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.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?
- 2.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?
- 3.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?
- 4.In quadrilateral WXYZ, which of the following correctly names the angle at vertex Y using standard notation?
- 5.In triangle XYZ, the side opposite vertex X is labelled using the standard lowercase-letter convention. Which label is correct?
- 6.A textbook contains the instruction 'Draw AB' and, on the next line, the statement 'AB = 5 cm'. Which statement about the notation AB is correct?
- 7.A sign-maker is told a metal bracket outline must have two pairs of adjacent sides equal in length (not opposite pairs), exactly one pair of opposite angles equal, and exactly one line of symmetry. Which quadrilateral matches this description?
- 8.A triangular prism has two triangular faces and three rectangular faces. How many vertices does a triangular prism have?
- 9.A garden designer describes a flower bed as a shape with four straight sides: exactly one pair of sides is parallel, those two parallel sides are different lengths, and the other two sides are equal in length to each other but not parallel to each other. Which shape should the designer draw?
- 10.In geometry, which of the following best describes a 'plane'?
- 11.Which of the following is the correct definition of a regular polygon?
- 12.Which of these correctly compares a 'line' with a 'line segment'?
- 13.A triangle has vertices X, Y and Z, with a right angle at Y. Side XY is twice as long as side YZ. Which of the following correctly describes this triangle?
- 14.A rhombus has all four sides equal in length, but it is not a square — its angles are not all 90°. How many lines of symmetry does this rhombus have?
- 15.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?
Answer key
- (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.
- (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) ∠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.
- (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) AB means the segment or its length, shown by context — Method: recall the standard convention for a two-letter label such as AB. Working: AB names the segment AND its length; the sentence around it shows which is meant — 'draw AB' means the segment, 'AB = 5 cm' means the length. Options: 'a different symbol is needed' and 'must be written as |AB|' both describe stricter rules than the convention GCSE actually uses; 'AB can only mean the segment' ignores that the same label is also used for the length. Answer: AB means the segment or its length, shown by context.
- (a) Kite — Method: check each named quadrilateral's properties against the three facts given, one at a time. Working: a kite has two pairs of adjacent sides equal (not opposite pairs), one pair of opposite angles equal (the two angles where an unequal pair of sides meet), and exactly one line of symmetry — matching all three facts. Options: a rhombus does have equal adjacent sides, but all four of its sides are equal, both pairs of its opposite angles are equal, and it has two lines of symmetry rather than exactly one; a parallelogram has its opposite sides equal rather than adjacent pairs, both pairs of opposite angles equal, and no line of symmetry at all; a trapezium does not generally have any pair of equal adjacent sides or a line of symmetry. Answer: kite.
- (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.
- (a) isosceles trapezium — One pair of parallel sides, plus a separate pair of equal non-parallel sides, is exactly the definition of an isosceles trapezium — the shape the designer should draw. Parallelogram is wrong because a parallelogram needs BOTH pairs of opposite sides parallel, but only one pair is parallel here. Kite is wrong because a kite has two separate pairs of adjacent equal sides and no requirement for any sides to be parallel, a different combination of properties. Rhombus is wrong because a rhombus needs all four sides equal, but the description only makes two of the four sides equal to each other.
- (a) A flat surface extending infinitely in two directions — Method: recall the precise geometric meaning of 'plane', versus 'line', 'point' and 'face'. Working: a plane is a flat, two-dimensional surface extending infinitely in every direction within it. Options: 'a straight line extending in one direction' describes a line, not a plane; 'a single fixed position with no size' describes a point; 'a flat, bounded face on a 3D shape' describes a face, a bounded piece of a plane, not the plane itself, which has no boundary. Answer: a flat surface extending infinitely in two directions.
- (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.
- (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.
- (b) A right-angled triangle — The right angle at Y makes this a right-angled triangle, so that description is correct. Since XY is twice YZ, those two sides cannot be equal. The third side XZ is opposite the right angle, so it is the hypotenuse and is longer than either XY or YZ, so it cannot equal either of them. No two sides are equal, which rules out 'an isosceles triangle' and 'a right-angled isosceles triangle', both of which wrongly assume two equal sides. 'An equilateral triangle' would need all three sides equal, which contradicts XY being twice YZ, so it is wrong too.
- (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.
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