?","back":"540°.","mcqOptions":[{"text":"540°.","correct":true,"explanation":"A pentagon has five sides, so its interior angle sum is (5 - 2) × 180° = 3 × 180° = 540°."},{"text":"360°.","correct":false,"explanation":"This effectively uses only two triangles instead of three. A pentagon splits into 5 - 2 = 3 triangles, so its interior angles total 540°."},{"text":"720°.","correct":false,"explanation":"This is the interior angle sum of a hexagon, not a pentagon. For a pentagon, (5 - 2) × 180° gives 540°."},{"text":"900°.","correct":false,"explanation":"This multiplies the five sides directly by 180° instead of subtracting two first. The correct calculation is (5 - 2) × 180° = 540°."}]},{"id":"bb6642db-b456-83c9-8a5a-9cc18fa8d56e","cardType":"basic","points":1,"front":"What is the interior angle sum of a hexagon?","back":"720°.","mcqOptions":[{"text":"720°.","correct":true,"explanation":"A hexagon has six sides, so its interior angle sum is (6 - 2) × 180° = 4 × 180° = 720°."},{"text":"540°.","correct":false,"explanation":"This is the interior angle sum of a pentagon and uses only three triangles. A hexagon splits into four triangles, giving a total of 720°."},{"text":"900°.","correct":false,"explanation":"This uses five triangles, as for a seven-sided polygon. A hexagon forms 6 - 2 = 4 triangles, so its interior angle sum is 720°."},{"text":"1080°.","correct":false,"explanation":"This multiplies the six sides directly by 180° instead of subtracting two. The correct calculation is (6 - 2) × 180° = 720°."}]},{"id":"03ea2ab7-cffe-1715-467a-6f44fecde5d3","cardType":"basic","points":2,"front":"A pentagon has angles 98°, 121°, 116°, 85° and x. Find x.","back":"120°.","mcqOptions":[{"text":"120°.","correct":true,"explanation":"A pentagon’s interior angles total 540°. The four known angles total 420°, so the missing angle is $540° - 420° = 120°$."},{"text":"110°.","correct":false,"explanation":"This results from using 530° as the pentagon’s angle sum. The correct sum is $(5 - 2) \\times 180° = 540°$, giving $540° - 420° = 120°$."},{"text":"130°.","correct":false,"explanation":"This comes from incorrectly using 550° for the total interior angle sum. A pentagon’s angles total 540°, so the missing angle is $540° - 420° = 120°$."},{"text":"140°.","correct":false,"explanation":"This commonly follows from adding the four given angles as 400° rather than 420°. Their correct total is 420°, leaving $540° - 420° = 120°$."}]},{"id":"7bebf229-0f6d-2dcf-20d9-d5708c1e7a02","cardType":"cloze","points":null,"front":"The exterior angles of a polygon, taking one at each vertex, add to [ ].","back":"The exterior angles of a polygon, taking one at each vertex, add to 360°.","mcqOptions":[{"text":"360°","correct":true,"explanation":"One exterior angle at each vertex represents the complete turn made when travelling around the polygon, so the exterior angles always add to 360°."},{"text":"180°","correct":false,"explanation":"180° is a half-turn and is also the sum of the angles on a straight line. A complete turn around any polygon is 360°."},{"text":"90°","correct":false,"explanation":"90° is a quarter-turn and may be one exterior angle of a square, but all the exterior angles of any polygon add to 360°."},{"text":"540°","correct":false,"explanation":"540° is the sum of the interior angles of a pentagon, not the universal exterior-angle sum. One exterior angle per vertex always totals 360°."}]},{"id":"84af33a5-e799-88f9-be26-9412cf42ac01","cardType":"basic","points":1,"front":"For a regular polygon with n sides, what is each exterior angle?","back":"360° ÷ n.","mcqOptions":[{"text":"360° ÷ n.","correct":true,"explanation":"The exterior angles of every polygon total 360°. In a regular polygon they are equal, so dividing 360° by the number of sides gives each exterior angle."},{"text":"180° ÷ n.","correct":false,"explanation":"This incorrectly uses a half-turn as the exterior-angle total. The exterior angles sum to 360°, so each exterior angle of a regular polygon is $360° \\div n$."},{"text":"360° × n.","correct":false,"explanation":"Multiplying would make each exterior angle increase as the number of sides increases. The fixed total of 360° must instead be divided equally among $n$ vertices."},{"text":"360° ÷ (n - 2).","correct":false,"explanation":"The expression $n - 2$ belongs to the formula for the interior-angle sum. Exterior angles total 360°, so each one is $360° \\div n$."}]},{"id":"04714e05-95f2-c7fa-b2ef-5206723dd3ca","cardType":"basic","points":2,"front":"What is the exterior angle of a regular nonagon?","back":"40°.","mcqOptions":[{"text":"40°.","correct":true,"explanation":"A nonagon has nine sides. Its equal exterior angles divide the full 360° turn into nine parts, so each measures $360° \\div 9 = 40°$."},{"text":"20°.","correct":false,"explanation":"This incorrectly divides 180° by nine, using a half-turn rather than the full exterior-angle total. The correct calculation is $360° \\div 9 = 40°$."},{"text":"45°.","correct":false,"explanation":"45° is the exterior angle of a regular octagon, which has eight sides. A nonagon has nine sides, so its exterior angle is $360° \\div 9 = 40°$."},{"text":"50°.","correct":false,"explanation":"This does not divide the complete 360° turn equally among nine vertices. For a regular nonagon, each exterior angle is $360° \\div 9 = 40°$."}]},{"id":"7247f2ab-3bcd-e156-cb24-f50c9a5b5a99","cardType":"cloze","points":null,"front":"At each corner, the interior angle and adjacent exterior angle add to [ ].","back":"At each corner, the interior angle and adjacent exterior angle add to 180°.","mcqOptions":[{"text":"180°","correct":true,"explanation":"An interior angle and its adjacent exterior angle lie on a straight line. Angles on a straight line are supplementary, so they add to 180°."},{"text":"360°","correct":false,"explanation":"360° is the total of one exterior angle at every vertex, not the pair at one corner. An adjacent interior and exterior angle form a straight line totalling 180°."},{"text":"90°","correct":false,"explanation":"90° is a right angle, but the adjacent interior and exterior angles form a straight line rather than a right angle. Their sum is therefore 180°."},{"text":"270°","correct":false,"explanation":"270° is a three-quarter turn and does not describe this linear pair. Adjacent interior and exterior angles lie on a straight line, so their sum is 180°."}]},{"id":"2dc378cd-8c0f-55f9-1974-7982e4ddc029","cardType":"basic","points":2,"front":"A regular 12-sided polygon has exterior angle 30°. What is its interior angle?","back":"150°.","mcqOptions":[{"text":"150°.","correct":true,"explanation":"The interior angle and adjacent exterior angle add to 180°. Therefore, the interior angle is $180° - 30° = 150°$."},{"text":"30°.","correct":false,"explanation":"This repeats the exterior angle instead of finding its supplement. The adjacent interior and exterior angles total 180°, so the interior angle is $180° - 30° = 150°$."},{"text":"120°.","correct":false,"explanation":"This may come from subtracting 30° twice from 180°. Only one adjacent exterior angle is subtracted, giving an interior angle of $180° - 30° = 150°$."},{"text":"330°.","correct":false,"explanation":"This subtracts the exterior angle from a full turn, but the interior angle forms a straight line with it. The correct calculation is $180° - 30° = 150°$."}]},{"id":"3174351a-95b0-2898-5e8a-947fdf8332b9","cardType":"basic","points":1,"front":"How do you find the number of sides from a regular polygon's exterior angle?","back":"n = 360° ÷ exterior angle.","mcqOptions":[{"text":"n = 360° ÷ exterior angle.","correct":true,"explanation":"The equal exterior angles total 360°. Dividing that total by the size of one exterior angle gives how many equal angles, and therefore how many sides, the polygon has."},{"text":"n = 180° ÷ exterior angle.","correct":false,"explanation":"This uses 180°, the angle on a straight line, instead of the complete exterior-angle total. The number of sides is $360° \\div$ the exterior angle."},{"text":"n = 360° × exterior angle.","correct":false,"explanation":"Multiplication does not determine how many equal exterior angles fit into 360°. The correct operation is division: $n = 360° \\div \\text{exterior angle}$."},{"text":"n = exterior angle ÷ 360°.","correct":false,"explanation":"This reverses the division and would give a value below one for ordinary exterior angles. Divide the full 360° turn by one exterior angle to find $n$."}]},{"id":"f896b6df-3b02-a5f1-66a1-75158aa554c7","cardType":"basic","points":2,"front":"A regular polygon has exterior angle 30°. How many sides does it have?","back":"12 sides.","mcqOptions":[{"text":"12 sides.","correct":true,"explanation":"The exterior angles total 360°, and each is 30°. Therefore, the number of equal angles and sides is $360° \\div 30° = 12$."},{"text":"6 sides.","correct":false,"explanation":"This comes from dividing 180° by 30°, using a half-turn instead of the full exterior-angle total. The correct calculation is $360° \\div 30° = 12$ sides."},{"text":"10 sides.","correct":false,"explanation":"Ten sides would give each exterior angle as $360° \\div 10 = 36°$, not 30°. An exterior angle of 30° gives $360° \\div 30° = 12$ sides."},{"text":"15 sides.","correct":false,"explanation":"Fifteen sides would produce an exterior angle of $360° \\div 15 = 24°$. For an exterior angle of 30°, the polygon has $360° \\div 30° = 12$ sides."}]},{"id":"b199d1b5-4efa-4605-c683-7543ca4aa670","cardType":"cloze","points":null,"front":"If given a regular polygon's interior angle, first find the exterior angle using [ ].","back":"If given a regular polygon's interior angle, first find the exterior angle using 180° - interior angle.","mcqOptions":[{"text":"180° - interior angle","correct":true,"explanation":"An interior angle and its corresponding exterior angle lie on a straight line, so they total 180°. Therefore, subtract the interior angle from 180°."},{"text":"360° - interior angle","correct":false,"explanation":"Using 360° confuses angles around a point with supplementary interior and exterior angles. The correct calculation is 180° - interior angle because the two form a straight line."},{"text":"180° + interior angle","correct":false,"explanation":"Adding the angles ignores that their total must be 180°. The exterior angle is found using 180° - interior angle because adjacent interior and exterior angles are supplementary."},{"text":"interior angle - 180°","correct":false,"explanation":"Reversing the subtraction produces a negative value for an ordinary convex polygon. The correct exterior angle is 180° - interior angle, giving a positive angle."}]},{"id":"88b73c37-f502-b7b6-c67b-cebaf275e480","cardType":"basic","points":2,"front":"A regular polygon has interior angle 156°. How many sides does it have?","back":"15 sides.","mcqOptions":[{"text":"15 sides.","correct":true,"explanation":"The exterior angle is 180° - 156° = 24°. Since regular exterior angles total 360°, the polygon has 360° ÷ 24° = 15 sides."},{"text":"24 sides.","correct":false,"explanation":"This mistakes the 24° exterior angle for the number of sides. Dividing 360° by 24° gives the correct number, which is 15 sides."},{"text":"14 sides.","correct":false,"explanation":"This is an off-by-one error. The exterior angle is 24°, and 360° ÷ 24° equals exactly 15, so the polygon has 15 sides."},{"text":"13 sides.","correct":false,"explanation":"This does not follow from the exterior-angle calculation. The exterior angle is 180° - 156° = 24°, so 360° ÷ 24° gives 15 sides."}]},{"id":"757d0aca-7212-7667-e7ea-524be3e60f49","cardType":"basic","points":2,"front":"Why must you not divide the interior angle sum by the number of sides for an irregular polygon?","back":"Its angles are not all equal.","mcqOptions":[{"text":"Its angles are not all equal.","correct":true,"explanation":"Dividing the angle sum by the number of sides gives only the mean angle. In an irregular polygon, the individual interior angles are not necessarily equal to that mean."},{"text":"Its angles are all equal.","correct":false,"explanation":"Equal interior angles would make division useful for finding each angle, as in a regular polygon. An irregular polygon’s angles are not all equal, so division gives only an average."},{"text":"Its sides are not all equal.","correct":false,"explanation":"Unequal side lengths do not directly prevent calculating individual angles from the sum. The relevant issue is that an irregular polygon’s interior angles are not all equal."},{"text":"Its angle sum is not fixed.","correct":false,"explanation":"The interior angle sum is fixed by the number of sides, even for an irregular polygon. Division fails because its individual angles are not all equal."}]},{"id":"3ed4826a-d68f-c6ed-96e8-0342684f5d15","cardType":"basic","points":2,"front":"Two missing angles are y and 2y and add to 210°. Find y.","back":"70°.","mcqOptions":[{"text":"70°.","correct":true,"explanation":"The two angles give y + 2y = 210°, so 3y = 210°. Dividing both sides by 3 gives y = 70°."},{"text":"105°.","correct":false,"explanation":"This comes from dividing 210° by 2, but the coefficients total 3 because y + 2y = 3y. Dividing by 3 gives y = 70°."},{"text":"140°.","correct":false,"explanation":"This treats 2y as the required value rather than y. Solving y + 2y = 210° gives y = 70°, while 2y equals 140°."},{"text":"210°.","correct":false,"explanation":"This uses the total of both missing angles as the value of y. Since y + 2y = 3y, the total must be divided by 3, giving 70°."}]},{"id":"614bf45f-b089-f092-823e-e22665835936","cardType":"basic","points":1,"front":"When polygons meet at a point, which angles usually make the 360° total?","back":"Interior angles touching the point.","mcqOptions":[{"text":"Interior angles touching the point.","correct":true,"explanation":"At a shared vertex, the polygons’ interior angles fit together around the point. Angles around a point total 360°, so these are the angles normally added."},{"text":"Exterior angles touching the point.","correct":false,"explanation":"The angles filling the space around the shared point are normally the polygons’ interior angles. Exterior angles lie outside each polygon and are not the angles being combined here."},{"text":"Central angles touching the point.","correct":false,"explanation":"Central angles are measured at a polygon’s centre, not at a vertex where polygons meet. The interior angles at the shared vertex make the 360° total."},{"text":"Reflex angles touching the point.","correct":false,"explanation":"Reflex angles are greater than 180° and are not normally the polygon angles fitted around a shared vertex. The relevant angles are the interior angles touching the point."}]},{"id":"a43a9557-2b41-6dec-938e-7b3072999e8b","cardType":"basic","points":3,"front":"A square, regular hexagon and another regular polygon meet at a point. How many sides does the other polygon have?","back":"12 sides.","mcqOptions":[{"text":"12 sides.","correct":true,"explanation":"The square and hexagon contribute 90° and 120°, leaving 150°. Its exterior angle is 30°, so the regular polygon has 360° ÷ 30° = 12 sides."},{"text":"6 sides.","correct":false,"explanation":"A regular hexagon has interior angle 120°, not the required 150°. The missing angle is 360° - 90° - 120° = 150°, corresponding to 12 sides."},{"text":"8 sides.","correct":false,"explanation":"A regular octagon has interior angle 135°, so the three angles would total only 345°. The required interior angle is 150°, which belongs to a 12-sided polygon."},{"text":"10 sides.","correct":false,"explanation":"A regular decagon has interior angle 144°, making the total 354° rather than 360°. The missing 150° angle gives exterior angle 30° and therefore 12 sides."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Angles in Polygons","title":"Angles in Polygons Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"f1aa81cf-e6df-463e-a45a-205e5f8f1827","walled":"$L37"}],"$L38","$L39"]