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.
Answer key: Geometric terms, notation and diagrams worksheet — GCSE Foundation
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- (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.
- (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.
- (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.
- (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) 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) 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) 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) 1 — An isosceles trapezium has one line of symmetry, running through the midpoints of the two parallel sides. 0 would be true for a scalene trapezium, whose non-parallel sides are unequal, but this trapezium's non-parallel sides are equal, so it is symmetrical. 2 is the number of lines of symmetry of a rectangle, not a trapezium. 4 comes from wrongly applying the 'number of sides equals number of lines of symmetry' rule, which only holds for REGULAR polygons — a trapezium is not regular.
- (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) 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) 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.
- (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.
- (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.
- (b) Nonagon — Method: match the number of sides given to the correct polygon name. Working: a polygon with 9 sides is called a nonagon. An octagon has 8 sides, a decagon has 10 sides, and a heptagon has 7 sides, so each names a different number of sides from the 9 given. Answer: nonagon.
- (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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