Printable · GCSE Foundation · ages 14-16
Properties of triangles and quadrilaterals worksheet — GCSE Foundation
Fifteen questions on "properties of triangles and quadrilaterals" — DfE statement G4. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Properties of triangles and quadrilaterals worksheet — GCSE Foundation
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- 1.In a kite, one pair of opposite angles are equal in size. In kite WXYZ, angle X = 40° and angle Z = 100° are the two angles that are NOT equal to each other. The other two angles, W and Y, are equal to each other. Work out the size of angle W.
- 2.The cross-section of a tent is an isosceles triangle. The angle at the top, between the two equal sloping sides, is 62°. The other two angles are equal to each other. Work out the size of each of those two angles.
- 3.A quadrilateral has exactly two lines of symmetry, its two pairs of opposite angles are equal, and its diagonals are not equal in length. Write down the name of this quadrilateral.
- 4.A triangular bunting flag has two equal sides. The angle between those two equal sides is 80°, and the other two angles of the flag are equal to each other. Work out the size of each of the two equal angles.
- 5.Write down how many right angles a right-angled triangle has.
- 6.In parallelogram ABCD the vertices are labelled in order round the shape, so angle A and angle B are at the two ends of the same side. Angle A is 70°. Work out the size of angle B.
- 7.A garden bed is designed as a trapezium. The two parallel sides are 8 m and 12 m, and the perpendicular distance between them is 5 m. Grass seed covers 4 m² per bag, sold only in whole bags. Work out how many bags of grass seed are needed.
- 8.Write down the mathematical name for a quadrilateral that has exactly one pair of parallel sides.
- 9.Write down the name given to a quadrilateral with exactly one pair of parallel sides.
- 10.Write down the name given to a quadrilateral that has exactly one pair of parallel sides.
- 11.Quadrilateral PQRS has angle P = 92°, angle Q = 84° and angle R = 106°. Work out the size of angle S.
- 12.The three angles of a triangle are 10°, 58° and 112°. Write down the name that describes this triangle.
- 13.A stained-glass window panel is designed as a kite. Two adjacent sides are each 25 cm, and the other two adjacent sides are each 40 cm. A frame is fitted around the whole panel. The frame costs £2.50 per metre, sold only in whole metres. Work out the total cost of the frame.
- 14.In an isosceles triangle each of the two base angles is 46°. Work out the size of the angle at the apex.
- 15.Write down how many right angles a rectangle has.
Answer key
- (a) 110° — The angles in a quadrilateral add up to 360°. So 40° + 100° + W + W = 360°, giving 2W = 360° − 140° = 220°, so W = 110°. A pupil who works out 2W = 220° but forgets to divide by 2, since there are two equal angles W, gives 220°. A pupil who mistakenly uses the angle sum of a triangle, 180°, instead of 360°, gets 180° − 140° = 40°. A pupil who simply adds the two given angles together instead of subtracting from 360° gets 40° + 100° = 140°. The correct answer is 110°.
- (a) 59° — Method: the three angles of a triangle add up to 180°, and in an isosceles triangle the two angles opposite the equal sides are equal, so take the known angle away from 180° and share what is left equally between the other two. Working: 180° − 62° = 118°, and 118° ÷ 2 = 59°. Answer: 59°. The distractors: 118° comes from taking 62° from 180° and stopping there, which gives the two angles together rather than one of them; 31° comes from halving the 62° that is given instead of halving what is left; 62° comes from assuming that all three angles of the triangle are equal to the one that is given.
- (b) Rhombus — A rhombus has exactly two lines of symmetry, formed by its two diagonals, and both pairs of opposite angles are equal, but its diagonals are unequal in length. A square also has opposite angles equal, but it has four lines of symmetry and its diagonals ARE equal, so it does not fit. A kite normally has only one line of symmetry and only one pair of opposite angles equal, so it does not fit. A general parallelogram has no lines of symmetry at all, so it does not fit. The correct answer is rhombus.
- (b) 50° — Method: the three angles of a triangle add up to 180°, and the two angles opposite the equal sides are equal, so subtract the given angle from 180° and halve the remainder. Working: 180° − 80° = 100°, and 100° ÷ 2 = 50°. Answer: 50°. The distractors: 100° comes from subtracting from 180° and forgetting to halve, so it is the two equal angles together; 40° comes from halving the 80° that is given rather than halving what is left of the 180°; 80° comes from assuming that the two base angles must match the angle between the equal sides.
- (c) 1 — Method: a right angle measures 90°, the three angles of any triangle add up to 180°, and a right-angled triangle is defined as a triangle that contains a right angle. Working: taking one right angle out of the total leaves 180° − 90° = 90° to be shared between the other two angles, so both of those must be acute; a second right angle would use the whole of that remaining 90° and leave nothing at all for the third angle, which is impossible. The definition therefore fixes the count at exactly one. Answer: 1. The distractors: 2 comes from counting the two sides that form the right angle instead of counting the angles themselves; 3 comes from reading the name as a description of the whole triangle, so that all three of its angles are taken to be right angles, which would need an angle sum of 3 × 90° = 270°; 0 comes from over-applying the angle sum — a candidate who works out that 90° + 90° = 180° leaves nothing for a third angle can conclude from that alone that no triangle may contain a right angle at all.
- (a) 110° — Method: in a parallelogram the two angles at the ends of one side are co-interior angles between a pair of parallel sides, so they add up to 180°. Working: angle A + angle B = 180°, so angle B = 180° − 70° = 110°. Answer: 110°. The distractors: 70° comes from applying the rule for opposite angles of a parallelogram, which are equal, to two angles that are next to each other instead; 20° comes from treating the two angles as complementary and working out 90° − 70°; 290° comes from using the 360° angle sum of a quadrilateral and taking away only the one angle that is given, 360° − 70°.
- (c) 13 — Area of the trapezium = 1/2 × (8 + 12) × 5 = 1/2 × 100 = 50 m². Number of bags = 50 ÷ 4 = 12.5, which rounds up to 13 bags since seed is sold only in whole bags. A student who mistakenly uses 2 m² of coverage per bag instead of 4 m² finds 50 ÷ 2 = 25 bags.
- (b) Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides. A parallelogram has two pairs of parallel sides, not one, so it does not fit the definition. A kite has no parallel sides at all. A rhombus, like a parallelogram, has two pairs of parallel sides. The correct name is trapezium.
- (a) Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides. A parallelogram has two pairs, and a kite and rhombus are defined by side lengths, not by having only one pair of parallel sides.
- (c) Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides.
- (c) 78° — The angles in any quadrilateral add up to 360°. Add the three given angles: 92° + 84° + 106° = 282°. Angle S = 360° − 282° = 78°. A pupil who only adds angle P and angle Q, forgetting angle R, gets 360° − (92° + 84°) = 184°. A pupil who only adds angle Q and angle R, forgetting angle P, gets 360° − (84° + 106°) = 170°. A pupil who makes a carrying slip adding the three angles, getting 292° instead of 282°, gets 360° − 292° = 68°. The correct answer is 78°.
- (c) Scalene — Method: equal angles in a triangle sit opposite equal sides, so compare the three angles with each other. Working: 10°, 58° and 112° are all different, so no two sides of the triangle are equal either, and a triangle with no equal sides is scalene. Answer: scalene. The distractors: isosceles is chosen by candidates who pair up the two acute angles, 10° and 58°, as base angles without checking that they are actually equal; equilateral is chosen by candidates who check that the three angles add to 180° and take that as meaning the triangle is regular; right-angled is chosen by candidates who see that 112° is larger than 90° and classify the triangle as containing a right angle, when in fact none of the three angles is 90°.
- (b) £5.00 — Perimeter of the kite = 25 + 25 + 40 + 40 = 130 cm = 1.3 m. The frame is sold only in whole metres, so 2 m must be bought. Cost = 2 × £2.50 = £5.00. A student who buys the exact 1.3 m instead of rounding up to whole metres gets 1.3 × £2.50 = £3.25. A student who never converts the perimeter from centimetres to metres and costs 130 × £2.50 gets £325.00.9 m, which rounds up to 3 whole metres, costing 3 × £2.50 = £7.50.
- (d) 88° — Method: the three angles add up to 180°, and here it is the two equal base angles that are known, so take both of them away from 180°. Working: the two base angles come to 2 × 46° = 92°, and 180° − 92° = 88°. Answer: 88°. The distractors: 134° comes from subtracting only one base angle, 180° − 46°, and forgetting that there are two of them; 92° comes from doubling the base angle and stopping there, which is the two base angles together rather than the apex; 46° comes from assuming that all three angles of the triangle are equal to the one that is given.
- (d) 4 — Method: use the definition of a rectangle — a quadrilateral in which every interior angle is a right angle. Working: a quadrilateral has four interior angles, and in a rectangle all four of them measure 90°, which is consistent with the 360° angle sum because 4 × 90° = 360°. Answer: 4. The distractors: 2 comes from applying the rule that opposite angles are equal and concluding that only one pair of corners is square; 1 comes from thinking that a single square corner is enough to make a shape a rectangle; 0 comes from confusing a rectangle with a general parallelogram, which has no right angles unless it is a rectangle.
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