Printable · GCSE Foundation · ages 14-16
Angle facts, parallel lines and polygons worksheet — GCSE Foundation
Fifteen questions on "angle facts, parallel lines and polygons" — DfE statement G3. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Answer key: Angle facts, parallel lines and polygons worksheet — GCSE Foundation
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- (a) 105° — The interior angles of a pentagon add up to (5 − 2) × 180° = 540°. Subtracting the three known angles, 540 − 100 − 110 − 120 = 210°, and this 210° is shared equally between the two angles marked x°, so each one is 210 ÷ 2 = 105°. 210° stops one step early, giving the total of the two unknown angles instead of one of them. 108° is the interior angle of a regular pentagon, which does not apply here since this pentagon's angles are not all equal. 55° comes from halving one of the given angles, 110°, instead of halving the remaining total.
- (b) 55° — Method: complementary angles are a pair that add up to 90°, so subtract the known angle from 90°. Working: 90 − 35 = 55. Answer: 55°. The distractors: 145° comes from subtracting from 180°, which is the rule for supplementary angles and not for complementary ones; 325° comes from subtracting from 360°, the total of the angles at a point; 90° is the total that the pair must make, written down in place of the missing angle.
- (d) They are always equal to each other — Method: label the four angles a, b, c, d in order around the crossing point, then use the fact that neighbouring angles lie on a straight line. Working: a and b lie on a straight line, so a + b = 180°; b and c also lie on a straight line, so b + c = 180°. Both a and c are therefore 180° minus b, which forces a = c. Answer: a pair of vertically opposite angles is always equal to each other. The distractors: the claim that each is 90° holds only when the two lines happen to be perpendicular, so it is not always true; the claim that they add to 180° confuses the opposite pair with the neighbouring pair that lies along a straight line, and again holds only in the perpendicular case; the claim that they add to 360° uses the total of all four angles at the point rather than of the opposite pair.
- (b) 117 — Method: two angles meeting at a point on a straight line add up to 180°. Working: 180 − 63 = 117. Answer: 117°. A candidate who thinks the two angles on a straight line must be equal gives 63. A candidate who uses 90° instead of 180°, working out 90 − 63, gets 27. A candidate who uses 360° instead of 180°, working out 360 − 63, gets 297.
- (d) 60° — Method: the six angles at the centre together make one complete turn of 360°, and because the hexagon is regular they are all equal, so divide 360° by 6. Working: 360 ÷ 6 = 60. Answer: 60°. The distractors: 120° is the interior angle of a regular hexagon, 720 ÷ 6, which is the angle at a vertex and not the angle at the centre; 45° comes from dividing 360 by 8, treating the hexagon as though it had eight sides; 30° comes from halving the angle at the centre, as though each of the six triangles were split again by a line of symmetry.
- (c) 32 — Method: alternate angles between parallel lines are equal, so 2x + 10 = 74. Working: subtracting 10 from both sides gives 2x = 64; dividing by 2 gives x = 32. Answer: x = 32. A candidate who forgets to subtract 10 first and divides 74 by 2 directly gets 37. A candidate who treats the angles as co-interior instead of alternate, so that the two expressions add to 180° rather than being equal, gets 48 after solving. A candidate who makes a sign error and treats the equation as 2x equalling 10 minus 74 instead of 74 minus 10 gets −32.
- (c) 8 — Method: the exterior angles of a polygon add up to 360°, so divide 360° by the size of one exterior angle. Working: 360 ÷ 45 = 8. Answer: 8 sides. A candidate who divides into a half turn instead of a full turn, working out 180 ÷ 45, gets 4. A candidate who reads off the given exterior angle as if it were the number of sides gets 45. A candidate who subtracts instead of dividing, working out 360 − 45, gets 315.
- (a) 60° — Method: the three angles of a triangle add up to 180°, so add the two known angles and subtract the total from 180°. Working: 50 + 70 = 120, then 180 − 120 = 60. Answer: 60°. The distractors: 120° is the sum of the two known angles, given as the answer instead of being subtracted from 180°; 110° comes from subtracting only the 70° angle from 180° and forgetting the 50° one; 240° comes from subtracting the sum from 360°, using the angles at a point rather than the angle sum of a triangle.
- (c) 62° — Co-interior (allied) angles on parallel lines sum to 180°, so the co-interior angle is 180° − 118° = 62°. 118° comes from treating the angles as corresponding angles, which are equal, instead of co-interior angles, which sum to 180°. 59° comes from halving the given angle. 236° comes from doubling the given angle.
- (d) 63.6 — Method: the three angles inside the triangle formed by the two ladders and the ground add up to 180°. Working: 180 − 58.2 − 58.2 = 63.6. Answer: 63.6°. A candidate who assumes the top angle equals the base angles gives 58.2. A candidate who subtracts only one base angle from 180°, working out 180 − 58.2, gets 121.8. A candidate who doubles the base angle instead of subtracting it twice from 180°, working out 2 × 58.2, gets 116.4.
- (a) 30° — Method: the exterior angles of any convex polygon add up to 360°, and in a regular polygon they are all equal, so divide 360° by the number of sides. Working: 360 ÷ 12 = 30. Answer: 30°. The distractors: 150° is the interior angle, 180 − 30, which answers for the wrong angle at the vertex; 15° comes from dividing 180 by 12, using the angles on a straight line instead of the full turn; 36° comes from dividing 360 by 12 − 2 = 10, carrying the subtraction of 2 out of the interior angle sum formula into a calculation that does not need it.
- (a) 70° — Method: the three angles of a triangle add up to 180°, so take the known corner away from 180° to find what is left for the other two corners, then share that remainder equally because those two corners are equal. Working: 180 − 40 = 140, and the two equal corners share that 140° between them, so 140 ÷ 2 = 70. Answer: 70°. The distractors: 140° comes from stopping after 180 − 40 and offering the combined total of the two equal corners as though it were the size of one of them; 20° comes from halving the 40° corner that was given instead of halving the 140° that the other two corners have to share; 50° comes from 90 − 40, using the two acute angles of a right-angled triangle as the fixed total rather than the 180° angle sum of the whole triangle.
- (b) 13 — Using the sum of interior angles formula, (n − 2) × 180° = 1980°, so n − 2 = 1980 ÷ 180 = 11, and n = 11 + 2 = 13. 11 stops after the division, forgetting to add 2 back to find n. 15 adds 2 twice by mistake, giving 11 + 2 + 2. 22 divides 1980 by 90 instead of 180.
- (b) 62 — Method: co-interior (allied) angles between parallel lines add up to 180°. Working: 180 − 118 = 62. Answer: 62°. A candidate who treats co-interior angles as equal, like corresponding angles, gives 118. A candidate who uses 360° instead of 180°, working out 360 − 118, gets 242. A candidate who subtracts as if the angles were complementary, working out 118 − 90, gets 28.
- (d) 20 — Alternate angles between parallel lines are equal, so 3x + 10 = 5x − 30. Rearranging, 10 + 30 = 5x − 3x, so 40 = 2x, and x = 20. −10 comes from a sign error when rearranging, moving a term to the wrong side and getting −20 = 2x instead. 25 comes from wrongly treating the two angles as co-interior and adding them to 180°: (3x + 10) + (5x − 30) = 180 gives 8x − 20 = 180, so x = 25. 47.5 makes the same co-interior mistake but sets the sum equal to 360° instead of 180°, giving 8x − 20 = 360 and x = 47.5.
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