Geometry: angles and shapes
By the end of this chapter you will be able to
- Find a missing angle at a point, on a straight line, or in a triangle or quadrilateral
- Use the angle sum of a polygon, and of a regular polygon
- Classify a triangle or quadrilateral by its properties, and say why
- Name the faces, edges and vertices of a 3-D shape from a picture or a description
- Say whether a statement is always, sometimes or never true, and give a reason
You should already know
- Right angles, straight lines and the names of 2-D shapes (Year 4 and Year 5 geometry)
- Angles in a triangle (Year 5 geometry)
Where this leads
- Position and coordinates (Chapter 6 of this book)
KS2 SATs practice for reasoning (Year 6). Original questions in the style of the national tests. Content: Use the properties of rectangles, triangles and regular polygons to deduce related facts (Year 5-6), Know that angles at a point add to 360 degrees, angles on a straight line add to 180 degrees, and vertically opposite angles are equal (Year 6), Calculate unknown angles in a triangle and in a quadrilateral (Year 5-6), Compare and classify triangles and quadrilaterals by their properties, and explain (Year 5-6), Identify 3-D shapes from 2-D representations, including nets, faces, edges and vertices (Year 5-6), Distinguish regular and irregular polygons, identify acute, obtuse and reflex angles, and name parts of a circle (Year 6), Prove or disprove a geometric statement with a reason (Year 5-6).
Exercise P Do you remember?
Non-calculatorMethod. A triangle has three sides and three angles.
- Count the sides. Three sides.3 sides
- Name the shape. A shape with three sides is a triangle.triangle
- Write the answer. triangle
Check. 3 sides make a triangle. ✓
Method. A right angle is a quarter turn.
- Use the quarter turn. A full turn is 360 degrees, and a quarter of 360 is 90.360 / 4 = 90
- Write the answer. 90
Check. 90 x 4 = 360. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. An acute angle is less than 90 degrees, and 45 is less than 90.45 degrees fits
- Reject the other choices. 120 is obtuse and 180 is a straight line.120 degrees and 180 degrees do not fit
- Write the answer. 45 degrees
Check. 45 degrees is the one that fits. ✓
Method. Hexa means six.
- Use the name. Hexagon has six sides.6
- Write the answer. 6
Check. A hexagon has six sides. ✓
Method. Subtract 75 from 180.
- Subtract. 180 - 75 = 105.105
- Write the answer. 105
Check. 105 + 75 = 180. ✓
Angles at a point, on a straight line and vertically opposite
Rule
Angles at a point add up to 360 degrees. Angles on a straight line add up to 180 degrees.
Vertically opposite angles are equal: when two straight lines cross, the angles opposite each other are the same.
Key ideas
- At a point: 360 degrees in a full turn.
- On a straight line: 180 degrees, half a turn.
- Vertically opposite angles are equal.
Method. Take the sum of the known angles from 360.
- Angles at a point add up to 360 degrees. A full turn is 360 degrees.360
- Add the known angles. Add the angles you know.120 + 85 + 60 = 265
- Take the total from 360. The missing angle is what is left.360 - 265 = 95
- Write the answer. 95
Check. 120 + 85 + 60 + 95 = 360. ✓
Watch out. Do not use 180 for angles at a point. A full turn is 360.
Method. Take the known angle from 180.
- Angles on a straight line add up to 180 degrees. A straight line is half a turn.180
- Subtract the known angle. Take 35 from 180.180 - 35 = 145
- Write the answer. 145
Check. 35 + 145 = 180. ✓
Watch out. Straight line: 180. Around a point: 360.
Method. Vertically opposite angles are equal.
- Find the opposite angle. The angle opposite a 65 degree angle is the same size.65 degrees
- Check on a straight line. The other two angles are 180 - 65 = 115 degrees each, which is consistent.65 + 115 = 180
- Write the answer. 65
Check. 65 + 115 = 180. ✓
Watch out
Watch out. A straight line is 180 degrees, not 360. Only a full turn at a point is 360.
Exam tip
Practice guide. Mark the known angles on the picture, then write the total next to the missing angle.
Exercise 5A Practice: angles at a point, on a line and opposite
Non-calculatorMethod. Take the sum of the known angles from 360.
- Angles at a point add up to 360 degrees. A full turn is 360 degrees.360
- Add the known angles. Add the angles you know.90 + 140 = 230
- Take the total from 360. The missing angle is what is left.360 - 230 = 130
- Write the answer. 130
Check. 90 + 140 + 130 = 360. ✓
Watch out. Do not use 180 for angles at a point. A full turn is 360.
Method. Take the sum of the known angles from 360.
- Angles at a point add up to 360 degrees. A full turn is 360 degrees.360
- Add the known angles. Add the angles you know.45 + 100 + 75 + 60 = 280
- Take the total from 360. The missing angle is what is left.360 - 280 = 80
- Write the answer. 80
Check. 45 + 100 + 75 + 60 + 80 = 360. ✓
Watch out. Do not use 180 for angles at a point. A full turn is 360.
Method. Take the sum of the known angles from 360.
- Angles at a point add up to 360 degrees. A full turn is 360 degrees.360
- Add the known angles. Add the angles you know.200 + 50 = 250
- Take the total from 360. The missing angle is what is left.360 - 250 = 110
- Write the answer. 110
Check. 200 + 50 + 110 = 360. ✓
Watch out. Do not use 180 for angles at a point. A full turn is 360.
Method. Take the known angle from 180.
- Angles on a straight line add up to 180 degrees. A straight line is half a turn.180
- Subtract the known angle. Take 72 from 180.180 - 72 = 108
- Write the answer. 108
Check. 72 + 108 = 180. ✓
Watch out. Straight line: 180. Around a point: 360.
Method. Take the known angle from 180.
- Angles on a straight line add up to 180 degrees. A straight line is half a turn.180
- Subtract the known angle. Take 115 from 180.180 - 115 = 65
- Write the answer. 65
Check. 115 + 65 = 180. ✓
Watch out. Straight line: 180. Around a point: 360.
Method. Vertically opposite angles are equal.
- Opposite angle. It is the same size.40 degrees
- Write the answer. 40
Check. 40 = 40. ✓
Method. Use the total of 180 and the ratio of 3 to 1.
- Write the total in parts. a + b = 180, and a = 3b, so 3b + b = 4b = 180.4b = 180
- Find b. 180 / 4 = 45, so b = 45.b = 45
- Find a. a = 3 x 45 = 135 degrees.135
- Write the answer. 135
Check. 135 + 45 = 180. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. A reflex angle is more than 180 degrees and less than 360 degrees.200 degrees fits
- Reject the other choices. 60 is acute, and 90 is a right angle.60 degrees and 90 degrees do not fit
- Write the answer. 200 degrees
Check. 200 degrees is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A straight line is half a turn, and vertically opposite angles are equal.Angles on a straight line add up to 180 degrees, Vertically opposite angles are equal
- Leave the others unticked. These do not match.Angles at a point add up to 180 degrees, Angles in a triangle add up to 360 degrees are not ticked
- Write the answer. Angles on a straight line add up to 180 degrees, Vertically opposite angles are equal
Check. Each ticked option is true. ✓
Watch out. Angles at a point add to 360 degrees, and a triangle has angles of 180 degrees.
Method. An obtuse angle is between 90 and 180 degrees.
- Compare with 90 and 180. 135 is more than 90 and less than 180.between 90 and 180
- Name it. An angle between 90 and 180 is obtuse.obtuse
- Write the answer. obtuse
Check. 90 < 135 < 180. ✓
Method. Think about a full turn.
- Think of a full turn. Turning all the way round is one whole turn.one full turn
- Use the size of a full turn. A full turn is 360 degrees.360 degrees
- Write the answer. Because they make a full turn, and a full turn is 360 degrees.
Check. A full turn is 360 degrees. ✓
Method. Angles a and b are on a straight line.
- Use the straight line. a and b are on a straight line, so a + b = 180.62 + b = 180
- Find b. 180 - 62 = 118 degrees.118
- Write the answer. 118
Check. 62 + 118 = 180. ✓
Method. Vertically opposite angles sit opposite each other where two lines cross.
- Check the position. If two lines cross, the angles opposite each other are vertically opposite.opposite each other
- Answer. So Ali is right if x and y are opposite at the crossing.yes
- Write the answer. yes
Check. Vertically opposite angles are equal. ✓
Method. Subtract the two known angles from 360.
- Add the known angles. 110 + 95 = 205.205
- Take from 360. 360 - 205 = 155.155
- Write the answer. 155
Check. 110 + 95 + 155 = 360. ✓
Angles in triangles
Rule
The angles in a triangle add up to 180 degrees.
In an isosceles triangle, two sides and two angles are equal. In an equilateral triangle, all three angles are 60 degrees.
Key ideas
- Angles in a triangle: 180 degrees in total.
- Isosceles: the two base angles are equal.
- Equilateral: each angle is 60 degrees.
Method. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.50 + 70 = 120
- Take the total from 180. Angle c is what is left.180 - 120 = 60
- Write the answer. 60
Check. 50 + 70 + 60 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 40 = 140
- Share it between the two base angles. Divide by 2.140 / 2 = 70
- Write the answer. 70
Check. 40 + 2 x 70 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Angles in a triangle add up to 180 degrees.
- Add the two known angles. 30 + 30 = 60.60
- Take from 180. 180 - 60 = 120 degrees.120
- Write the answer. 120
Check. 30 + 30 + 120 = 180. ✓
Watch out
Watch out. In a triangle the angles add up to 180 degrees, not 360. Only a quadrilateral has 360 degrees.
Exam tip
Practice guide. Label the equal sides or angles first, so you can see which angles are the same.
Exercise 5B Practice: angles in triangles
Non-calculatorMethod. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.35 + 85 = 120
- Take the total from 180. Angle x is what is left.180 - 120 = 60
- Write the answer. 60
Check. 35 + 85 + 60 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.90 + 28 = 118
- Take the total from 180. Angle y is what is left.180 - 118 = 62
- Write the answer. 62
Check. 90 + 28 + 62 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.60 + 60 = 120
- Take the total from 180. Angle z is what is left.180 - 120 = 60
- Write the answer. 60
Check. 60 + 60 + 60 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 30 = 150
- Share it between the two base angles. Divide by 2.150 / 2 = 75
- Write the answer. 75
Check. 30 + 2 x 75 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 100 = 80
- Share it between the two base angles. Divide by 2.80 / 2 = 40
- Write the answer. 40
Check. 100 + 2 x 40 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 50 = 130
- Share it between the two base angles. Divide by 2.130 / 2 = 65
- Write the answer. 65
Check. 50 + 2 x 65 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Share 180 equally between the three angles.
- Share 180. 180 / 3 = 60.60
- Write the answer. 60
Check. 60 x 3 = 180. ✓
Method. One angle is 90 degrees, the right angle.
- Add the known angles. 90 + 38 = 128.128
- Take from 180. 180 - 128 = 52 degrees.52
- Write the answer. 52
Check. 90 + 38 + 52 = 180. ✓
Method. Add the parts to find the total number of parts, then find one part.
- Total parts. 1 + 2 + 3 = 6 parts.6 parts
- One part. 180 / 6 = 30 degrees.30
- Largest angle. The largest is 3 parts: 3 x 30 = 90 degrees.90
- Write the answer. 90
Check. 30 + 60 + 90 = 180. ✓
Method. Add two obtuse angles.
- Pick two obtuse angles. Two angles of 100 degrees.100 + 100
- Compare with 180. 200 is more than 180, so it is not possible.200 > 180
- Write the answer. No, because two obtuse angles are each more than 90 degrees, which makes more than 180 degrees.
Check. 100 + 100 = 200 > 180. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 60 + 60 + 70 = 190, which is more than 180, so it cannot be a triangle.No, the angles add up to 190 degrees fits
- Reject the other choices. The total is 190, not 180, so the third option is wrong too.Yes, the angles add up to 190 degrees and No, the angles add up to 180 degrees do not fit
- Write the answer. No, the angles add up to 190 degrees
Check. No, the angles add up to 190 degrees is the one that fits. ✓
Method. Take the top angle from 180, then halve.
- Take from 180. 180 - 84 = 96.96
- Halve. 96 / 2 = 48 degrees.48
- Write the answer. 48
Check. 84 + 48 + 48 = 180. ✓
Method. Use the total of the angles in a triangle.
- Add A and B. 50 + 60 = 110.110
- Find C. 180 - 110 = 70 degrees.70
- Write the answer. 70
Check. 50 + 60 + 70 = 180. ✓
Method. Find the total of the base angles first.
- Total of base angles. 180 - 40 = 140.140
- Each base angle. 140 / 2 = 70, and they are equal.70
- Write the answer. The base angles are both 70 degrees, because (180 - 40) / 2 = 70 and the two angles are equal in an isosceles triangle.
Check. 70 + 70 + 40 = 180. ✓
Quadrilaterals and polygons
Rule
The angles in a quadrilateral add up to 360 degrees. A polygon with n sides has (n - 2) x 180 degrees of angles inside it.
A regular polygon has all sides equal and all angles equal. Each interior angle = (n - 2) x 180 / n.
Key ideas
- Quadrilateral: 360 degrees.
- Sum of interior angles: (n - 2) x 180.
- Regular polygon: divide the total by the number of sides.
Method. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.80 + 95 + 110 = 285
- Take the total from 360. The fourth angle is what is left.360 - 285 = 75
- Write the answer. 75
Check. 80 + 95 + 110 + 75 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 8 sides splits into 6 triangles.8 - 2 = 6
- Multiply by 180. Each triangle has angles of 180 degrees.6 x 180 = 1080
- Write the answer. 1080
Check. 6 triangles x 180 = 1080. ✓
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 5 sides has 3 triangles inside it, so the total is (5 - 2) x 180.(5 - 2) x 180 = 540
- Share the total equally between the angles. All 5 angles of a regular shape are the same size.540 / 5 = 108
- Write the answer. 108
Check. 108 x 5 = 540. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Watch out
Watch out. A regular pentagon has interior angles of 108 degrees, not 72. The exterior angle is 72.
Exam tip
Practice guide. Split the shape into triangles from one corner, then add up the triangles.
Exercise 5C Practice: quadrilaterals and polygons
Non-calculatorMethod. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.90 + 90 + 60 = 240
- Take the total from 360. The fourth angle is what is left.360 - 240 = 120
- Write the answer. 120
Check. 90 + 90 + 60 + 120 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.100 + 70 + 130 = 300
- Take the total from 360. The fourth angle is what is left.360 - 300 = 60
- Write the answer. 60
Check. 100 + 70 + 130 + 60 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 6 sides splits into 4 triangles.6 - 2 = 4
- Multiply by 180. Each triangle has angles of 180 degrees.4 x 180 = 720
- Write the answer. 720
Check. 4 triangles x 180 = 720. ✓
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 10 sides splits into 8 triangles.10 - 2 = 8
- Multiply by 180. Each triangle has angles of 180 degrees.8 x 180 = 1440
- Write the answer. 1440
Check. 8 triangles x 180 = 1440. ✓
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 6 sides has 4 triangles inside it, so the total is (6 - 2) x 180.(6 - 2) x 180 = 720
- Share the total equally between the angles. All 6 angles of a regular shape are the same size.720 / 6 = 120
- Write the answer. 120
Check. 120 x 6 = 720. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 8 sides has 6 triangles inside it, so the total is (8 - 2) x 180.(8 - 2) x 180 = 1080
- Share the total equally between the angles. All 8 angles of a regular shape are the same size.1080 / 8 = 135
- Write the answer. 135
Check. 135 x 8 = 1080. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 3 sides has 1 triangles inside it, so the total is (3 - 2) x 180.(3 - 2) x 180 = 180
- Share the total equally between the angles. All 3 angles of a regular shape are the same size.180 / 3 = 60
- Write the answer. 60
Check. 60 x 3 = 180. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Divide 360 by the number of sides.
- Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.360
- Share equally for a regular shape. Each of the 5 turns is the same size.360 / 5 = 72
- Write the answer. 72
Check. 72 x 5 = 360. ✓
Watch out. Exterior angle plus interior angle = 180.
Method. Divide 360 by the number of sides.
- Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.360
- Share equally for a regular shape. Each of the 12 turns is the same size.360 / 12 = 30
- Write the answer. 30
Check. 30 x 12 = 360. ✓
Watch out. Exterior angle plus interior angle = 180.
Method. The exterior angle is 180 - 150 = 30 degrees, and exterior angles add up to 360.
- Exterior angle. 180 - 150 = 30 degrees.30
- Number of sides. 360 / 30 = 12 sides.12
- Write the answer. 12
Check. 12 x 30 = 360. ✓
Method. The angles of a pentagon add to (5 - 2) x 180 = 540.
- Total of a pentagon. (5 - 2) x 180 = 540.540
- Add the four known angles. 100 + 110 + 95 + 120 = 425.425
- Find x. 540 - 425 = 115 degrees.115
- Write the answer. 115
Check. 425 + 115 = 540. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. A heptagon has 7 sides, so (7 - 2) x 180 = 900 degrees.heptagon fits
- Reject the other choices. A hexagon has 720 degrees and an octagon has 1080 degrees.hexagon and octagon do not fit
- Write the answer. heptagon
Check. heptagon is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. In a regular polygon all sides and angles are equal, and a regular pentagon has 5 lines of symmetry.All sides and all angles are equal in a regular polygon, A regular pentagon has 5 lines of symmetry
- Leave the others unticked. These do not match.A regular hexagon has interior angles of 100 degrees, Every regular polygon has an even number of sides are not ticked
- Write the answer. All sides and all angles are equal in a regular polygon, A regular pentagon has 5 lines of symmetry
Check. Each ticked option is true. ✓
Watch out. A regular hexagon has interior angles of 120 degrees, and a triangle is a regular polygon with 3 sides.
Method. Draw a diagonal.
- Split the shape. Draw a diagonal to make two triangles.two triangles
- Add the triangles. Each triangle is 180 degrees, so 2 x 180 = 360.360 degrees
- Write the answer. Because a quadrilateral can be split into two triangles, and two triangles have 2 x 180 = 360 degrees.
Check. 2 x 180 = 360. ✓
Method. Use (n - 2) x 180 with n = 8.
- Use the formula. (8 - 2) x 180 = 6 x 180 = 1080.1080
- Write the answer. 1080
Check. 1080 / 8 = 135. ✓
Method. Work out the interior angle for 9 sides.
- Total. (9 - 2) x 180 = 1260 degrees.1260
- Each angle. 1260 / 9 = 140 degrees, not 120.140
- Write the answer. no
Check. 140 x 9 = 1260. ✓
Classifying shapes, and proving or disproving
Rule
A rectangle has four right angles and opposite sides equal. A square is a rectangle with four equal sides. A rhombus has four equal sides. A parallelogram has two pairs of parallel sides. A trapezium has one pair of parallel sides.
To disprove a statement, find one example where it is false.
Key ideas
- A square is a special rectangle and a special rhombus.
- An isosceles triangle has two equal sides, and an equilateral triangle has three.
- To disprove a statement, give one counter-example.
Method. A rhombus has four equal sides, and a square has four equal sides.
- Check the sides. A square has four equal sides.4 equal sides
- Compare with a rhombus. A rhombus has four equal sides, so every square is a rhombus.always
- Write the answer. always
Check. A square has four equal sides. ✓
Method. Find an isosceles triangle that is not equilateral.
- Pick a triangle. A triangle with sides 5, 5 and 7 has two equal sides.5, 5, 7
- Check it is not equilateral. Its sides are not all equal, so it is not equilateral.not equilateral
- Write the answer. no
Check. The triangle 5, 5, 7 is isosceles, not equilateral. ✓
Method. Know which quadrilateral has one pair of parallel sides.
- Parallelogram. A parallelogram has two pairs of parallel sides.two pairs
- Trapezium. A trapezium has exactly one pair of parallel sides.one pair
- Write the answer. trapezium
Check. One pair of parallel sides is a trapezium. ✓
Watch out
Watch out. A square is always a rhombus, but a rhombus is not always a square.
Exam tip
Practice guide. To disprove a statement, one counter-example is enough.
Exercise 5D Practice: classifying and proving
Non-calculatorMethod. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. A rhombus has four equal sides, but its angles are not all 90 degrees.rhombus fits
- Reject the other choices. A rectangle has four right angles but not equal sides. A square has four right angles.rectangle and square do not fit
- Write the answer. rhombus
Check. rhombus is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A rectangle has equal opposite sides and four right angles, and its diagonals cut each other in half.Opposite sides are equal, All angles are 90 degrees, The diagonals bisect each other
- Leave the others unticked. These do not match.All sides are equal are not ticked
- Write the answer. Opposite sides are equal, All angles are 90 degrees, The diagonals bisect each other
Check. Each ticked option is true. ✓
Watch out. All sides are equal only in a square, not in every rectangle.
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. The opposite sides of a parallelogram are equal.always fits
- Reject the other choices. It is not sometimes or never: the opposite sides are always equal.sometimes and never do not fit
- Write the answer. always
Check. always is the one that fits. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. Two right angles would already make 180 degrees, leaving nothing for the third angle.always fits
- Reject the other choices. A triangle cannot have two right angles, so it is not sometimes true.sometimes and never do not fit
- Write the answer. always
Check. always is the one that fits. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. Only an isosceles triangle with all three sides equal is equilateral.sometimes fits
- Reject the other choices. Not every isosceles triangle is equilateral, so it is not always true.always and never do not fit
- Write the answer. sometimes
Check. sometimes is the one that fits. ✓
Method. Compare the properties of a square and a rectangle.
- Properties of a rectangle. A rectangle has four right angles and opposite sides equal.four right angles
- Square. A square has four right angles and four equal sides, so it has the rectangle properties.yes
- Write the answer. yes
Check. A square has four right angles. ✓
Method. Find a triangle without a right angle.
- Pick a triangle. An equilateral triangle has angles of 60 degrees.60, 60, 60
- Check. None of the angles is 90 degrees, so it has no right angle.no right angle
- Write the answer. A triangle with angles 60, 60 and 60 has no right angle, so the statement is false.
Check. 60 + 60 + 60 = 180. ✓
Method. Think of a shape with four equal sides and angles that are not right angles.
- Pick a shape. A rhombus has four equal sides.four equal sides
- Check the angles. A rhombus can have angles of 60 and 120 degrees, so it is not a square.not 90 degrees
- Write the answer. Disproved, for example a rhombus with sides of 5 cm and angles of 60 and 120 degrees has four equal sides but is not a square.
Check. A rhombus with angles of 60 and 120 has four equal sides. ✓
Method. The three angles add up to 180 degrees.
- Add the expressions. 2x + 3x + 4x = 9x.9x
- Solve. 9x = 180, so x = 180 / 9 = 20.20
- Write the answer. 20
Check. 40 + 60 + 80 = 180. ✓
Method. Check which angle is bigger.
- Compare. 80 is bigger than 50.80
- Write the answer. 80
Check. 50 + 50 + 80 = 180. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. One angle is 90 degrees, and all three angles are different, so the sides are different.right-angled and scalene fits
- Reject the other choices. It is not equilateral, and it is not acute because of the 90 degree angle.equilateral and isosceles and acute do not fit
- Write the answer. right-angled and scalene
Check. right-angled and scalene is the one that fits. ✓
Method. Draw a parallelogram that leans over.
- Pick a parallelogram. A parallelogram with angles 60 and 120 degrees.60 and 120
- Check. Its angles are not all 90 degrees, so it is not a rectangle.not a rectangle
- Write the answer. No, because a parallelogram with angles 60 and 120 degrees is not a rectangle.
Check. 60 + 120 = 180, so the angles are not right angles. ✓
Method. Check which shape has two pairs of equal sides with the angles not all equal.
- Rectangle. A rectangle has equal opposite sides and all four angles equal (90 degrees).all angles equal
- Kite. A kite has two pairs of equal sides that are next to each other, and its angles are not all equal.kite
- Write the answer. kite
Check. The angles are not all equal, so it is not a rectangle. ✓
3-D shapes, circles and angle types
Rule
A face is a flat surface, an edge is where two faces meet, and a vertex is a corner.
In a circle, the radius goes from the centre to the edge, and the diameter goes across the whole circle through the centre. Diameter = 2 x radius.
Acute: less than 90 degrees. Right: 90. Obtuse: between 90 and 180. Reflex: more than 180 and less than 360.
Key ideas
- Diameter = 2 x radius.
- Acute < 90; obtuse between 90 and 180; reflex between 180 and 360.
- A cube has 6 faces, 12 edges and 8 vertices.
Method. Count the edges: the lines where two faces meet.
- Picture the shape. A cube has 6 faces, 12 edges and 8 vertices.faces 6, edges 12, vertices 8
- Count the edges. An edge is where two faces meet.12 edges
- Write the answer. 12
Check. 6 faces + 8 vertices - 12 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Use diameter = 2 x radius.
- Write the rule. Diameter = 2 x radius.2 x 5
- Multiply. Double the radius.10
- Write the answer. 10
Check. diameter = 2 x radius, so 20 = 2 x 10 or 5. ✓
Watch out. The radius is half the diameter. Do not double it when you need the radius.
Method. Compare the size with 90 and 180 degrees.
- Compare with 90 and 180. 250 degrees is compared with the rules for each type.250 is between 180 and 360 degrees
- Name the angle. An angle that is between 180 and 360 degrees is reflex.reflex
- Write the answer. reflex
Check. 250 is between 180 and 360 degrees. ✓
Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.
Watch out
Watch out. The radius is half the diameter, not twice it. A diameter of 10 cm has a radius of 5 cm.
Exam tip
Practice guide. Count the faces, edges and vertices of a solid from a labelled picture before you answer.
Exercise 5E Practice: 3-D shapes, circles and angle types
Non-calculatorMethod. Count the edges: the lines where two faces meet.
- Picture the shape. A triangular prism has 5 faces, 9 edges and 6 vertices.faces 5, edges 9, vertices 6
- Count the edges. An edge is where two faces meet.9 edges
- Write the answer. 9
Check. 5 faces + 6 vertices - 9 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Count the edges: the lines where two faces meet.
- Picture the shape. A square-based pyramid has 5 faces, 8 edges and 5 vertices.faces 5, edges 8, vertices 5
- Count the edges. An edge is where two faces meet.8 edges
- Write the answer. 8
Check. 5 faces + 5 vertices - 8 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Count the edges: the lines where two faces meet.
- Picture the shape. A cuboid has 6 faces, 12 edges and 8 vertices.faces 6, edges 12, vertices 8
- Count the edges. An edge is where two faces meet.12 edges
- Write the answer. 12
Check. 6 faces + 8 vertices - 12 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Use diameter = 2 x radius.
- Write the rule. Radius = diameter / 2.14 / 2
- Divide. The radius is half the diameter.7
- Write the answer. 7
Check. diameter = 2 x radius, so 14 = 2 x 7 or 14. ✓
Watch out. The radius is half the diameter. Do not double it when you need the radius.
Method. Use diameter = 2 x radius.
- Write the rule. Diameter = 2 x radius.2 x 7
- Multiply. Double the radius.14
- Write the answer. 14
Check. diameter = 2 x radius, so 28 = 2 x 14 or 7. ✓
Watch out. The radius is half the diameter. Do not double it when you need the radius.
Method. Compare the size with 90 and 180 degrees.
- Compare with 90 and 180. 40 degrees is compared with the rules for each type.40 is less than 90 degrees
- Name the angle. An angle that is less than 90 degrees is acute.acute
- Write the answer. acute
Check. 40 is less than 90 degrees. ✓
Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.
Method. Compare the size with 90 and 180 degrees.
- Compare with 90 and 180. 90 degrees is compared with the rules for each type.90 is exactly 90 degrees
- Name the angle. An angle that is exactly 90 degrees is right.right
- Write the answer. right
Check. 90 is exactly 90 degrees. ✓
Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.
Method. Compare the size with 90 and 180 degrees.
- Compare with 90 and 180. 305 degrees is compared with the rules for each type.305 is between 180 and 360 degrees
- Name the angle. An angle that is between 180 and 360 degrees is reflex.reflex
- Write the answer. reflex
Check. 305 is between 180 and 360 degrees. ✓
Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.
Method. A cube has six square faces.
- Count the faces. Each face is one square, and a cube has 6 square faces.6
- Write the answer. 6
Check. 6 faces. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 60 degrees is exactly 60, so it is the closest.60 degrees fits
- Reject the other choices. 45 and 75 are 15 degrees away from 60, and 60 is 0 away.45 degrees and 75 degrees do not fit
- Write the answer. 60 degrees
Check. 60 degrees is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. The radius is half of the diameter, and the diameter passes through the centre.The radius is half the diameter, The diameter goes through the centre
- Leave the others unticked. These do not match.The diameter is half the radius, The circumference is the distance across the middle are not ticked
- Write the answer. The radius is half the diameter, The diameter goes through the centre
Check. Each ticked option is true. ✓
Watch out. The distance around a circle is the circumference, not the diameter.
Method. Compare the size of the two angles.
- Right angle. A right angle is 90 degrees.90
- Obtuse angle. An obtuse angle is more than 90 degrees.more than 90
- Write the answer. obtuse
Check. obtuse is more than 90 degrees. ✓
Method. Check each angle against the rule for a reflex angle.
- Check 170. 170 is less than 180, so it is not reflex.not reflex
- Check 90. 90 is a right angle, so it is not reflex.not reflex
- Check 350. 350 is between 180 and 360, so it is reflex.350 is reflex
- Write the answer. 350
Check. 180 < 350 < 360. ✓
Method. Diameter = 2 x radius.
- Double the radius. 2 x 9 = 18.18
- Write the answer. 18
Check. 18 / 2 = 9. ✓
Method. Draw a circle with a line from the centre to the edge.
- Radius. The radius goes from the centre to the edge.centre to edge
- Diameter. Two radii joined together make a line across the circle, which is the diameter.two radii = diameter
- Write the answer. Because the diameter goes across the circle through the centre, and the radius is from the centre to the edge, so two radii make a diameter.
Check. 2 x radius = diameter. ✓
Chapter summary
| Skill | Method |
|---|---|
| Angles at a point | Add to 360 degrees. |
| Angles on a straight line | Add to 180 degrees. |
| Vertically opposite | Angles opposite each other at a crossing are equal. |
| Triangle | The three angles add to 180 degrees; isosceles has two equal base angles. |
| Quadrilateral | The four angles add to 360 degrees. |
| Polygon | Angles inside add to (n - 2) x 180; regular: all sides and angles equal. |
| Circle | Diameter = 2 x radius. |
| Angle types | Acute < 90; right = 90; obtuse 90 to 180; reflex 180 to 360. |
Can you…?
- I can find missing angles at a point, on a straight line and vertically opposite
- I can find missing angles in a triangle and in a quadrilateral
- I can use the angle sum of a polygon, and of a regular polygon
- I can classify shapes by their properties and say why
- I can name faces, edges and vertices of a 3-D shape
- I can disprove a statement with a counter-example
Key terms
- acute angle
- an angle less than 90 degrees
- obtuse angle
- an angle between 90 and 180 degrees
- reflex angle
- an angle between 180 and 360 degrees
- regular polygon
- a shape with all sides and all angles equal
- isosceles triangle
- a triangle with two equal sides and two equal angles
- counter-example
- one example that shows a statement is false
Exercise 5Z Check-up
Non-calculatorMethod. Take the sum of the known angles from 360.
- Angles at a point add up to 360 degrees. A full turn is 360 degrees.360
- Add the known angles. Add the angles you know.100 + 120 = 220
- Take the total from 360. The missing angle is what is left.360 - 220 = 140
- Write the answer. 140
Check. 100 + 120 + 140 = 360. ✓
Watch out. Do not use 180 for angles at a point. A full turn is 360.
Method. Take the known angle from 180.
- Angles on a straight line add up to 180 degrees. A straight line is half a turn.180
- Subtract the known angle. Take 58 from 180.180 - 58 = 122
- Write the answer. 122
Check. 58 + 122 = 180. ✓
Watch out. Straight line: 180. Around a point: 360.
Method. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.47 + 73 = 120
- Take the total from 180. Angle w is what is left.180 - 120 = 60
- Write the answer. 60
Check. 47 + 73 + 60 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 70 = 110
- Share it between the two base angles. Divide by 2.110 / 2 = 55
- Write the answer. 55
Check. 70 + 2 x 55 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.75 + 110 + 95 = 280
- Take the total from 360. The fourth angle is what is left.360 - 280 = 80
- Write the answer. 80
Check. 75 + 110 + 95 + 80 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 10 sides has 8 triangles inside it, so the total is (10 - 2) x 180.(10 - 2) x 180 = 1440
- Share the total equally between the angles. All 10 angles of a regular shape are the same size.1440 / 10 = 144
- Write the answer. 144
Check. 144 x 10 = 1440. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Divide 360 by the number of sides.
- Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.360
- Share equally for a regular shape. Each of the 10 turns is the same size.360 / 10 = 36
- Write the answer. 36
Check. 36 x 10 = 360. ✓
Watch out. Exterior angle plus interior angle = 180.
Method. Count the edges: the lines where two faces meet.
- Picture the shape. A cube has 6 faces, 12 edges and 8 vertices.faces 6, edges 12, vertices 8
- Count the edges. An edge is where two faces meet.12 edges
- Write the answer. 12
Check. 6 faces + 8 vertices - 12 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Check the angles of a rhombus.
- Compare. A square is a rhombus with four right angles, but a rhombus can have other angles.not always
- Answer. So Ben is wrong.no
- Write the answer. no
Check. A rhombus with angles 60 and 120 is not right-angled. ✓
Method. Compare the size with 90 and 180 degrees.
- Compare with 90 and 180. 160 degrees is compared with the rules for each type.160 is between 90 and 180 degrees
- Name the angle. An angle that is between 90 and 180 degrees is obtuse.obtuse
- Write the answer. obtuse
Check. 160 is between 90 and 180 degrees. ✓
Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.
Total 16 marks. Allow about 16 minutes.
Exercise 5X Practice test questions
Non-calculatorMethod. Take the known angle from 180.
- Angles on a straight line add up to 180 degrees. A straight line is half a turn.180
- Subtract the known angle. Take 48 from 180.180 - 48 = 132
- Write the answer. 132
Check. 48 + 132 = 180. ✓
Watch out. Straight line: 180. Around a point: 360.
Method. Take the sum of the two known angles from 180.
- Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.180
- Add the two known angles. Add the angles you know.65 + 55 = 120
- Take the total from 180. Angle q is what is left.180 - 120 = 60
- Write the answer. 60
Check. 65 + 55 + 60 = 180. ✓
Watch out. Check that the three angles add up to 180.
Method. Take the apex from 180, then share the rest equally between the two base angles.
- The two base angles are equal. In an isosceles triangle the base angles are the same.base angles are equal
- Find the total of the two base angles. Take the apex angle from 180.180 - 56 = 124
- Share it between the two base angles. Divide by 2.124 / 2 = 62
- Write the answer. 62
Check. 56 + 2 x 62 = 180. ✓
Watch out. The two base angles are equal, not the apex and a base angle.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 12 sides has 10 triangles inside it, so the total is (12 - 2) x 180.(12 - 2) x 180 = 1800
- Share the total equally between the angles. All 12 angles of a regular shape are the same size.1800 / 12 = 150
- Write the answer. 150
Check. 150 x 12 = 1800. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 9 sides splits into 7 triangles.9 - 2 = 7
- Multiply by 180. Each triangle has angles of 180 degrees.7 x 180 = 1260
- Write the answer. 1260
Check. 7 triangles x 180 = 1260. ✓
Method. Count the edges: the lines where two faces meet.
- Picture the shape. A triangular prism has 5 faces, 9 edges and 6 vertices.faces 5, edges 9, vertices 6
- Count the edges. An edge is where two faces meet.9 edges
- Write the answer. 9
Check. 5 faces + 6 vertices - 9 edges = 2. ✓
Watch out. Edges are not faces. A cube has 12 edges, not 6.
Method. Use diameter = 2 x radius.
- Write the rule. Radius = diameter / 2.6 / 2
- Divide. The radius is half the diameter.3
- Write the answer. 3
Check. diameter = 2 x radius, so 6 = 2 x 3 or 6. ✓
Watch out. The radius is half the diameter. Do not double it when you need the radius.
Method. Compare a square and a rectangle.
- Compare. A square has four right angles, like a rectangle.four right angles
- Conclude. So a square is a special rectangle, and Sam is wrong.Sam is wrong
- Write the answer. a square has four right angles and opposite sides equal
Check. A square is a special rectangle. ✓
Method. Find an obtuse angle that is less than 120 degrees.
- Pick an obtuse angle. An obtuse angle is between 90 and 180 degrees.between 90 and 180
- Find a counter-example. 100 degrees is obtuse and is less than 120.100 degrees
- Write the answer. An angle of 100 degrees is obtuse but is not more than 120 degrees.
Check. 90 < 100 < 120. ✓
Method. Use the exterior angle, 180 - 108 = 72, and 360 / 72.
- Exterior angle. 180 - 108 = 72 degrees.72
- Number of sides. 360 / 72 = 5 sides.5
- Write the answer. 5
Check. 5 x 108 = 540 = (5 - 2) x 180. ✓