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.
Geometric terms, notation and diagrams worksheet — GCSE Foundation
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- 1.A tile manufacturer cuts a hexagonal tile so that all six sides measure exactly 4 cm, but a check shows only two pairs of the six interior angles are equal to each other, not all six angles. The customer's order specifies regular hexagonal tiles. Based on the check, is this tile a regular hexagon?
- 2.In pentagon PQRST, which of the following correctly names the interior angle at vertex R, using standard three-letter angle notation?
- 3.A regular polygon has 5 lines of symmetry. What is the order of rotational symmetry of this polygon?
- 4.What does the statement AB = CD mean, where A, B, C and D are all points?
- 5.A quadrilateral has all four sides equal in length and all four interior angles equal to 90°. Its diagonals are equal in length and bisect each other at right angles. Which quadrilateral matches this description exactly?
- 6.A polygon has one more side than a heptagon. What is the polygon called?
- 7.In quadrilateral WXYZ, which of the following correctly names the angle at vertex Y using standard notation?
- 8.A patio plan states: 'Side AB of the patio is parallel to side DC, and AB is longer than DC. Sides BC and AD are not parallel to each other and are not equal in length.' The four sides measure AB = 5.1 m, BC = 3.2 m, DC = 2.8 m and AD = 4.5 m. The tiler needs to order edging strip only for the sides that are not parallel to any other side. What total length of edging strip, in metres, should the tiler order?
- 9.Which of these correctly compares a 'line' with a 'line segment'?
- 10.In triangle XYZ, the side opposite vertex X is labelled using the standard lowercase-letter convention. Which label is correct?
- 11.Point M lies exactly halfway between point A and point B, with A, M and B all on the same straight line. Which term correctly describes point M?
- 12.Lines AB and CD cross at a point, forming four right angles where they meet. Which notation correctly describes the relationship between AB and CD?
- 13.A quadrilateral has two pairs of parallel sides. All four of its interior angles are right angles, but its sides are not all the same length. Which quadrilateral is this?
- 14.A surveyor is marking out field ABCD for a new crop. The plan states that side AB is parallel to side DC, but AB and DC are not equal in length, and sides BC and AD are not parallel to each other. Which term correctly describes the shape of field ABCD?
- 15.A written description of a diagram states: 'AB is marked with a single arrow, CD is marked with a single arrow, and EF is marked with a double arrow.' Using the standard convention for marking parallel lines, what can be concluded?
Answer key
- (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.
- (a) ∠QRS — Method: the middle letter in three-letter angle notation is always the vertex of the angle, and the outer two letters are the neighbouring vertices along the shape's sides. Working: at vertex R, the two adjacent vertices along the pentagon are Q and S, so the interior angle is written ∠QRS, with R in the middle. Options: ∠PQR names the angle at Q, not R, since Q is the middle letter there; ∠RST puts R first rather than in the middle, so it actually names the angle at S; ∠TRP does have R in the middle, but T and P are not the vertices adjacent to R along the pentagon's sides, so it does not describe R's interior angle. Answer: ∠QRS.
- (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.
- (c) AB and CD are equal in length — AB = CD states that the line segments AB and CD are equal in length; it says nothing about their direction or position. 'AB is parallel to CD' would be written AB ∥ CD, not AB = CD. 'A, B, C and D all lie on one line' is not what an equals sign between two segment names states at all. 'AB is perpendicular to CD' would be written AB ⊥ CD, not AB = CD.
- (b) Square — A square has all four sides equal, all four angles equal to 90°, and diagonals that are equal in length and bisect each other at right angles — every part of the description matches, so Square is correct. A rhombus has all four sides equal and diagonals bisecting at right angles, but its interior angles are not generally 90° (only a square, a special rhombus, has that), so it does not fully match. A rectangle has four 90° angles and equal diagonals, but its sides are not all equal in general, so it fails the equal-sides condition. A kite has two pairs of adjacent equal sides rather than all four sides equal, and its diagonals are not generally equal in length, so it fails both conditions.
- (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.
- (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) 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) A line has no endpoints; a segment has two — A line extends without end in both directions, whereas a line segment is the part of a line between two specific fixed endpoints, so 'a line has no endpoints; a segment has two' is correct. 'A line has two endpoints; a segment has none' reverses these two definitions, so it is wrong. 'A line segment is always curved' is wrong because a line segment is straight, not curved, and does not extend infinitely. 'A line segment is a closed shape' is wrong because a line segment is a straight length between two points, not a polygon.
- (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.
- (c) The midpoint of AB — M is a single point lying exactly halfway along the straight line AB, so 'the midpoint of AB' correctly describes it. 'The vertex of AB' is wrong because a vertex is a corner point where two edges or lines meet, which does not apply to a point on a single straight line. 'A perpendicular of AB' is wrong because perpendicular describes two lines meeting at right angles, not a single point. 'A plane of AB' is wrong because a plane is a flat two-dimensional surface, not a point.
- (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) 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.
- (b) trapezium — A quadrilateral with exactly one pair of parallel sides is a trapezium; field ABCD has AB parallel to DC and no other pair of parallel sides, so trapezium is correct. Parallelogram requires BOTH pairs of opposite sides to be parallel, but the plan says only AB and DC are parallel. Rhombus requires all four sides to be equal in length, which is not stated here. A kite is defined by two pairs of adjacent equal sides, not by having a pair of parallel sides, so it does not match this description either.
- (c) AB ∥ CD only; EF not confirmed — By convention, lines marked with the same number of arrows are parallel to each other, but lines marked with a different number of arrows belong to a different, unrelated family of parallel lines. AB and CD both have a single arrow, so AB is parallel to CD. EF has a double arrow, showing it is not part of the same family as AB and CD; it may be parallel to some other line marked with a double arrow, but nothing here confirms it is parallel to AB or CD, so 'AB ∥ CD only; EF not confirmed' is correct. 'AB, CD and EF are all parallel' and 'EF is parallel to AB' both wrongly treat every arrow mark as showing the same relationship. 'None of the lines are parallel' wrongly assumes a different arrow count rules out any parallel relationship at all, when it actually just signals a different pairing.
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