Printable · GCSE Foundation · ages 14-16
Geometric terms, notation and diagrams worksheet — GCSE Foundation
Fifteen questions on "geometric terms, notation and diagrams" — DfE statement G1. 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: Geometric terms, notation and diagrams worksheet — GCSE Foundation
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- (a) 7.7 m — AB is parallel to DC, so those two sides are each parallel to another side. BC and AD are stated to be not parallel to each other, so neither one is parallel to any other side — these are the two sides that need edging. Adding these: 3.2 + 4.5 = 7.7 m, so 7.7 m is correct. 7.6 m comes from an arithmetic slip when adding 3.2 and 4.5. 15.4 m comes from doubling the correct total, mistakenly assuming edging strip is needed along both faces of each side. 4.5 m comes from using only the longer of the two non-parallel sides and forgetting to add the shorter one.
- (c) ∠XYZ — The angle at a named vertex is written with that vertex's letter in the middle, flanked by its two neighbouring vertices. The angle at Y sits between X and Z, its neighbours in quadrilateral WXYZ, so it is written ∠XYZ. ∠WXY names the angle at X, since X is the middle letter, not Y. ∠YZW names the angle at Z, since Z is the middle letter. ∠ZWX names the angle at W, since W is the middle letter.
- (a) 5 — For any regular polygon, the order of rotational symmetry is always equal to the number of sides, which is also equal to the number of lines of symmetry. Since this polygon has 5 lines of symmetry, it has 5 sides, so its order of rotational symmetry is 5. 4 comes from subtracting one from the number of sides by mistake. 6 comes from adding one to the number of sides by mistake. 10 comes from doubling the number of lines of symmetry instead of using it directly.
- (a) 2 — Method: recall that the diagonals of a rhombus are always lines of symmetry, whatever its angles are. Working: a rhombus (all sides equal) always has its two diagonals as lines of symmetry, giving 2 lines of symmetry, whether or not the angles are 90°. Options: 0 wrongly assumes a non-square rhombus has no symmetry at all; 4 comes from the number of lines of symmetry a square has, mistaking this rhombus for a square; 1 comes from treating the rhombus like a kite, which has only one diagonal as a line of symmetry. Answer: 2.
- (c) Rectangle — A rectangle has two pairs of parallel sides and four right angles, but does not require all sides to be equal — this matches exactly, so Rectangle is correct. A square also has four right angles and parallel sides, but additionally requires all four sides to be equal, which contradicts 'not all the same length', so it is wrong. A rhombus has two pairs of parallel sides and all four sides equal, but its angles are not generally 90° unless it is also a square, so it does not match the right-angle condition here. A kite has no pairs of parallel sides at all, so it does not match the first condition given.
- (c) x — By convention, the side opposite a vertex is labelled with the lowercase version of that vertex's letter, so the side opposite X is labelled x. y is the label for the side opposite Y, not X. z is the label for the side opposite Z, not X. X is the vertex's own uppercase letter — the convention specifically switches to lowercase for the side, so the uppercase letter on its own is not correct.
- (d) No — angles must be equal too — A regular polygon must have both all sides equal and all angles equal. This tile has all six sides equal, but its interior angles are not all equal, so it fails the angle condition and is not regular — 'No — angles must be equal too' is correct. 'Yes — all sides are equal' is wrong because equal sides alone are not enough; a shape can have equal sides but unequal angles, as here. 'Yes — six equal sides means regular' is wrong for the same reason: equal sides do not automatically guarantee equal angles. 'No — hexagons can't be regular' is wrong because regular hexagons certainly exist (six equal sides and six equal 120° angles); it is this particular tile that fails to be regular, not hexagons in general.
- (b) 6 — Method: count the corner points (vertices) of the shape directly. Working: a triangular prism has two triangular ends, each with 3 corners, and no other corners elsewhere on the shape, giving 3 + 3 = 6 vertices. Options: 5 comes from counting the faces of the prism (2 triangular + 3 rectangular = 5) instead of the vertices; 8 comes from confusing the prism with a cube, which has 8 vertices; 9 comes from counting the edges of the prism (3 on each triangular end, plus 3 connecting them, giving 9) instead of the vertices. Answer: 6.
- (c) Equal sides and equal interior angles — Method: recall the full definition of 'regular' as applied to a polygon. Working: a regular polygon must have both equal side lengths and equal interior angles at the same time. Options: 'all sides equal' alone describes an equilateral but not necessarily equiangular shape, such as a rhombus, which is not regular; 'all angles equal' alone describes an equiangular but not necessarily equilateral shape, such as a rectangle, which is not regular; 'at least one line of symmetry' is a much weaker condition that many irregular shapes also satisfy. Answer: equal sides and equal interior angles.
- (b) Its diagonals cross at right angles — In a rhombus, the diagonals always bisect each other at right angles, because a rhombus is a parallelogram with all four sides equal. Its diagonals are not always equal in length — that is a property of a rectangle, and only holds for a rhombus in the special case where it is also a square. It does not always have four right angles — again, that is only true when the rhombus is also a square. Its order of rotational symmetry is generally 2, not 4; order 4 only happens when the rhombus is a square.
- (c) Dodecagon — Deca- means 10, so Decagon names a 10-sided polygon, two sides short of this one, so it is wrong. Hendeca- means 11, one side short of 12, so Hendecagon is also wrong. Icosa- means 20, double the number of sides given here, so Icosagon is wrong too. Dodeca- means 12, so Dodecagon correctly names this polygon.
- (c) It is a parallelogram but not a rectangle — Method: match the given properties to the definition of each named quadrilateral. Working: having both pairs of opposite sides parallel and equal in length is exactly the definition of a parallelogram; since none of the angles are right angles, it cannot also be a rectangle, which needs four right angles. Options: 'must be a rectangle' wrongly assumes every parallelogram has right angles; 'must be a rhombus' wrongly assumes equal opposite sides means all four sides are equal, but only the opposite pairs are stated as equal here; 'trapezium' is wrong because a trapezium has exactly one pair of parallel sides, while this shape has two pairs, so it is a parallelogram and not a trapezium. Answer: it is a parallelogram but not a rectangle.
- (c) AB ∥ DC — The symbol ∥ means ‘is parallel to’, so AB is parallel to DC is written AB ∥ DC. AB = DC would mean the two sides are equal in length, which is not what the question states. AB ⊥ DC would mean the two sides are perpendicular to each other. AB ∥ BC does use the parallel symbol, but it claims AB is parallel to BC — two sides that meet at vertex B, so they cannot be parallel; the question states AB is parallel to DC.
- (c) AB ⊥ CD — Method: match the description of the lines to the correct symbol. Working: the symbol ⊥ means 'is perpendicular to', used when two lines meet at right angles, which is exactly what is described. Options: ∥ means 'is parallel to', for lines that never meet, so it does not apply here; = compares two lengths as equal, but no lengths are mentioned; ≅ means 'is congruent to', used for whole shapes, not for describing how two lines cross. Answer: AB ⊥ CD.
- (c) octagon — A heptagon has 7 sides. One more than 7 is 8, and the polygon with 8 sides is an octagon. hexagon (6 sides) comes from subtracting one instead of adding one. heptagon (7 sides) is just the number of sides already given in the question, not one more than that. nonagon (9 sides) comes from adding two instead of one.
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