State whether each angle is acute, right angle, obtuse, straight or reflex.
(a)35 degrees(1)
(b)90 degrees(1)
(c)128 degrees(1)
(d)200 degrees(1)
2
Angles on a straight line always add up to 180 degrees.
(a)A straight line is split into two angles by a ray. One angle is 65 degrees. Work out the size of the other angle, x.(2)
(b)A straight line is split into three angles: 42 degrees, 96 degrees and y degrees. Work out y.(2)
3
Angles that meet at a single point always add up to 360 degrees.
(a)Four angles meet at a point. Three of the angles are 110 degrees, 95 degrees and 88 degrees. Work out the fourth angle, x.(2)
(b)At a different point, four angles of 3x, 2x, x and 4x degrees meet, with no other angles present. Find x, and hence work out the size of the largest angle.(3)
4
Two straight lines AB and CD cross at point O. Angle AOC = 72 degrees.
Diagram NOT accurately drawn
(a)Write down the size of angle BOD, giving a reason for your answer.(2)
(b)Work out the size of angle AOD, giving a reason for your answer.(2)
5
The angles in any triangle add up to 180 degrees.
(a)In triangle PQR, angle P = 54 degrees and angle Q = 67 degrees. Work out angle R.(2)
(b)Triangle STU is right angled at S. Angle T is twice the size of angle U. Work out angle U.(3)
6
In an isosceles triangle, the two base angles are equal.
(a)Triangle ABC is isosceles with AB = AC. Angle BAC = 48 degrees. Work out angle ABC.(2)
(b)A different isosceles triangle has a base angle of 72 degrees. Work out the size of the apex angle.(2)
7
The side BC of triangle ABC is extended to a point D. Angle ABC = 58 degrees and angle BCA = 71 degrees.
Diagram NOT accurately drawn
(a)Work out the exterior angle, ACD.(2)
(b)State the geometric fact about the exterior angle of a triangle that gives the same answer without first finding angle BCA.(1)
8
The angles in any quadrilateral add up to 360 degrees.
(a)A quadrilateral has angles of 95 degrees, 108 degrees, 84 degrees and x degrees. Work out x.(2)
(b)A kite has one pair of equal angles of 115 degrees each. Work out the size of each of the other pair of equal angles.(2)
9
Lines AB and CD are parallel, cut by a transversal EF. One angle formed at the intersection with AB is 63 degrees.
Diagram NOT accurately drawn
(a)Write down the size of the corresponding angle at the intersection with CD, giving a reason.(2)
(b)Write down the size of the alternate angle at the intersection with CD, giving a reason.(2)
10
Two parallel lines are cut by a transversal. One of the co-interior (allied) angles is 71 degrees.
Diagram NOT accurately drawn
(a)Work out the other co-interior angle.(2)
(b)Explain why co-interior angles between parallel lines always sum to 180 degrees.(1)
11
Answer the following about quadrilaterals and their properties.
(a)A quadrilateral has exactly one pair of parallel sides and no equal angles. Name this quadrilateral.(1)
(b)A different quadrilateral has all four sides equal in length but no right angles. Name this quadrilateral and state its number of lines of symmetry.(2)
12
The sum of the interior angles of a polygon with n sides is given by (n - 2) x 180 degrees.
(a)Use the formula to work out the sum of the interior angles of a hexagon.(2)
(b)A hexagon has interior angles 130, 118, 122, 108, 130 and x degrees. Work out x.(2)
13
A regular polygon has an exterior angle of 24 degrees.
(a)Work out the number of sides of the polygon.(2)
(b)Work out the size of one interior angle of this polygon.(2)
14
For each shape, state the number of lines of symmetry and the order of rotational symmetry.
(a)A square(2)
(b)An isosceles triangle that is not equilateral(2)
(c)A regular pentagon(2)
15
Bearings are always given as three-figure angles measured clockwise from north.
(a)The bearing of a lighthouse L from a boat B is 128 degrees. Work out the bearing of the boat B from the lighthouse L.(2)
(b)A ship sails on a bearing of 072 degrees from port. State the three-figure bearing needed to sail directly back to port.(1)
16
Sumaya says that regular pentagons will tessellate (fit together with no gaps or overlaps) because 360 is close to a multiple of their interior angle.
(a)Work out the interior angle of a regular pentagon.(2)
(b)Determine, with a reason, whether Sumaya is correct.(2)
17
The three angles of a triangle are (2x + 20) degrees, (3x + 10) degrees and (x + 30) degrees.
(a)Show that 6x + 60 = 180.(2)
(b)Solve the equation to find x, and hence work out the size of the largest angle in the triangle.(3)
18
The interior angle of a regular polygon is 156 degrees.
(a)Work out the exterior angle of the polygon.(1)
(b)Hence work out the number of sides of the polygon.(2)
(c)A different regular polygon has n sides. Show that its interior angle can be written as ((n - 2) x 180) / n degrees, and use this to find the interior angle when n = 20.(3)
19
Lines AB and CD are parallel. A point X lies between the two lines. X is joined to a point P on line AB and to a point Q on line CD. Angle APX (between PA and PX, on the side facing X) is 35 degrees, and angle CQX (between QC and QX, on the side facing X) is 48 degrees.
Diagram NOT accurately drawn
(a)By drawing a line through X parallel to both AB and CD, show that angle PXQ = 83 degrees.(4)
20
A boat sails from point A to point B on a bearing of 048 degrees. At B, the boat changes course and sails to point C on a bearing of 140 degrees. Points A, B and C form a triangle.
Diagram NOT accurately drawn
(a)State the bearing of A from B.(1)
(b)Hence work out angle ABC, the interior angle of the triangle at B, using the bearing of C from B (140 degrees).(2)
(c)Given that angle BAC = 55 degrees, work out angle ACB.(2)
Mark scheme · KS3.M-G1 Geometry: Angles and Shapes
Question 1
(a) B1 acute
(a) Answer: Acute
(b) B1 right angle
(b) Answer: Right angle
(c) B1 obtuse
(c) Answer: Obtuse
(d) B1 reflex
(d) Answer: Reflex
Question 2
(a) M1 180 - 65 oe
(a) A1 x = 115 cao
(a) Answer: x = 115 degrees
(b) M1 180 - 42 - 96 oe
(b) A1 y = 42 cao
(b) Answer: y = 42 degrees
Question 3
(a) M1 360 - 110 - 95 - 88 oe
(a) A1 x = 67 cao
(a) Answer: x = 67 degrees
(b) M1 3x + 2x + x + 4x = 360, i.e. 10x = 360
(b) A1 x = 36
(b) A1 largest angle = 144, ft from their x
(b) Answer: x = 36, largest angle = 144 degrees
Question 4
(a) B1 angle BOD = 72 degrees
(a) B1 correct reason given, e.g. vertically opposite angles are equal
(a) Answer: Angle BOD = 72 degrees (vertically opposite to angle AOC)
(b) B1 angle AOD = 108 degrees
(b) B1 correct reason given, e.g. angles on a straight line (CD) add up to 180 degrees, or vertically opposite to angle BOC
(b) Answer: Angle AOD = 108 degrees
Question 5
(a) M1 180 - 54 - 67 oe
(a) A1 angle R = 59 cao
(a) Answer: Angle R = 59 degrees
(b) M1 angle T + angle U = 90 (since angle S = 90)
(b) M1 2u + u = 90 oe, using T = 2U
(b) A1 angle U = 30 cao
(b) Answer: Angle U = 30 degrees
Question 6
(a) M1 (180 - 48) / 2 oe
(a) A1 angle ABC = 66 cao
(a) Answer: Angle ABC = 66 degrees
(b) M1 180 - 72 - 72 oe
(b) A1 apex angle = 36 cao
(b) Answer: Apex angle = 36 degrees
Question 7
(a) M1 180 - 71 oe, or 180 - 58 - 71 then add 58
(a) A1 angle ACD = 109 cao
(a) Answer: Angle ACD = 109 degrees
(b) B1 the exterior angle of a triangle equals the sum of the two interior opposite angles (here angle BAC + angle ABC)
(b) Answer: The exterior angle equals the sum of the two interior opposite angles
Question 8
(a) M1 360 - 95 - 108 - 84 oe
(a) A1 x = 73 cao
(a) Answer: x = 73 degrees
(b) M1 (360 - 2 x 115) / 2 oe
(b) A1 65 cao
(b) Answer: 65 degrees each
Question 9
(a) B1 63 degrees
(a) B1 reason: corresponding angles on parallel lines are equal
(a) Answer: 63 degrees
(b) B1 63 degrees
(b) B1 reason: alternate angles on parallel lines are equal
(b) Answer: 63 degrees
Question 10
(a) M1 180 - 71 oe
(a) A1 109 cao
(a) Answer: 109 degrees
(b) B1 correct explanation, e.g. the co-interior angle is supplementary to the corresponding angle, and the corresponding angle sits on a straight line with the other co-interior angle, so together they total 180 degrees
(b) Answer: A co-interior angle and the corresponding angle to its partner lie on a straight line together, and a straight line totals 180 degrees, so co-interior angles are supplementary
Question 11
(a) B1 trapezium
(a) Answer: Trapezium
(b) B1 rhombus
(b) B1 2 lines of symmetry
(b) Answer: Rhombus, 2 lines of symmetry
Question 12
(a) M1 (6 - 2) x 180 oe
(a) A1 720 cao
(a) Answer: 720 degrees
(b) M1 720 - 130 - 118 - 122 - 108 - 130 oe, ft from part (a)
(b) A1 x = 112 cao
(b) Answer: x = 112 degrees
Question 13
(a) M1 360 / 24 oe
(a) A1 15 cao
(a) Answer: 15 sides
(b) M1 180 - 24 oe
(b) A1 156 cao
(b) Answer: 156 degrees
Question 14
(a) B1 4 lines of symmetry
(a) B1 rotational symmetry of order 4
(a) Answer: 4 lines of symmetry, order 4
(b) B1 1 line of symmetry
(b) B1 rotational symmetry of order 1
(b) Answer: 1 line of symmetry, order 1
(c) B1 5 lines of symmetry
(c) B1 rotational symmetry of order 5
(c) Answer: 5 lines of symmetry, order 5
Question 15
(a) M1 128 + 180 oe, since the given bearing is less than 180
(a) A1 308 cao
(a) Answer: 308 degrees
(b) B1 252 cao
(b) Answer: 252 degrees
Question 16
(a) M1 (5 - 2) x 180 / 5 oe
(a) A1 108 cao
(a) Answer: 108 degrees
(b) B1 no, Sumaya is not correct
(b) B1 correct reason, e.g. 360 / 108 = 3.33 (not a whole number), so a whole number of pentagons cannot meet exactly at a point, leaving a gap
(b) Answer: No, Sumaya is not correct: 360 / 108 is not a whole number, so regular pentagons alone cannot tessellate
Question 17
(a) M1 (2x + 20) + (3x + 10) + (x + 30) = 180 set up, using angle sum of a triangle
(a) A1 cso: correctly simplified to 6x + 60 = 180
(a) Answer: 6x + 60 = 180 (shown)
(b) M1 6x = 120
(b) A1 x = 20
(b) A1 largest angle = 70 degrees, ft from their x
(b) Answer: x = 20, largest angle = 70 degrees
Question 18
(a) B1 24 cao
(a) Answer: 24 degrees
(b) M1 360 / 24 oe, ft from part (a)
(b) A1 15 cao
(b) Answer: 15 sides
(c) M1 correct reasoning that the sum of interior angles is (n-2) x 180, shared equally between the n equal angles of a regular polygon
(c) A1 cso: interior angle = ((n-2) x 180) / n stated
(c) A1 n = 20 gives 162 degrees, cao
(c) Answer: ((n - 2) x 180) / n; when n = 20, interior angle = 162 degrees
Question 19
(a) M1 a line through X parallel to AB and CD is constructed, splitting angle PXQ into two parts
(a) M1 each part correctly identified as alternate to one of the given angles (35 and 48 degrees) using the parallel lines
(a) A1 35 + 48 used to combine the two parts
(a) A1 cso: angle PXQ = 83 degrees, matching the given statement