Mathematics questions, chapter 9

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Basic polygons

40 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Do not use a protractor: the diagrams are drawn to scale, but you must calculate the angles. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer.

Question 1

What is the name of a polygon with 8 sides?

Hexagon
Heptagon
Octagon
Nonagon

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An octagon has 8 sides (think of an octopus with 8 legs). A hexagon has 6, a heptagon 7 and a nonagon 9.

For the teacher: Remembering, low difficulty, standard 9.1.2

Textbook p. 204

Question 2

How many diagonals does a polygon with 9 sides have?

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Number of diagonals = 9(9 − 3) ÷ 2 = 54 ÷ 2 = 27. 54 forgets to divide by 2; 6 is only the number of diagonals from one vertex; 36 counts the sides as well.

For the teacher: Applying, low difficulty, standard 9.1.1

Textbook p. 203

Question 3

Diagram 1 shows a pentagon. Which is a correct name for the pentagon?

Pentagon with vertices P at the top, then Q, R, S, T going clockwise — Diagram 1

PQSRT
PRTQS
PQRTS
TSRQP

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The letters must go round the shape in order. TSRQP goes anticlockwise from T, so it is correct. The other names jump across the pentagon (for example Q to S).

For the teacher: Understanding, low difficulty, standard 9.1.2

Textbook p. 204

Question 4

Which triangle has exactly one axis of symmetry?

Isosceles triangle (two equal sides only)
Equilateral triangle
Scalene triangle
Obtuse-angled scalene triangle

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An isosceles triangle folds onto itself along the line from the apex to the middle of the base: 1 axis. An equilateral triangle has 3 axes, and a scalene triangle has none.

For the teacher: Remembering, low difficulty, standard 9.2.1

Textbook p. 207

Question 5

Diagram 2 shows triangle ABC. What type of triangle is ABC, based on its angles?

Triangle ABC with angle B = 25 degrees and angle C = 35 degrees — Diagram 2

Acute-angled triangle
Obtuse-angled triangle
Right-angled triangle
Equilateral triangle

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∠BAC = 180° − 25° − 35° = 120°, which is obtuse. Choosing acute-angled comes from looking only at the two angles given.

For the teacher: Understanding, medium difficulty, standard 9.2.1

Textbook p. 207

Question 6

In Diagram 3, BCD is a straight line. Find the value of x.

Triangle ABC with angle A = 68 degrees and angle B = 47 degrees; BC extended to D; x is angle ACD — Diagram 3

65°
21°
115°
133°

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The exterior angle equals the sum of the two opposite interior angles: x = 47° + 68° = 115°. 65° is the interior angle ∠ACB, and 133° is 180° − 47°.

For the teacher: Applying, low difficulty, standard 9.2.2

Textbook p. 209

Question 7

Diagram 4 shows an isosceles triangle PQR with PQ = PR. Find the value of x.

Isosceles triangle PQR with PQ = PR (tick marks), angle P = 40 degrees, x = angle Q — Diagram 4

70°
140°
40°
100°

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The base angles are equal: 2x + 40° = 180°, so 2x = 140° and x = 70°. 140° forgets to halve, and 100° wrongly takes a base angle as 40°.

For the teacher: Applying, low difficulty, standard 9.2.3

Textbook p. 210

Question 8

The diagonals of a quadrilateral are perpendicular bisectors of each other but are not equal in length. What is the quadrilateral?

Rectangle
Square
Kite
Rhombus

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A rhombus has diagonals that are perpendicular bisectors of each other. A square's diagonals are also equal; a rectangle's are not perpendicular; in a kite only one diagonal bisects the other.

For the teacher: Remembering, low difficulty, standard 9.3.1

Textbook p. 213

Question 9

Which statement is TRUE?

A trapezium has two pairs of parallel sides.
A square is also a rhombus.
A parallelogram has two axes of symmetry.
The diagonals of a rectangle are always perpendicular to each other.

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A square has all the properties of a rhombus (four equal sides, opposite sides parallel, opposite angles equal, perpendicular diagonals). A trapezium has only one pair of parallel sides, and a parallelogram has no axis of symmetry.

For the teacher: Understanding, medium difficulty, standard 9.3.1

Textbook p. 213

Question 10

In Diagram 5, CDE is a straight line. Find the value of x.

Quadrilateral ABCD with angle A = 95 degrees, angle B = 110 degrees, angle C = 70 degrees; CD extended to E; x = angle ADE — Diagram 5

85°
275°
95°
105°

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∠ADC = 360° − 95° − 110° − 70° = 85°, so x = 180° − 85° = 95°. 85° is the interior angle, not the exterior angle x.

For the teacher: Applying, medium difficulty, standard 9.3.2

Textbook p. 215

Question 11

Diagram 6 shows a rhombus PQRS with diagonal PR. Find ∠PQR.

Rhombus PQRS with all sides equal; diagonal PR; angle QPR = 28 degrees — Diagram 6

124°
152°
56°
62°

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PQ = QR, so ΔPQR is isosceles and ∠QRP = ∠QPR = 28°. ∠PQR = 180° − 28° − 28° = 124°. 152° uses only one 28°; 56° is ∠QPS.

For the teacher: Analysing, medium difficulty, standard 9.3.3

Textbook p. 216

Question 12

In Diagram 7, QRTU is a parallelogram and PQRS is a straight line. Find the value of y.

Parallelogram QRTU on the line PQRS with angle PQU = 115 degrees; triangle RST with angle TSR = 50 degrees; y = angle RTS — Diagram 7

50°
115°
15°
65°

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∠UQR = 180° − 115° = 65°. RT ∥ QU, so ∠TRS = ∠UQR = 65° (corresponding angles). y = 180° − 65° − 50° = 65°. 15° comes from using ∠QRT = 115° inside ΔRST.

For the teacher: Analysing, high difficulty, standard 9.3.4

Textbook p. 217

Section B: structured questions

28 marks

Answer all questions. Show your working.

Question 13

Answer these questions about polygons.

[1 mark] (a)A polygon with 11 sides has 11 vertices.

ATrueBFalse

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True

Every polygon has the same number of vertices as sides.

[2 marks] (b)Calculate the number of diagonals of a heptagon.

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A heptagon has 7 sides: 7(7 − 3) ÷ 2 [1]1
= 14 diagonals [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 203

Question 14

Diagram 8 shows three triangles A, B and C.

Triangle A has three equal sides. Triangle B has a right angle and two equal sides. Triangle C has angles 30 degrees and 45 degrees and three different sides — Diagram 8

[2 marks] (a)Name the type of triangle C based on (i) its sides, (ii) its angles.

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(i) Scalene triangle [1]1
(ii) Third angle = 180° − 30° − 45° = 105°, so obtuse-angled triangle [1]1

Tick each point you got.

[2 marks] (b)State the number of axes of symmetry of triangle A and of triangle B.

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Triangle A (equilateral): 3 [1]1
Triangle B (right-angled isosceles): 1 [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 9.2.1

Textbook p. 207

Question 15

In Diagram 9, BCD is a straight line.

Triangle ABC with angle A = x and angle B = 2x; BC extended to D; exterior angle ACD = 126 degrees — Diagram 9

[2 marks] (a)Find the value of x.

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x + 2x = 126° (exterior angle = sum of opposite interior angles), so 3x = 126° [1]1
x = 42° [1]1

Tick each point you got.

[1 mark] (b)Find ∠ACB.

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∠ACB = 180° − 126° = 54° [1]1

Check: 42° + 84° + 54° = 180°.

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 9.2.2

Textbook p. 209

Question 16

In Diagram 10, PQRS is a straight line and TQ = TS.

Points P, Q, R, S on a straight line; T above; TQ = TS with tick marks; angle TSR = 52 degrees; angle TRS = 96 degrees; x = angle PQT; y = angle QTR — Diagram 10

[2 marks] (a)Calculate the value of x.

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∠TQS = ∠TSQ = 52° (base angles of isosceles ΔQST) [1]1
x = 180° − 52° = 128° [1]1

Tick each point you got.

[2 marks] (b)Calculate the value of y.

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∠QRT = 180° − 96° = 84°, or ∠QTS = 180° − 52° − 52° = 76° and ∠RTS = 180° − 96° − 52° = 32° [1]1
y = 180° − 52° − 84° = 44°, or y = 76° − 32° = 44° [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 9.2.3

Textbook p. 210

Question 17

Name each quadrilateral from its properties.

[1 mark] (a)Only one pair of opposite sides is parallel.

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Trapezium [1]1

Tick each point you got.

[1 mark] (b)Two pairs of adjacent sides are equal (but not all four sides are equal), and only one diagonal is the perpendicular bisector of the other.

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Kite [1]1

Tick each point you got.

[1 mark] (c)Every rectangle is a square.

ATrueBFalse

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False

A rectangle needs only its opposite sides equal; a square needs all four sides equal. (But every square is a rectangle.)

For the teacher: Understanding, low difficulty, standard 9.3.1

Textbook p. 213

Question 18

In Diagram 11, PST is a straight line.

Quadrilateral PQRS with angle P = 88 degrees, angle Q = 104 degrees, x = angle R; PS extended to T; exterior angle RST = 70 degrees — Diagram 11

[1 mark] (a)Find ∠PSR.

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∠PSR = 180° − 70° = 110° [1]1

Tick each point you got.

[2 marks] (b)Hence, find the value of x.

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x + 88° + 104° + 110° = 360° [1]1
x = 58° [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 9.3.2

Textbook p. 215

Question 19

Diagram 12 shows a rhombus PQRS with diagonal QS. Diagram 13 shows Amir's working to find ∠QRS.

Rhombus PQRS with all sides equal; diagonal QS; angle PQS = 35 degrees — Diagram 12

Amir's working. Line 1: angle PSQ = 35 degrees (PQ = PS, base angles of an isosceles triangle). Line 2: angle QPS = 180 − 35 − 35 = 110 degrees. Line 3: angle QRS = 180 − 110 = 70 degrees — Diagram 12

[2 marks] (a)Amir's answer is wrong. In which line is his mistake? Explain the mistake.

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Line 3 [1]1
∠QRS is opposite ∠QPS, and opposite angles of a rhombus are equal; they do not add up to 180° [1]1

Tick each point you got.

[2 marks] (b)Find the correct value of ∠QRS and the value of ∠PQR.

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∠QRS = ∠QPS = 110° [1]1
∠PQR = 180° − 110° = 70° (or 2 × 35° = 70°) [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 9.3.3

Textbook p. 216

Question 20

Nadia is designing tiles shaped like quadrilaterals.

[2 marks] (a)The four interior angles of her first tile are in the ratio 3 : 4 : 5 : 6. Find the largest angle.

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One part = 360° ÷ (3 + 4 + 5 + 6) = 360° ÷ 18 = 20° [1]1
Largest angle = 6 × 20° = 120° [1]1

Tick each point you got.

[2 marks] (b)Her second tile is a kite whose largest angle is 120°. Suggest the sizes of all four of its interior angles and show that they are possible.

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A set with one pair of equal angles, largest angle 120°, e.g. 120°, 95°, 95°, 50° [1]1
Shows the sum is 360°: 120° + 95° + 95° + 50° = 360° [1]1

Other correct answers: 120°, 70°, 70°, 100° or 120°, 100°, 100°, 40°. The two equal angles must be opposite each other, and the other two angles must be different (otherwise the tile is a rhombus).

Tick each point you got.

For the teacher: Creating, high difficulty, standard 9.3.2

Textbook p. 221

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