State whether each angle below is acute, right angle, obtuse, straight or reflex.
(a)An angle of 35 degrees.(1)
A) acute
B) obtuse
C) reflex
D) right angle
(b)An angle of 92 degrees.(1)
A) acute
B) obtuse
C) reflex
D) straight
(c)An angle of 180 degrees.(1)
A) acute
B) right angle
C) straight
D) reflex
(d)An angle of 300 degrees.(1)
A) acute
B) obtuse
C) reflex
D) straight
(Total for Question 1 is 4 marks)
2
A straight line is divided into two angles by a single ray. One angle is 118 degrees and the other is x degrees. Calculate the value of x.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
Three angles meet at a point: 145 degrees, 95 degrees and y degrees. Calculate the value of y.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
Two straight lines cross at a point, forming four angles. One of the angles is 72 degrees.
Diagram NOT accurately drawn
(a)State the size of the angle vertically opposite the 72 degree angle.(1)
(b)Calculate the size of angle x, which is adjacent to the 72 degree angle on the same straight line.(2)
(Total for Question 4 is 3 marks)
5
Triangle PQR is isosceles, with PQ = PR. The angle at P (the apex) is 38 degrees. Calculate the size of each base angle, angle PQR and angle PRQ.
Diagram NOT accurately drawn
(Total for Question 5 is 3 marks)
6
Quadrilateral ABCD has angle A = 92 degrees, angle B = 88 degrees, angle C = 105 degrees and angle D = x degrees. Calculate the value of x.
Diagram NOT accurately drawn
(Total for Question 6 is 2 marks)
7
Complete each statement about polygons.
(a)A polygon with 5 sides is called a _______.(1)
(b)A polygon with 7 sides is called a _______.(1)
(c)A hexagon has _______ sides.(1)
(d)A polygon with 9 sides is called a _______.(1)
(e)A decagon has _______ sides.(1)
(Total for Question 7 is 5 marks)
8
Identify the quadrilateral described in each statement.
(a)A quadrilateral with four equal sides but no right angles.(1)
A) square
B) rhombus
C) kite
D) trapezium
(b)A quadrilateral with exactly one pair of parallel sides.(1)
A) parallelogram
B) rhombus
C) trapezium
D) kite
(c)A quadrilateral with two pairs of adjacent equal sides and exactly one pair of equal opposite angles.(1)
A) kite
B) rhombus
C) square
D) parallelogram
(Total for Question 8 is 3 marks)
9
In triangle ABC, angle A = 48 degrees and angle B = 67 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)Calculate the size of angle ACB, the interior angle at C.(2)
(b)Hence calculate the size of the exterior angle ACD.(2)
(Total for Question 9 is 4 marks)
10
Two parallel lines PQ and RS are cut by a transversal. One of the angles formed, at the intersection with PQ, is 73 degrees.
Diagram NOT accurately drawn
(a)State the size of the alternate angle to the 73 degree angle at line RS, giving a reason.(1)
(b)Calculate the size of the co-interior (allied) angle to the 73 degree angle at line RS, giving a reason.(2)
(Total for Question 10 is 3 marks)
11
A regular hexagon has six equal sides and six equal angles.
Diagram NOT accurately drawn
(a)Calculate the size of one exterior angle of the hexagon.(2)
(b)Hence calculate the size of one interior angle of the hexagon.(2)
(Total for Question 11 is 4 marks)
12
Calculate the sum of the interior angles of a heptagon (a 7-sided polygon).
(Total for Question 12 is 2 marks)
13
Identify the type of triangle described by each set of angles.
(a)A triangle with angles 60 degrees, 60 degrees and 60 degrees.(1)
A) scalene
B) isosceles
C) equilateral
D) right-angled
(b)A triangle with angles 90 degrees, 45 degrees and 45 degrees.(1)
A) equilateral
B) right-angled isosceles
C) scalene
D) obtuse-angled
(c)A triangle with angles 40 degrees, 65 degrees and 75 degrees.(1)
A) isosceles
B) equilateral
C) scalene
D) right-angled
(Total for Question 13 is 3 marks)
14
The bearing of a church C from a school S is 072 degrees. Calculate the bearing of the school S from the church C (the back bearing).
Diagram NOT accurately drawn
(Total for Question 14 is 2 marks)
15
Complete the table by stating the number of lines of symmetry and the order of rotational symmetry for each shape.
(a)A square: state the number of lines of symmetry and the order of rotational symmetry.(2)
(b)An equilateral triangle: state the number of lines of symmetry and the order of rotational symmetry.(2)
(c)A regular pentagon: state the number of lines of symmetry and the order of rotational symmetry.(2)
(Total for Question 15 is 6 marks)
16
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 triangular prism has 5 faces, 9 edges and 6 vertices. Show that this is consistent with Euler's formula.(2)
(b)A different solid (for example, a hexagonal prism) has 8 faces and 12 vertices. Calculate the number of edges it must have.(2)
(Total for Question 16 is 4 marks)
17
The three angles of a triangle are x degrees, 2x degrees and 3x degrees. Calculate the value of x and hence find the size of each angle.
Diagram NOT accurately drawn
(Total for Question 17 is 3 marks)
18
Triangle ABC is isosceles with AB = AC. Angle BAC = 56 degrees. Side BC is extended to a point D. Calculate the size of angle ACD, the exterior angle at C.
Diagram NOT accurately drawn
(Total for Question 18 is 3 marks)
19
Two regular hexagons and an equilateral triangle are placed together so that they all meet at a single point, with no overlap. Calculate the size of the gap that remains at that point.
Diagram NOT accurately drawn
(Total for Question 19 is 3 marks)
20
A quadrilateral has four angles: 90 degrees, 90 degrees, 2x degrees and (3x - 10) 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 the two unknown angles, 2x degrees and (3x - 10) degrees.(2)
(Total for Question 20 is 5 marks)
Mark scheme · EP.M12 Angles and Shape
Question 1
(a) B1 acute cao
(a) Answer: acute
(b) B1 obtuse cao
(b) Answer: obtuse
(c) B1 straight (line) cao
(c) Answer: straight
(d) B1 reflex cao
(d) Answer: reflex
Question 2
M1 180 - 118 seen (angles on a straight line sum to 180 degrees)
A1 62 degrees cao
Answer: x = 62 degrees
Question 3
M1 360 - 145 - 95 seen (angles around a point sum to 360 degrees)
A1 120 degrees cao
Answer: y = 120 degrees
Question 4
(a) B1 72 degrees cao (vertically opposite angles are equal)
(a) Answer: 72 degrees
(b) M1 180 - 72 seen
(b) A1 108 degrees cao
(b) Answer: x = 108 degrees
Question 5
M1 180 - 38 seen
M1 142 / 2 seen (ft)
A1 71 degrees cao (for each base angle)
Answer: angle PQR = angle PRQ = 71 degrees
Question 6
M1 360 - 92 - 88 - 105 seen (angles in a quadrilateral sum to 360 degrees)
A1 75 degrees cao
Answer: x = 75 degrees
Question 7
(a) B1 pentagon cao
(a) Answer: pentagon
(b) B1 heptagon oe (accept septagon)
(b) Answer: heptagon
(c) B1 6 cao
(c) Answer: 6
(d) B1 nonagon cao
(d) Answer: nonagon
(e) B1 10 cao
(e) Answer: 10
Question 8
(a) B1 rhombus cao
(a) Answer: rhombus
(b) B1 trapezium cao
(b) Answer: trapezium
(c) B1 kite cao
(c) Answer: kite
Question 9
(a) M1 180 - 48 - 67 seen
(a) A1 65 degrees cao
(a) Answer: angle ACB = 65 degrees
(b) M1 180 - their (a) seen, or 48 + 67 (exterior angle = sum of two opposite interior angles)
(b) A1 115 degrees cao (ft)
(b) Answer: angle ACD = 115 degrees
Question 10
(a) B1 73 degrees oe with reason 'alternate angles are equal'
(a) Answer: 73 degrees (alternate angles are equal)
(b) M1 180 - 73 seen
(b) A1 107 degrees cao with reason 'co-interior angles sum to 180 degrees'
(b) Answer: 107 degrees (co-interior angles sum to 180 degrees)
Question 11
(a) M1 360 / 6 seen
(a) A1 60 degrees cao
(a) Answer: 60 degrees
(b) M1 180 - their (a) seen (ft)
(b) A1 120 degrees cao (ft)
(b) Answer: 120 degrees
Question 12
M1 (7 - 2) x 180 seen
A1 900 degrees cao
Answer: 900 degrees
Question 13
(a) B1 equilateral cao
(a) Answer: equilateral
(b) B1 right-angled isosceles cao
(b) Answer: right-angled isosceles
(c) B1 scalene cao
(c) Answer: scalene
Question 14
M1 72 + 180 seen
A1 252 degrees cao
Answer: 252 degrees
Question 15
(a) B1 4 lines of symmetry cao
(a) B1 order of rotational symmetry 4 cao
(a) Answer: 4 lines of symmetry, order 4 rotational symmetry
(b) B1 3 lines of symmetry cao
(b) B1 order of rotational symmetry 3 cao
(b) Answer: 3 lines of symmetry, order 3 rotational symmetry
(c) B1 5 lines of symmetry cao
(c) B1 order of rotational symmetry 5 cao
(c) Answer: 5 lines of symmetry, order 5 rotational symmetry
Question 16
(a) M1 5 + 6 - 9 substituted correctly
(a) A1 = 2 cso (matches Euler's formula)
(a) Answer: 2 (consistent with Euler's formula)
(b) M1 8 + 12 - E = 2 seen, rearranged to E = 8 + 12 - 2
(b) A1 18 cao
(b) Answer: 18 edges
Question 17
M1 x + 2x + 3x = 180 seen (oe)
A1 x = 30 cao
A1 angles 30 degrees, 60 degrees and 90 degrees all correct (ft)
Answer: x = 30; angles are 30 degrees, 60 degrees and 90 degrees
Question 18
M1 (180 - 56) / 2 seen (base angles of isosceles triangle)
A1 base angle ABC = ACB = 62 degrees (ft)
A1 180 - 62 = 118 degrees cao (ft), or 56 + 62 = 118
Answer: angle ACD = 118 degrees
Question 19
M1 interior angle of a regular hexagon = 120 degrees found or used
M1 120 + 120 + 60 = 300 seen (oe)
A1 360 - 300 = 60 degrees cao
Answer: 60 degrees
Question 20
(a) M1 90 + 90 + 2x + (3x - 10) = 360 seen (oe)
(a) M1 5x + 170 = 360 simplified (oe), leading to 5x = 190
(a) A1 x = 38 cao
(a) Answer: x = 38
(b) M1 their x substituted into 2x and 3x - 10 (ft)
(b) A1 76 degrees and 104 degrees both correct cao