A straight line PQ passes through point R. A ray RS is drawn from R, splitting angle PRQ into two angles: angle PRS = 96 degrees and angle SRQ = x degrees. Calculate the value of x.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
At a point O, two angles are formed that together make a complete turn. One angle is 254 degrees (marked with a dashed reflex arc) and the other is m degrees. Calculate the value of m.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
Which of these is NOT a property of a square?
A) It has four right angles
B) It has four equal sides
C) It has exactly one pair of parallel sides
D) Its diagonals bisect each other
(Total for Question 4 is 1 mark)
5
For each pair of angles, state whether they are complementary, supplementary, or neither.
(a)38 degrees and 52 degrees.(1)
(b)112 degrees and 68 degrees.(1)
(c)35 degrees and 45 degrees.(1)
(Total for Question 5 is 3 marks)
6
Two angles lie on a straight line, in the ratio 2 : 3. Calculate the size of each angle.
Diagram NOT accurately drawn
(Total for Question 6 is 3 marks)
7
Triangle STU is isosceles with ST = SU. Each base angle, angle STU and angle SUT, is 71 degrees. Calculate the size of the apex angle, angle TSU.
Diagram NOT accurately drawn
(Total for Question 7 is 3 marks)
8
Quadrilateral JKLM has angle J = 84 degrees, angle K = 97 degrees, angle L = 112 degrees and angle M = x degrees. Calculate the value of x.
Diagram NOT accurately drawn
(Total for Question 8 is 2 marks)
9
A parallelogram has two pairs of parallel sides. For each statement, write True or False.
(a)Opposite sides of a parallelogram are always equal in length.(1)
(b)All four angles of a parallelogram are always right angles.(1)
(c)The diagonals of a parallelogram are always equal in length.(1)
(Total for Question 9 is 3 marks)
10
In triangle FGH, angle F = 56 degrees and angle G = 71 degrees. Side GH is extended to a point I, so that angle FHI is the exterior angle of the triangle at H.
Diagram NOT accurately drawn
(a)Calculate the size of angle FHG, the interior angle at H.(2)
(b)Hence calculate the size of the exterior angle FHI.(2)
(Total for Question 10 is 4 marks)
11
Two parallel lines are cut by a transversal. This creates a co-interior (allied) pair of angles: (2x + 5) degrees and (3x - 15) degrees, as shown in the diagram.
Diagram NOT accurately drawn
(a)Form an equation using the fact that co-interior angles sum to 180 degrees, and solve it to find the value of x.(2)
(b)Hence calculate the size of each of the two angles.(2)
(Total for Question 11 is 4 marks)
12
A regular dodecagon has twelve equal sides and twelve equal angles.
Diagram NOT accurately drawn
(a)Calculate the size of one exterior angle of the dodecagon.(2)
(b)Hence calculate the size of one interior angle of the dodecagon.(2)
(Total for Question 12 is 4 marks)
13
Calculate the sum of the interior angles of a nonagon (a 9-sided polygon).
(Total for Question 13 is 2 marks)
14
State the type of triangle described by each set of angles.
(a)42 degrees, 68 degrees and 70 degrees.(1)
(b)90 degrees, 63 degrees and 27 degrees.(1)
(c)55 degrees, 55 degrees and 70 degrees.(1)
(Total for Question 14 is 3 marks)
15
The bearing of a reservoir R from a campsite C is 205 degrees. Calculate the bearing of the campsite C from the reservoir R (the back bearing).
Diagram NOT accurately drawn
(Total for Question 15 is 2 marks)
16
Complete the table by stating the number of lines of symmetry and the order of rotational symmetry for each shape.
(a)A regular hexagon: state the number of lines of symmetry and the order of rotational symmetry.(2)
(b)A kite that is not a rhombus: state the number of lines of symmetry and the order of rotational symmetry.(2)
(c)A rhombus that is not a square: state the number of lines of symmetry and the order of rotational symmetry.(2)
(Total for Question 16 is 6 marks)
17
Euler's formula for solids states that F + V - E = 2, where F is the number of faces, V is the number of vertices and E is the number of edges.
(a)A cuboid has 6 faces, 8 vertices and 12 edges. Show that this is consistent with Euler's formula.(2)
(b)A different solid has 7 faces and 15 edges. Calculate the number of vertices it must have.(2)
(Total for Question 17 is 4 marks)
18
The diagram shows a net made from four identical equilateral triangles, arranged with one central triangle and a further triangle attached to each of its three sides. Which 3D shape does this net fold up to make?
Diagram NOT accurately drawn
A) Square-based pyramid
B) Triangular prism
C) Tetrahedron (triangular-based pyramid)
D) Octahedron
(Total for Question 18 is 1 mark)
19
A cylinder has one curved surface and two flat circular faces (count the curved surface as one face in total).
Diagram NOT accurately drawn
(a)State the total number of faces of a cylinder.(1)
(b)State the number of vertices of a cylinder.(1)
(c)State the number of edges of a cylinder.(1)
(Total for Question 19 is 3 marks)
20
A 3D shape has one flat circular face, one curved face, one vertex and one edge. What is this shape?
A) Cylinder
B) Cone
C) Sphere
D) Square-based pyramid
(Total for Question 20 is 1 mark)
21
Points A and B have coordinates A(2, 7) and B(8, -3).
(a)Calculate the coordinates of the midpoint of the line segment AB.(2)
(b)Point C has coordinates (-3, 4). It is translated 5 units right and 7 units down. Write down the coordinates of the image of C after this translation.(2)
(Total for Question 21 is 4 marks)
22
The points (1, 1), (5, 1), (5, 4) and (1, 4) are joined in order to form a quadrilateral. State the mathematical name of the quadrilateral formed, giving a reason.
(Total for Question 22 is 2 marks)
23
Two regular octagons and one square are placed together so that they all meet exactly at a single point, with no overlap and no gaps.
Diagram NOT accurately drawn
(a)Calculate the size of one interior angle of a regular octagon.(2)
(b)Show that two regular octagons and one square fit together exactly around the point, with no gap remaining.(2)
(Total for Question 23 is 4 marks)
24
A quadrilateral has four angles: 2x degrees, 3x degrees, 4x degrees and (x + 30) degrees.
Diagram NOT accurately drawn
(a)Form an equation using the fact that the angles of a quadrilateral sum to 360 degrees, and solve it to find the value of x.(3)
(b)Hence calculate the size of all four angles.(2)
(Total for Question 24 is 5 marks)
25
In triangle ABC, angle A = 2y degrees, angle B = 3y degrees and angle C = (5y - 10) degrees. Side BC is extended to a point D, so that angle ACD is the exterior angle of the triangle at C.
Diagram NOT accurately drawn
(a)Form an equation using the fact that the angles in a triangle sum to 180 degrees, and solve it to find the value of y.(3)
(b)Hence calculate the size of the exterior angle ACD.(2)
(Total for Question 25 is 5 marks)
Mark scheme · EP.M54 Angles and Shape: Practice Set 3
Question 1
(a) B1 acute cao
(a) Answer: acute
(b) B1 right angle cao
(b) Answer: right angle
(c) B1 obtuse cao
(c) Answer: obtuse
(d) B1 straight (line) cao
(d) Answer: straight
(e) B1 reflex cao
(e) Answer: reflex
Question 2
M1 180 - 96 seen (angles on a straight line sum to 180 degrees)
A1 84 degrees cao
Answer: x = 84 degrees
Question 3
M1 360 - 254 seen (angles around a point sum to 360 degrees)
A1 106 degrees cao
Answer: m = 106 degrees
Question 4
B1 C cao
Answer: C) It has exactly one pair of parallel sides
Question 5
(a) B1 complementary cao
(a) Answer: complementary
(b) B1 supplementary cao
(b) Answer: supplementary
(c) B1 neither cao
(c) Answer: neither
Question 6
M1 180 / 5 = 36 seen (dividing into 2 + 3 = 5 equal parts)
A1 72 degrees cao
A1 108 degrees cao
Answer: 72 degrees and 108 degrees
Question 7
M1 71 + 71 = 142 seen
M1 180 - 142 seen
A1 38 degrees cao
Answer: angle TSU = 38 degrees
Question 8
M1 360 - 84 - 97 - 112 seen (angles in a quadrilateral sum to 360 degrees)
A1 67 degrees cao
Answer: x = 67 degrees
Question 9
(a) B1 True cao
(a) Answer: True
(b) B1 False cao
(b) Answer: False
(c) B1 False cao
(c) Answer: False
Question 10
(a) M1 180 - 56 - 71 seen
(a) A1 53 degrees cao
(a) Answer: angle FHG = 53 degrees
(b) M1 180 - their (a) seen, or 56 + 71 (exterior angle = sum of the two opposite interior angles)
(b) B1 right-angled scalene cao (accept right-angled)
(b) Answer: right-angled scalene
(c) B1 isosceles cao
(c) Answer: isosceles
Question 15
M1 205 - 180 seen
A1 025 degrees cao (accept 25 degrees)
Answer: 025 degrees
Question 16
(a) B1 6 lines of symmetry cao
(a) B1 order of rotational symmetry 6 cao
(a) Answer: 6 lines of symmetry, order 6 rotational symmetry
(b) B1 1 line of symmetry cao
(b) B1 order of rotational symmetry 1 cao
(b) Answer: 1 line of symmetry, order 1 rotational symmetry
(c) B1 2 lines of symmetry cao
(c) B1 order of rotational symmetry 2 cao
(c) Answer: 2 lines of symmetry, order 2 rotational symmetry
Question 17
(a) M1 6 + 8 - 12 seen
(a) A1 = 2, consistent with Euler's formula
(a) Answer: 6 + 8 - 12 = 2, consistent
(b) M1 V = 2 - F + E seen, i.e. 2 - 7 + 15
(b) A1 10 cao
(b) Answer: 10 vertices
Question 18
B1 C cao
Answer: C) Tetrahedron (triangular-based pyramid)
Question 19
(a) B1 3 cao
(a) Answer: 3
(b) B1 0 cao
(b) Answer: 0
(c) B1 2 cao
(c) Answer: 2
Question 20
B1 B cao
Answer: B) Cone
Question 21
(a) M1 ((2 + 8) / 2, (7 + (-3)) / 2) seen
(a) A1 (5, 2) cao
(a) Answer: (5, 2)
(b) M1 (-3 + 5, 4 - 7) seen
(b) A1 (2, -3) cao
(b) Answer: (2, -3)
Question 22
B1 rectangle cao
B1 valid reason, e.g. opposite sides are equal and parallel (4 units and 3 units) and all angles are 90 degrees, but adjacent sides are not equal so it is not a square
Answer: rectangle
Question 23
(a) M1 360 / 8 = 45 seen (exterior angle), then 180 - 45
(a) A1 135 degrees cao
(a) Answer: 135 degrees
(b) M1 135 + 135 + 90 seen
(b) A1 = 360, so they fit exactly with no gap
(b) Answer: 135 + 135 + 90 = 360, confirmed
Question 24
(a) M1 2x + 3x + 4x + x + 30 = 360 seen
(a) M1 10x + 30 = 360 seen
(a) A1 x = 33 cao
(a) Answer: x = 33
(b) B1 66, 99 and 132 degrees cao (ft) (for 2x, 3x and 4x)