Angles, polygons and constructions
By the end of this chapter you will be able to
- Recognise and use angle facts at a point, on a straight line and vertically opposite, and solve for a missing angle
- Use alternate and corresponding angles with parallel lines
- Use the angle sum of a triangle and a quadrilateral, and the properties of isosceles and equilateral triangles
- Work out the interior and exterior angles of regular polygons, and the angle sum of any polygon
- Describe ruler and compass constructions, and label triangles ABC
You should already know
- Measuring and drawing angles, and the names of acute, right, obtuse and reflex angles (Year 6, Chapter 7)
- Angles in a triangle and quadrilateral, and angles at a point (Year 5 and Year 6, geometry chapters)
- Solving equations (Chapter 6)
Where this leads
- Congruence and similarity of triangles (Chapter 10)
- Pythagoras and trigonometry in right-angled triangles (Chapter 11, and Year 8)
National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: G3 draw and measure line segments and angles, and interpret scale drawings, G4 ruler and compass constructions: perpendicular bisector, perpendicular from or at a point, angle bisector, G5 points, lines, parallel and perpendicular lines, right angles, regular and symmetric polygons, conventional terms, G6 labelling conventions for triangles (ABC) and the sides and angles opposite them, G7 derive and illustrate properties of triangles, quadrilaterals and plane figures, G10 angles at a point, on a straight line and vertically opposite angles, G11 parallel lines: alternate and corresponding angles, G12 the angle sum of a triangle and of any polygon; the properties of regular polygons.
Exercise P Do you remember?
Non-calculatorMethod. A straight line is a half turn.
- Think about the angle. A straight line makes a half turn.180°
- Write the answer. Check it against the question and give the unit.180
Check. Two right angles make a straight line: 2 × 90 = 180. ✓
Watch out. A straight line is 180°.
Method. A right angle is a quarter turn.
- Think about the turn. A right angle is a quarter of a full turn.360 ÷ 4 = 90
- Write the answer. Check it against the question and give the unit.90
Check. 4 × 90 = 360. ✓
Watch out. A right angle is 90°.
Method. A full turn is a complete circle.
- Think about the circle. A full turn goes all the way round.360°
- Write the answer. Check it against the question and give the unit.360
Check. 4 right angles make 360. ✓
Watch out. A full turn is 360°.
Method. Compare the angle with 90°.
- Compare. An obtuse angle is between 90° and 180°.90 < 120 < 180
- Write the answer. Check it against the question and give the unit.obtuse
Check. 120 is more than 90 and less than 180. ✓
Watch out. Acute is less than 90°; obtuse is between 90° and 180°.
Method. Use the name for each number of sides.
- Count the sides. Five sides make a pentagon.5 sides
- Write the answer. Check it against the question and give the unit.pentagon
Check. Penta means five. ✓
Watch out. Hexagon is six sides; pentagon is five.
Angles at a point, on a line and vertically opposite
Angles at a point add up to 360°. Angles on a straight line add up to 180°. When two lines cross, the angles opposite each other are vertically opposite, and they are equal.
Key ideas
- Angles at a point add up to 360°.
- Angles on a straight line add up to 180°.
- Vertically opposite angles are equal.
- Angles are measured in degrees (°). A right angle is 90°.
Method. Angles on a straight line add up to 180°.
- Write the equation. The two angles make a straight line.62 + x = 180
- Subtract 62. This leaves x on its own.x = 180 − 62 = 118
- Write the answer. Check it against the question and give the unit.118°
Check. 62 + 118 = 180. ✓
Watch out. Take the known angle away from 180.
Method. Angles at a point add up to 360°.
- Write the equation. Three angles meet at a point.125 + 85 + x = 360
- Add the two known angles. Add 125 and 85.210 + x = 360
- Subtract 210. This gives x.x = 150
- Write the answer. Check it against the question and give the unit.150°
Check. 125 + 85 + 150 = 360. ✓
Watch out. Add the known angles first, then subtract from 360.
Method. Vertically opposite angles are equal.
- Identify the opposite angle. The angle opposite the 47° angle is vertically opposite.opposite angle = 47
- Check with the angle next to it. The angle next to 47° on the straight line is 180 − 47 = 133, which is not the answer.180 − 47 = 133
- Write the answer. Check it against the question and give the unit.47°
Check. The angles on a straight line: 47 + 133 = 180. ✓
Watch out. Opposite angles are equal; the angles next to each other are not.
Watch out
Watch out: an angle at a point is 360°, not 180°. Only angles on a straight line add up to 180°.
Exam tip
Grown-up tip. Ask the pupil to measure one angle with a protractor and check it against the sum: a line of 180° split into two parts.
Exercise 9A Angles at a point, on a line and vertically opposite
Non-calculatorMethod. Angles on a line add up to 180°.
- Write. 65 + x = 180.65 + x = 180
- Subtract. x = 180 − 65.x = 115
- Write the answer. Check it against the question and give the unit.115°
Check. 65 + 115 = 180. ✓
Watch out. Subtract from 180.
Method. Angles on a line add up to 180°.
- Write. 140 + y = 180.140 + y = 180
- Subtract. y = 180 − 140.y = 40
- Write the answer. Check it against the question and give the unit.40°
Check. 140 + 40 = 180. ✓
Watch out. Subtract from 180.
Method. Angles at a point add up to 360°.
- Write. 90 + 120 + z = 360.210 + z = 360
- Subtract. z = 360 − 210.z = 150
- Write the answer. Check it against the question and give the unit.150°
Check. 90 + 120 + 150 = 360. ✓
Watch out. Use 360, not 180.
Method. Angles at a point add up to 360°.
- Add the known angles. 75 + 80 + 105 = 260.260
- Subtract. x = 360 − 260.x = 100
- Write the answer. Check it against the question and give the unit.100°
Check. 260 + 100 = 360. ✓
Watch out. Add all the known angles.
Method. Vertically opposite angles are equal.
- Use the rule. Vertically opposite angles are equal.58
- Write the answer. Check it against the question and give the unit.58°
Check. The angle next to it is 122°, which is 180 − 58. ✓
Watch out. Opposite, not next to.
Method. Vertically opposite angles are equal.
- Write. 2x + 30 = x + 60.2x + 30 = x + 60
- Solve. Subtract x, then subtract 30.x = 30
- Write the answer. Check it against the question and give the unit.x = 30
Check. 2 × 30 + 30 = 90 = 30 + 60. ✓
Watch out. Subtract x from both sides first.
Method. Angles on a line add up to 180°.
- Write. 3x + 2x = 180.5x = 180
- Divide by 5. x = 36.x = 36
- Find the angles. 3 × 36 = 108 and 2 × 36 = 72.108° and 72°
- Write the answer. Check it against the question and give the unit.108° and 72°
Check. 108 + 72 = 180. ✓
Watch out. Find x first, then both angles.
Method. Compare each with 90° and 180°.
- Compare. 35 is less than 90; 90 is a right angle; 150 is between 90 and 180; 200 is more than 180.acute, right, obtuse, reflex
- Write the answer. Check it against the question and give the unit.acute, right, obtuse, reflex
Check. Each angle is in the right group. ✓
Watch out. Reflex is more than 180°.
Method. A right angle is 90°.
- Read the clock. At 3 o'clock the hands make a right angle.90
- Write the answer. Check it against the question and give the unit.90°
Check. A quarter turn is 90°. ✓
Watch out. Three o'clock is a quarter of a turn from twelve.
Method. On a line add to 180°.
- Subtract. 180 − 50.x = 130
- Write the answer. Check it against the question and give the unit.130°
Check. 50 + 130 = 180. ✓
Watch out. Subtract from 180.
Method. Vertically opposite angles are equal.
- Equal angles. 2y = 80.y = 40 for 2y = 80
- Write the answer. Check it against the question and give the unit.40
Check. 2 × 40 = 80. ✓
Watch out. Divide by 2.
Method. At a point they add to 360°.
- Write. x + x + 160 = 360.2x = 200
- Divide. x = 100.x = 100
- Write the answer. Check it against the question and give the unit.100°
Check. 100 + 100 + 160 = 360. ✓
Watch out. Two x terms make 2x.
Method. Angles on a straight line add up to 180°.
- Write. x + (x + 10) + (x + 20) = 180.3x + 30 = 180
- Solve. Subtract 30, then divide by 3.x = 50
- Find the angles. 50, 60 and 70.50°, 60° and 70°
- Write the answer. Check it against the question and give the unit.50°, 60° and 70°
Check. 50 + 60 + 70 = 180. ✓
Watch out. Write the three angles as x and then add 10 and 20.
Method. Angles on a straight line add up to 180°.
- Add the angles. Add the two given angles.100 + 90 = 190
- Write the answer. Check it against the question and give the unit.No, the angles add to 190°, not 180°
Check. 190 is not 180, so Sam is wrong. ✓
Watch out. A straight line is always 180°.
Parallel lines: alternate and corresponding angles
When a transversal crosses two parallel lines, the corresponding angles (in the same position at each crossing) are equal. The alternate angles (on opposite sides of the transversal, between the parallel lines) are also equal.
Key ideas
- Parallel lines are marked with arrows and never meet.
- Corresponding angles (F-shape, same position) are equal.
- Alternate angles (Z-shape, opposite sides) are equal.
- Angles on a straight line still add up to 180°.
Method. Alternate angles are equal.
- Identify the alternate angle. The angle on the other side of the transversal, between the parallel lines.alternate angle
- Use the rule. Alternate angles are equal.65 = 65
- Write the answer. Check it against the question and give the unit.65°
Check. Alternate angles lie on a Z shape and are equal. ✓
Watch out. Alternate angles are equal only for parallel lines.
Method. Alternate angles are equal.
- Write. The two alternate angles are equal.3x = 75
- Divide by 3. This gives x.x = 25
- Write the answer. Check it against the question and give the unit.x = 25
Check. 3 × 25 = 75. ✓
Watch out. Equal angles give equal expressions.
Method. Corresponding angles are equal.
- Write. The two corresponding angles are equal.y = 2y − 40
- Solve. Subtract y, then add 40.y = 40
- Write the answer. Check it against the question and give the unit.y = 40
Check. 2 × 40 − 40 = 40. ✓
Watch out. Check the angle, not just the letter.
Watch out
Watch out: the angles are equal only when the lines are parallel. The marking with arrows tells you the lines are parallel.
Exam tip
Grown-up tip. Ask the pupil to trace the letter F for corresponding and Z for alternate, to find the matching angles.
Exercise 9B Parallel lines and transversals
Non-calculatorMethod. Corresponding angles are equal.
- Use the rule. Corresponding angles are equal.65
- Write the answer. Check it against the question and give the unit.65°
Check. 65 = 65. ✓
Watch out. Corresponding means in the same position.
Method. Alternate angles are equal.
- Use the rule. Alternate angles are equal.65
- Write the answer. Check it against the question and give the unit.65°
Check. Z shape: 65 = 65. ✓
Watch out. Alternate means on opposite sides.
Method. Corresponding angles are equal.
- Use the rule. x and 112 are equal.x = 112
- Write the answer. Check it against the question and give the unit.112°
Check. x = 112 by the corresponding rule. ✓
Watch out. Equal, not supplementary.
Method. Angles on a straight line add up to 180°.
- Subtract. The angle next to 58 on a straight line is 180 − 58.180 − 58 = 122
- Write the answer. Check it against the question and give the unit.122°
Check. 58 + 122 = 180. ✓
Watch out. Subtract from 180.
Method. Corresponding angles are equal, and so are alternate angles.
- Find the other angle. Each parallel line has the same angle, so the alternate angle is 74.74
- Write the answer. Check it against the question and give the unit.74°
Check. The corresponding and alternate angles are equal. ✓
Watch out. Both are equal to the same angle.
Method. Alternate angles are equal.
- Write. 3x = 75.3x = 75
- Divide. x = 25.x = 25
- Write the answer. Check it against the question and give the unit.x = 25
Check. 3 × 25 = 75. ✓
Watch out. Divide by 3.
Method. Corresponding angles are equal.
- Write. y = 2y − 40.y = 2y − 40
- Solve. Subtract y, then add 40.y = 40
- Write the answer. Check it against the question and give the unit.y = 40
Check. 2 × 40 − 40 = 40. ✓
Watch out. Subtract y from both sides first.
Method. Alternate angles are equal.
- Write. 2x + 10 = 4x − 10.2x + 10 = 4x − 10
- Solve. Subtract 2x, then add 10, then divide by 2.x = 10
- Find the angle. 2 × 10 + 10.30
- Write the answer. Check it against the question and give the unit.x = 10, angle 30°
Check. 4 × 10 − 10 = 30. ✓
Watch out. Find the angle by substituting x.
Method. Alternate angles are equal.
- Use the rule. Alternate angles are equal.47
- Write the answer. Check it against the question and give the unit.47°
Check. 47 = 47. ✓
Watch out. Equal.
Method. Corresponding angles are equal.
- Use the rule. Corresponding angles are equal.102
- Write the answer. Check it against the question and give the unit.102°
Check. 102 = 102. ✓
Watch out. Equal.
Method. On a straight line add to 180°.
- Subtract. 180 − 102 = 78.78
- Write the answer. Check it against the question and give the unit.78°
Check. 102 + 78 = 180. ✓
Watch out. Subtract from 180.
Method. Angles on a straight line add up to 180°.
- Subtract. The angle on the other side is 180 − 58.122
- Write the answer. Check it against the question and give the unit.122°
Check. 58 + 122 = 180. ✓
Watch out. Use the straight line rule.
Method. Think about the direction of the two lines.
- Look at the lines. Parallel lines point in the same direction.same direction
- Compare the angles. The transversal makes the same angle with each line, on opposite sides.Z shape: equal
- Write the answer. Check it against the question and give the unit.Parallel lines make equal alternate angles
Check. The angles are the same but on opposite sides. ✓
Watch out. The rule depends on the lines being parallel.
Angles in triangles and quadrilaterals
The three angles of any triangle add up to 180°. The four angles of any quadrilateral add up to 360°, because a quadrilateral can be split into two triangles.
An isosceles triangle has two equal sides, and its two base angles are equal. An equilateral triangle has three equal sides and three equal angles of 60°.
Key ideas
- The angles in a triangle add up to 180°.
- The angles in a quadrilateral add up to 360°.
- Isosceles: two equal sides and two equal base angles. Equilateral: all angles 60°.
- A right-angled triangle has one angle of 90°, so the other two add up to 90°.
Method. Angles in a triangle add up to 180°.
- Write the equation. The three angles add to 180.50 + 60 + x = 180
- Add and subtract. Add 50 and 60, then subtract from 180.110 + x = 180, x = 70
- Write the answer. Check it against the question and give the unit.70°
Check. 50 + 60 + 70 = 180. ✓
Watch out. Add the known angles first.
Method. The two base angles are equal and add up to 180 − 40.
- Find the two base angles together. Take the top angle from 180.180 − 40 = 140
- Share equally. The two base angles are equal, so divide by 2.140 ÷ 2 = 70
- Write the answer. Check it against the question and give the unit.70° each
Check. 70 + 70 + 40 = 180. ✓
Watch out. The base angles are equal, so divide by 2.
Method. Angles in a quadrilateral add up to 360°.
- Add the known angles. 80 + 95 + 110 = 285.285
- Subtract from 360. x = 360 − 285.x = 75
- Write the answer. Check it against the question and give the unit.75°
Check. 285 + 75 = 360. ✓
Watch out. Use 360, not 180, for four angles.
Watch out
Watch out: a quadrilateral has 360° of angles, not 180°. Only triangles have a sum of 180°.
Exam tip
Grown-up tip. Ask the pupil to tear the corners off a paper triangle and put them together on a straight line, to see the 180° sum.
Exercise 9C Angles in triangles and quadrilaterals
Non-calculatorMethod. Angles in a triangle add up to 180°.
- Add the known angles. 35 + 80 = 115.115
- Subtract. x = 180 − 115.x = 65
- Write the answer. Check it against the question and give the unit.65°
Check. 35 + 80 + 65 = 180. ✓
Watch out. Subtract from 180.
Method. The angles add to 180, and one is 90.
- Subtract the right angle. 180 − 90 = 90 left for the other two.90
- Subtract the 37°. The other acute angle is 90 − 37.90 − 37 = 53
- Write the answer. Check it against the question and give the unit.53°
Check. 90 + 37 + 53 = 180. ✓
Watch out. Two angles share the 90 that is left.
Method. An equilateral triangle has three equal angles.
- Share. Three equal angles share 180.180 ÷ 3 = 60
- Write the answer. Check it against the question and give the unit.60°
Check. 60 + 60 + 60 = 180. ✓
Watch out. Divide by 3.
Method. Base angles are equal.
- Remove the top angle. The two base angles add up to 150.180 − 30 = 150
- Share equally. Divide by 2.75
- Write the answer. Check it against the question and give the unit.75° each
Check. 75 + 75 + 30 = 180. ✓
Watch out. Divide by 2.
Method. Angles in a triangle add up to 180.
- Add the base angles. 65 + 65 = 130.130
- Subtract. Top angle = 180 − 130.50
- Write the answer. Check it against the question and give the unit.50°
Check. 65 + 65 + 50 = 180. ✓
Watch out. Use the sum of 180.
Method. Angles in a quadrilateral add up to 360.
- Add the known angles. 90 + 90 + 60 = 240.240
- Subtract. x = 360 − 240.120
- Write the answer. Check it against the question and give the unit.120°
Check. 240 + 120 = 360. ✓
Watch out. Four angles add to 360.
Method. Angles in a quadrilateral add up to 360.
- Collect the terms. 2x + x + x + x = 5x.5x = 360
- Divide. x = 72.x = 72
- Write the answer. Check it against the question and give the unit.x = 72
Check. 144 + 72 + 72 + 72 = 360. ✓
Watch out. Collect the x terms.
Method. Adjacent angles in a parallelogram add up to 180°.
- Write. x + 2x = 180.3x = 180
- Divide. x = 60.x = 60
- Write the answer. Check it against the question and give the unit.60°
Check. 60 + 120 = 180. ✓
Watch out. Adjacent angles of a parallelogram sum to 180°.
Method. Angles in a triangle add up to 180.
- Subtract. 180 − 90 − 45.45
- Write the answer. Check it against the question and give the unit.45°
Check. 90 + 45 + 45 = 180. ✓
Watch out. Subtract both known angles.
Method. Triangle angles add to 180.
- Subtract. 180 − 120.60
- Write the answer. Check it against the question and give the unit.60°
Check. 20 + 100 + 60 = 180. ✓
Watch out. Add the two known angles first.
Method. Triangle angles add to 180.
- Subtract. 180 − 135.45
- Write the answer. Check it against the question and give the unit.45°
Check. 90 + 45 + 45 = 180. ✓
Watch out. Add the known angles first.
Method. Triangle angles add to 180.
- Subtract. 180 − 140.40
- Write the answer. Check it against the question and give the unit.40°
Check. 70 + 70 + 40 = 180. ✓
Watch out. Add the known angles first.
Method. Share 180 into 6 parts.
- Total parts. 1 + 2 + 3 = 6.6
- One part. 180 ÷ 6 = 30.30
- Largest. The largest is 3 parts: 3 × 30.90
- Write the answer. Check it against the question and give the unit.90°
Check. 30 + 60 + 90 = 180. ✓
Watch out. Share the 180 in the ratio.
Method. Add the three angles.
- Add. 90 + 50 + 45 = 185.185
- Write the answer. Check it against the question and give the unit.No, the angles add to 185°
Check. 185 is not 180. ✓
Watch out. A triangle always adds to 180°.
Polygons and regular polygons
A polygon is a closed shape with straight sides. A regular polygon has all sides equal and all angles equal. The sum of the interior (inside) angles of a polygon with n sides is (n − 2) × 180°.
For a regular polygon, each exterior angle is 360° ÷ n, and each interior angle is 180° minus the exterior angle.
Key ideas
- Polygon with n sides: sum of interior angles = (n − 2) × 180°.
- Regular polygon: each exterior angle = 360° ÷ n.
- Each interior angle = 180° − exterior angle (they lie on a straight line).
- Names: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.
Method. Use (n − 2) × 180 with n = 5.
- Write the formula. A pentagon has 5 sides.(5 − 2) × 180
- Work out. Subtract 2, then multiply by 180.3 × 180 = 540
- Write the answer. Check it against the question and give the unit.540°
Check. The triangle has 180; three triangles fit in a pentagon: 3 × 180 = 540. ✓
Watch out. Subtract 2 from the number of sides first.
Method. Find the sum, then divide by the number of angles.
- Find the sum. A hexagon has 6 sides: (6 − 2) × 180 = 720.720
- Share equally. All six angles are equal in a regular hexagon.720 ÷ 6 = 120
- Write the answer. Check it against the question and give the unit.120°
Check. 6 × 120 = 720. ✓
Watch out. Share the sum equally for a regular polygon only.
Method. Use exterior angle = 360 ÷ n.
- Write the equation. 360 ÷ n = 30.360 ÷ 30 = n
- Solve. Divide 360 by 30.n = 12
- Write the answer. Check it against the question and give the unit.12 sides
Check. Each exterior angle is 360 ÷ 12 = 30. ✓
Watch out. Exterior angles add up to 360.
Watch out
Watch out: the sum of the interior angles is NOT 360° for every polygon. It is 360° only for quadrilaterals; a pentagon has 540°.
Exam tip
Grown-up tip. Ask the pupil to split a polygon into triangles from one corner, and to count the triangles: n − 2 of them.
Exercise 9D Polygons
Non-calculatorMethod. (5 − 2) × 180.
- Calculate. 3 × 180.540
- Write the answer. Check it against the question and give the unit.540°
Check. 540 ÷ 180 = 3. ✓
Watch out. Use n − 2 = 3 triangles.
Method. (8 − 2) × 180.
- Calculate. 6 × 180.1080
- Write the answer. Check it against the question and give the unit.1080°
Check. 1080 ÷ 180 = 6. ✓
Watch out. An octagon has 8 sides.
Method. Sum = 540, then share equally.
- Find the sum. (5 − 2) × 180 = 540.540
- Share. 540 ÷ 5 = 108.108
- Write the answer. Check it against the question and give the unit.108°
Check. 5 × 108 = 540. ✓
Watch out. Regular means equal angles.
Method. Exterior angle = 360 ÷ 8.
- Divide. 360 ÷ 8.45
- Write the answer. Check it against the question and give the unit.45°
Check. 8 × 45 = 360. ✓
Watch out. Exterior angles total 360.
Method. 360 ÷ 30.
- Divide. 360 ÷ 30.12
- Write the answer. Check it against the question and give the unit.12 sides
Check. 12 × 30 = 360. ✓
Watch out. Divide 360 by the exterior angle.
Method. Sum = (10 − 2) × 180 = 1440, then share.
- Find the sum. 8 × 180 = 1440.1440
- Share. 1440 ÷ 10 = 144.144
- Write the answer. Check it against the question and give the unit.144°
Check. 10 × 144 = 1440. ✓
Watch out. Use 10 sides in the formula.
Method. Count the sides.
- Use the name. Hepta means seven.heptagon
- Write the answer. Check it against the question and give the unit.heptagon
Check. 7 sides make a heptagon. ✓
Watch out. Heptagon has seven sides.
Method. Exterior = 180 − 150 = 30, then 360 ÷ 30.
- Exterior angle. The exterior angle is 180 − 150 = 30.30
- Sides. 360 ÷ 30 = 12.12
- Write the answer. Check it against the question and give the unit.12 sides
Check. 12 × 30 = 360. ✓
Watch out. Find the exterior angle first.
Method. (n − 2) × 180 = 900.
- Divide by 180. n − 2 = 900 ÷ 180 = 5.5
- Add 2. n = 5 + 2.7
- Write the answer. Check it against the question and give the unit.7 sides
Check. (7 − 2) × 180 = 900. ✓
Watch out. Add 2 to the answer of the division.
Method. All sides and angles are equal.
- Name it. A four-sided shape with equal sides and right angles.square
- Write the answer. Check it against the question and give the unit.square
Check. A square is a regular quadrilateral. ✓
Watch out. A rectangle has equal angles but sides that are not all equal.
Method. All sides equal.
- Name it. Equal sides make an equilateral triangle.equilateral
- Write the answer. Check it against the question and give the unit.equilateral triangle
Check. Each angle is 60°. ✓
Watch out. Equilateral means equal sides.
Method. Equal sides, but angles can differ.
- Name it. A rhombus has four equal sides but not right angles.rhombus
- Write the answer. Check it against the question and give the unit.rhombus
Check. A square would have all angles equal. ✓
Watch out. A rhombus is not regular.
Method. Use the exterior angle.
- Exterior angle. The exterior angle is 180 − 100 = 80.80
- Sides. 360 ÷ 80 = 4.5.4.5
- Write the answer. Check it against the question and give the unit.4.5 sides is not possible
Check. A regular polygon has a whole number of sides. ✓
Watch out. The number of sides must be a whole number.
Constructions and labelling triangles
Label a triangle with capital letters at its corners, ABC. The side opposite corner A is labelled a (side BC), and the angle at A is ∠A. The perpendicular bisector of a line cuts it in half at right angles. A perpendicular from a point to a line is the shortest distance to that line.
Key ideas
- Label a triangle ABC: the side opposite A is a, opposite B is b, opposite C is c.
- Perpendicular bisector: a line at 90° through the midpoint of a line segment.
- Angle bisector: a line that divides an angle into two equal parts.
- The shortest distance from a point to a line is along the perpendicular.
Method. Find the side that does not touch A.
- Look at the corners. The side opposite A goes from B to C.BC
- Check. The sides AB and AC both touch A, so BC is the one opposite.AB and AC touch A
- Write the answer. Check it against the question and give the unit.BC
Check. BC is the side with no corner A. ✓
Watch out. The opposite side is the one that does not touch the corner.
Method. The midpoint is halfway along.
- Halve the length. The midpoint is half of the length from A.8 ÷ 2 = 4
- Say where. It is 4 cm from A, and 4 cm from B.4 cm from A
- Write the answer. Check it against the question and give the unit.4 cm from A
Check. 4 + 4 = 8. ✓
Watch out. The midpoint is the same distance from both ends.
Method. Multiply by the scale.
- Read the scale. Each cm on the plan is 20 cm in real life.1 cm : 20 cm
- Multiply. Six lots of 20 cm.6 × 20 = 120
- Write the answer. Check it against the question and give the unit.120 cm
Check. 120 ÷ 20 = 6. ✓
Watch out. Use the scale factor.
Watch out
Watch out: the perpendicular bisector is at 90° to the line, not at 45°. Use a set square to check the right angle.
Exam tip
Grown-up tip. Ask the pupil to follow each construction with a ruler and compasses on paper, and to check the result with a protractor.
Exercise 9E Constructions and labels
Non-calculatorMethod. The side opposite a corner is the one that does not touch that corner.
- Look at the corners. B is not on side AC.AC
- Write the answer. Check it against the question and give the unit.AC
Check. AC does not touch B. ✓
Watch out. Use the corner letter.
Method. The side opposite C does not touch C.
- Look at the corners. The side AB does not touch C.AB
- Write the answer. Check it against the question and give the unit.AB
Check. AB is opposite C. ✓
Watch out. Check the corners.
Method. The midpoint is halfway.
- Halve the length. 10 ÷ 2 = 5.5 cm from A
- Write the answer. Check it against the question and give the unit.5 cm from A
Check. 5 + 5 = 10. ✓
Watch out. Halve the length.
Method. Use compasses, and draw arcs of the same radius from both ends.
- Set the compasses. Use a radius bigger than half of AB.radius > half of AB
- Draw arcs. Draw arcs above and below the line from A and from B.arcs from A and B
- Join the crossing points. The line through the two crossings is the bisector.line through the crossings
- Write the answer. Check it against the question and give the unit.Set the compasses, draw arcs from both ends, and join the crossings
Check. The line is at 90° and passes through the midpoint. ✓
Watch out. Keep the compass setting the same for all arcs.
Method. Use equal arcs on both arms.
- Draw the first arc. Draw an arc across both arms from the vertex.two points on the arms
- Draw equal arcs. From each point, draw arcs of the same radius that cross inside the angle.crossing point
- Draw the bisector. Join the vertex to the crossing.the bisector line
- Write the answer. Check it against the question and give the unit.Draw equal arcs from both arms, then join the vertex to the crossing
Check. The two halves are equal. ✓
Watch out. Keep the compass setting the same.
Method. Multiply by 50 to get cm, then change to m.
- Multiply. 4 × 50 = 200 cm.200 cm
- Change to m. 200 cm = 2 m.2
- Write the answer. Check it against the question and give the unit.2 m
Check. 200 ÷ 100 = 2. ✓
Watch out. Change cm to m at the end.
Method. Use a set square to draw a line at 90° from P to AB.
- Draw the perpendicular. Draw a line from P meeting AB at 90°.perpendicular from P
- Measure it. The length of this line is the shortest distance.the shortest distance
- Write the answer. Check it against the question and give the unit.The perpendicular from P to AB
Check. The perpendicular is always the shortest distance. ✓
Watch out. Any other line from P is longer.
Method. An equilateral triangle has three equal angles.
- Share. 180 ÷ 3 = 60.60
- Write the answer. Check it against the question and give the unit.60°
Check. 60 + 60 + 60 = 180. ✓
Watch out. Equal angles share 180.
Method. Think about the arcs from the arms.
- Equal arcs. Both arcs have the same radius from the vertex.same radius
- Equal triangles. The triangles formed share sides and are equal.congruent triangles
- Write the answer. Check it against the question and give the unit.The arcs have the same radius, so the angles are equal
Check. Equal sides make equal angles. ✓
Watch out. Equal radius gives equal triangles.
Method. The side opposite A is BC.
- Look. It does not touch A.BC
- Write the answer. Check it against the question and give the unit.BC
Check. BC does not touch A. ✓
Watch out. Use the corner letters.
Method. The angle at C is between the sides CA and CB.
- Name it. Use three letters with C in the middle.ACB
- Write the answer. Check it against the question and give the unit.angle ACB
Check. The middle letter is the corner. ✓
Watch out. The corner letter goes in the middle.
Chapter summary
| Skill | Method |
|---|---|
| Angle facts | At a point: 360°; on a straight line: 180°; vertically opposite: equal |
| Parallel lines | Corresponding and alternate angles are equal |
| Triangles and quadrilaterals | Triangle angles add to 180°; quadrilateral angles add to 360°; isosceles base angles equal |
| Polygons | Interior sum (n − 2) × 180°; regular exterior angle 360° ÷ n |
| Constructions | Perpendicular bisector; angle bisector; perpendicular from a point is the shortest distance |
| Labelling | Triangle ABC: side a is opposite A, and so on |
Can you…?
- I can find missing angles at a point, on a straight line and vertically opposite
- I can use alternate and corresponding angles with parallel lines
- I can use the angle sums of triangles and quadrilaterals
- I can work out the angle sum and the angles of regular polygons
- I can describe the constructions of a perpendicular bisector and an angle bisector
- I can label a triangle ABC and name the sides opposite its angles
Key terms
- transversal
- a line that crosses two or more other lines
- alternate angles
- equal angles on opposite sides of a transversal, between two parallel lines
- corresponding angles
- equal angles in the same position at two crossings of a transversal
- polygon
- a closed shape with straight sides
- regular polygon
- a polygon with all sides equal and all angles equal
- perpendicular bisector
- a line at 90° that cuts a line segment in half
- vertically opposite
- the angles opposite each other where two lines cross; they are equal
Exercise 9Z Check-up
Non-calculatorMethod. On a straight line: 180.
- Subtract. 180 − 72.108
- Write the answer. Check it against the question and give the unit.108°
Check. 72 + 108 = 180. ✓
Watch out. Subtract from 180.
Method. At a point: 360.
- Add the known angles. 90 + 60 + 110 = 260.260
- Subtract. 360 − 260.100
- Write the answer. Check it against the question and give the unit.100°
Check. 260 + 100 = 360. ✓
Watch out. Add all the known angles.
Method. Vertically opposite angles are equal.
- Use the rule. They are equal.48
- Write the answer. Check it against the question and give the unit.48°
Check. The next angle is 132, and 48 + 132 = 180. ✓
Watch out. Opposite means equal.
Method. Triangle: 180.
- Add the known angles. 40 + 75 = 115.115
- Subtract. 180 − 115.65
- Write the answer. Check it against the question and give the unit.65°
Check. 40 + 75 + 65 = 180. ✓
Watch out. Add, then subtract from 180.
Method. Alternate angles are equal.
- Write. 5x = 100.5x = 100
- Divide. x = 20.x = 20
- Write the answer. Check it against the question and give the unit.x = 20
Check. 5 × 20 = 100. ✓
Watch out. Divide by 5.
Method. (6 − 2) × 180.
- Calculate. 4 × 180.720
- Write the answer. Check it against the question and give the unit.720°
Check. 720 ÷ 180 = 4. ✓
Watch out. Hexagon has 6 sides.
Method. Exterior = 360 ÷ 6.
- Divide. 360 ÷ 6.60
- Write the answer. Check it against the question and give the unit.60°
Check. 6 × 60 = 360. ✓
Watch out. Use 6 sides.
Method. Quadrilateral: 360.
- Add the known angles. 110 + 80 + 95 = 285.285
- Subtract. 360 − 285.75
- Write the answer. Check it against the question and give the unit.75°
Check. 285 + 75 = 360. ✓
Watch out. Four angles total 360.
Method. Triangle: 180.
- Add the base angles. 50 + 50 = 100.100
- Subtract. 180 − 100.80
- Write the answer. Check it against the question and give the unit.80°
Check. 50 + 50 + 80 = 180. ✓
Watch out. Use the sum of 180.
Method. 360 ÷ 45.
- Divide. 360 ÷ 45.8
- Write the answer. Check it against the question and give the unit.8 sides
Check. 8 × 45 = 360. ✓
Watch out. Divide 360 by the exterior angle.
Total 22 marks. Allow about 22 minutes.
Exercise 9X Practice test questions
Non-calculatorMethod. On a straight line: 180.
- Subtract. 180 − 135.45
- Write the answer. Check it against the question and give the unit.45°
Check. 135 + 45 = 180. ✓
Watch out. Subtract from 180.
Method. (5 − 2) × 180.
- Calculate. 3 × 180.540
- Write the answer. Check it against the question and give the unit.540°
Check. 540 ÷ 180 = 3. ✓
Watch out. Use n − 2.
Method. Sum = 540, then share equally.
- Find the sum. 540 (from the pentagon sum).540
- Share. 540 ÷ 5 = 108.108
- Write the answer. Check it against the question and give the unit.108°
Check. 5 × 108 = 540. ✓
Watch out. Regular means equal angles.
Method. Alternate angles are equal.
- Write. 3x = 75.3x = 75
- Divide. x = 25.x = 25
- Write the answer. Check it against the question and give the unit.x = 25
Check. 3 × 25 = 75. ✓
Watch out. Equal angles.
Method. Quadrilateral: 360.
- Add the known angles. 80 + 95 + 110 = 285.285
- Subtract. 360 − 285.75
- Write the answer. Check it against the question and give the unit.75°
Check. 285 + 75 = 360. ✓
Watch out. Four angles total 360.
Method. Exterior = 360 ÷ 10.
- Divide. 360 ÷ 10.36
- Write the answer. Check it against the question and give the unit.36°
Check. 10 × 36 = 360. ✓
Watch out. Use the number of sides.
Method. Exterior = 180 − 140 = 40, then 360 ÷ 40.
- Exterior angle. 180 − 140 = 40.40
- Sides. 360 ÷ 40 = 9.9
- Write the answer. Check it against the question and give the unit.9 sides
Check. 9 × 40 = 360. ✓
Watch out. Find the exterior angle first.
Method. Angles on a straight line add to 180.
- Write. x + x + 10 + x + 20 = 180.3x + 30 = 180
- Solve. Subtract 30, then divide by 3.x = 50
- Largest. The largest is x + 20 = 70.70
- Write the answer. Check it against the question and give the unit.70°
Check. 50 + 60 + 70 = 180. ✓
Watch out. Find x first.
Method. Add the angles.
- Add. 90 + 50 + 45 = 185.185
- Write the answer. Check it against the question and give the unit.The angles add to 185°, not 180°
Check. 185 is not 180. ✓
Watch out. A triangle always adds to 180°.
Method. Use the perpendicular from P.
- Draw. Use a set square to draw a line from P at 90° to AB.perpendicular from P
- Measure. The length of the perpendicular is the shortest distance.shortest distance
- Write the answer. Check it against the question and give the unit.Draw the perpendicular from P to AB and measure it
Check. Other lines from P to AB are longer. ✓
Watch out. The perpendicular is the shortest line from a point to a line.