Pythagoras and angles
By the end of this chapter you will be able to
- Use Pythagoras' theorem to find the longest side of a right-angled triangle, and give an exact answer
- Use Pythagoras' theorem to find a shorter side, by subtracting the squares
- Find missing angles in triangles, quadrilaterals and at a point, and explain each angle fact
- Identify corresponding, alternate and co-interior angles, and find angles in parallel lines
- Find the interior and exterior angles of regular polygons, and the number of sides from an angle
You should already know
- Squares and square roots, and square numbers (Year 7 and Year 8, Chapter 2)
- Angles on a straight line, at a point and in a triangle (Year 7)
- Parallel lines and the names of the angles they make (Year 7)
Where this leads
- Trigonometry in right-angled triangles, and the exact values of sine, cosine and tangent (Year 9 and GCSE)
- Three-dimensional Pythagoras and its uses in coordinates (Year 9)
- Circle theorems and proofs using angle facts (GCSE)
National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Find the hypotenuse of a right-angled triangle using Pythagoras' theorem, Find a shorter side of a right-angled triangle using Pythagoras' theorem, Use angles on a straight line, at a point, vertically opposite, in a triangle and in a quadrilateral, Use angles in parallel lines: corresponding, alternate and co-interior angles, Use the interior and exterior angles of polygons, and the angle sum of a polygon.
Exercise P Do you remember?
Non-calculatorMethod. Six squared means 6 × 6.
- Write the square. The index 2 means two 6s.6 × 6
- Multiply 6 by itself. Six sixes make 36.6 × 6 = 36
Check. √36 = 6.
Method. Find the number that multiplied by itself is 49.
- Think of square numbers. 7 × 7 = 49.7 × 7 = 49
- Write the root. 7
Check. 7 squared is 49.
Method. This is a fact about triangles.
- Recall the fact. The three angles of any triangle add up to 180°.3 angles = 180°
- Write the answer. 180°
Check. Check with a triangle that has angles 60, 60 and 60.
Method. A straight line is a half turn.
- Recall the fact. A straight line is half a turn, which is 180°.180°
- Write the answer. 180°
Check. A full turn is 360°, and half of that is 180°.
Method. Work out the power first, then the multiplication, then add.
- Work out the power. 32 = 9.32 = 9
- Multiply. 12 × 0 = 0.12 × 0 = 0
- Add. 5 + 0 + 9 = 14.14
Check. Order: powers, then multiplication, then addition.
Finding the hypotenuse with Pythagoras' theorem
In a right-angled triangle, the longest side is the hypotenuse. It is always opposite the right angle.
Pythagoras’ theorem says: c2 = a2 + b2, where c is the hypotenuse and a and b are the other two sides.
To find the hypotenuse: square the two shorter sides, add the squares, then take the square root.
Key ideas
- Pythagoras’ theorem only works for a right-angled triangle.
- The hypotenuse is opposite the right angle, and it is the longest side.
- To find the hypotenuse, add the squares of the two shorter sides, then take the square root.
- Give an exact answer, such as √13, if the square root is not a whole number.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 3 and 4.32 + 42 = 9 + 16
- Add. Add the two squares to find c2.c2 = 25
- Take the square root. c is the square root of the total.c = √25 = 5
- Write the answer. 5
Check. Check: 5 squared is 25, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 2 and 3.22 + 32 = 4 + 9
- Add. Add the two squares to find c2.c2 = 13
- Take the square root. c is the square root of the total.c = √13 = √13
- Write the answer. √13
Check. Check: (√13)2 = 13, and 22 + 32 = 13.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 6 and 6.62 + 62 = 36 + 36
- Add. Add the two squares to find c2.c2 = 72
- Take the square root. c is the square root of the total.c = √72 = 6√2
- Write the answer. 6√2
Check. Check: 6√2 squared is 36 × 2 = 72, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Watch out
Watch out: do not add the two shorter sides. 3 + 4 = 7, but 32 + 42 = 25, and √25 = 5.
Exam tip
Exam tip. Write the formula c2 = a2 + b2 first, then put in the numbers. Examiners often give a mark for the correct formula.
Exercise 9A Finding the hypotenuse
Non-calculatorMethod. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 3 and 4.32 + 42 = 9 + 16
- Add. Add the two squares to find c2.c2 = 25
- Take the square root. c is the square root of the total.c = √25 = 5
- Write the answer. 5
Check. Check: 5 squared is 25, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 5 and 12.52 + 122 = 25 + 144
- Add. Add the two squares to find c2.c2 = 169
- Take the square root. c is the square root of the total.c = √169 = 13
- Write the answer. 13
Check. Check: 13 squared is 169, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 6 and 8.62 + 82 = 36 + 64
- Add. Add the two squares to find c2.c2 = 100
- Take the square root. c is the square root of the total.c = √100 = 10
- Write the answer. 10
Check. Check: 10 squared is 100, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 2 and 3.22 + 32 = 4 + 9
- Add. Add the two squares to find c2.c2 = 13
- Take the square root. c is the square root of the total.c = √13 = √13
- Write the answer. √13
Check. Check: (√13)2 = 13, and 22 + 32 = 13.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 1 and 4.12 + 42 = 1 + 16
- Add. Add the two squares to find c2.c2 = 17
- Take the square root. c is the square root of the total.c = √17 = √17
- Write the answer. √17
Check. Check: (√17)2 = 17, and 12 + 42 = 17.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 2 and 5.22 + 52 = 4 + 25
- Add. Add the two squares to find c2.c2 = 29
- Take the square root. c is the square root of the total.c = √29 = √29
- Write the answer. √29
Check. Check: (√29)2 = 29, and 22 + 52 = 29.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 4 and 6.42 + 62 = 16 + 36
- Add. Add the two squares to find c2.c2 = 52
- Take the square root. c is the square root of the total.c = √52 = 2√13
- Write the answer. 2√13
Check. Check: 2√13 squared is 4 × 13 = 52, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 3 and 3.32 + 32 = 9 + 9
- Add. Add the two squares to find c2.c2 = 18
- Take the square root. c is the square root of the total.c = √18 = 3√2
- Write the answer. 3√2
Check. Check: 3√2 squared is 9 × 2 = 18, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 5 and 5.52 + 52 = 25 + 25
- Add. Add the two squares to find c2.c2 = 50
- Take the square root. c is the square root of the total.c = √50 = 5√2
- Write the answer. 5√2
Check. Check: 5√2 squared is 25 × 2 = 50, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 7 and 24.72 + 242 = 49 + 576
- Add. Add the two squares to find c2.c2 = 625
- Take the square root. c is the square root of the total.c = √625 = 25
- Write the answer. 25
Check. Check: 25 squared is 625, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 8 and 15.82 + 152 = 64 + 225
- Add. Add the two squares to find c2.c2 = 289
- Take the square root. c is the square root of the total.c = √289 = 17
- Write the answer. 17
Check. Check: 17 squared is 289, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 9 and 12.92 + 122 = 81 + 144
- Add. Add the two squares to find c2.c2 = 225
- Take the square root. c is the square root of the total.c = √225 = 15
- Write the answer. 15
Check. Check: 15 squared is 225, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 1 and 1.12 + 12 = 1 + 1
- Add. Add the two squares to find c2.c2 = 2
- Take the square root. c is the square root of the total.c = √2 = √2
- Write the answer. √2
Check. Check: (√2)2 = 2, and 12 + 12 = 2.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 6 and 6.62 + 62 = 36 + 36
- Add. Add the two squares to find c2.c2 = 72
- Take the square root. c is the square root of the total.c = √72 = 6√2
- Write the answer. 6√2
Check. Check: 6√2 squared is 36 × 2 = 72, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 12 and 5.122 + 52 = 144 + 25
- Add. Add the two squares to find c2.c2 = 169
- Take the square root. c is the square root of the total.c = √169 = 13
- Write the answer. 13
Check. Check: 13 squared is 169, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras’ theorem to check the hypotenuse.
- Write the rule. c2 = 32 + 42.c2 = 9 + 16
- Work out c2. 9 + 16 = 25.c2 = 25
- Take the square root. √25 = 5, not 7.c = 5
- Write the answer. Ali is not right.No
Check. The hypotenuse must be bigger than 4, but 7 is too big to fit as the longest side.
Method. Work out c2, then decide whether it is a square number.
- Work out c2. 22 + 22 = 4 + 4 = 8.c2 = 8
- Check for a square. 8 is not a square number, so its square root is not a whole number.8 is not a square number
- Write the answer. No
Check. √8 = 2√2, which is about 2.83.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 10 and 6, then subtract.102 − 62 = 100 − 36 = 64
- Take the square root. b is the square root of the result.b = √64 = 8
- Write the answer. 8
Check. Check: 102 = 62 + (8)2, which is 100 = 36 + 64.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 13 and 5, then subtract.132 − 52 = 169 − 25 = 144
- Take the square root. b is the square root of the result.b = √144 = 12
- Write the answer. 12
Check. Check: 132 = 52 + (12)2, which is 169 = 25 + 144.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 8 and 15.82 + 152 = 64 + 225
- Add. Add the two squares to find c2.c2 = 289
- Take the square root. c is the square root of the total.c = √289 = 17
- Write the answer. 17
Check. Check: 17 squared is 289, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 4 and 5.42 + 52 = 16 + 25
- Add. Add the two squares to find c2.c2 = 41
- Take the square root. c is the square root of the total.c = √41 = √41
- Write the answer. √41
Check. Check: (√41)2 = 41, and 42 + 52 = 41.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Finding a shorter side with Pythagoras' theorem
When a shorter side is missing, rearrange the rule: b2 = c2 − a2.
Subtract the square of the known shorter side from the square of the hypotenuse. Then take the square root.
Key ideas
- When a shorter side is missing: b2 = c2 − a2.
- The hypotenuse is always the biggest side, so the answer must be smaller than c.
- Subtract the squares; do not subtract the lengths.
- Give an exact answer, such as 2√10, if the number is not a square.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 13 and 5, then subtract.132 − 52 = 169 − 25 = 144
- Take the square root. b is the square root of the result.b = √144 = 12
- Write the answer. 12
Check. Check: 132 = 52 + (12)2, which is 169 = 25 + 144.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 10 and 6, then subtract.102 − 62 = 100 − 36 = 64
- Take the square root. b is the square root of the result.b = √64 = 8
- Write the answer. 8
Check. Check: 102 = 62 + (8)2, which is 100 = 36 + 64.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 7 and 3, then subtract.72 − 32 = 49 − 9 = 40
- Take the square root. b is the square root of the result.b = √40 = 2√10
- Write the answer. 2√10
Check. Check: 72 = 32 + (2√10)2, which is 49 = 9 + 40.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Watch out
Watch out: do not add the squares when a shorter side is missing. Subtract: 132 − 52, not 132 + 52.
Exam tip
Exam tip. Check your answer is smaller than the hypotenuse. If it is bigger, you have added instead of subtracted.
Exercise 9B Finding a shorter side
Non-calculatorMethod. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 13 and 5, then subtract.132 − 52 = 169 − 25 = 144
- Take the square root. b is the square root of the result.b = √144 = 12
- Write the answer. 12
Check. Check: 132 = 52 + (12)2, which is 169 = 25 + 144.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 10 and 6, then subtract.102 − 62 = 100 − 36 = 64
- Take the square root. b is the square root of the result.b = √64 = 8
- Write the answer. 8
Check. Check: 102 = 62 + (8)2, which is 100 = 36 + 64.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 17 and 8, then subtract.172 − 82 = 289 − 64 = 225
- Take the square root. b is the square root of the result.b = √225 = 15
- Write the answer. 15
Check. Check: 172 = 82 + (15)2, which is 289 = 64 + 225.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 7 and 3, then subtract.72 − 32 = 49 − 9 = 40
- Take the square root. b is the square root of the result.b = √40 = 2√10
- Write the answer. 2√10
Check. Check: 72 = 32 + (2√10)2, which is 49 = 9 + 40.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 6 and 4, then subtract.62 − 42 = 36 − 16 = 20
- Take the square root. b is the square root of the result.b = √20 = 2√5
- Write the answer. 2√5
Check. Check: 62 = 42 + (2√5)2, which is 36 = 16 + 20.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 15 and 9, then subtract.152 − 92 = 225 − 81 = 144
- Take the square root. b is the square root of the result.b = √144 = 12
- Write the answer. 12
Check. Check: 152 = 92 + (12)2, which is 225 = 81 + 144.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 5 and 2, then subtract.52 − 22 = 25 − 4 = 21
- Take the square root. b is the square root of the result.b = √21 = √21
- Write the answer. √21
Check. Check: 52 = 22 + (√21)2, which is 25 = 4 + 21.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 25 and 7, then subtract.252 − 72 = 625 − 49 = 576
- Take the square root. b is the square root of the result.b = √576 = 24
- Write the answer. 24
Check. Check: 252 = 72 + (24)2, which is 625 = 49 + 576.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 9 and 4, then subtract.92 − 42 = 81 − 16 = 65
- Take the square root. b is the square root of the result.b = √65 = √65
- Write the answer. √65
Check. Check: 92 = 42 + (√65)2, which is 81 = 16 + 65.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 13 and 12, then subtract.132 − 122 = 169 − 144 = 25
- Take the square root. b is the square root of the result.b = √25 = 5
- Write the answer. 5
Check. Check: 132 = 122 + (5)2, which is 169 = 144 + 25.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 12 and 5, then subtract.122 − 52 = 144 − 25 = 119
- Take the square root. b is the square root of the result.b = √119 = √119
- Write the answer. √119
Check. Check: 122 = 52 + (√119)2, which is 144 = 25 + 119.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 10 and 6, then subtract.102 − 62 = 100 − 36 = 64
- Take the square root. b is the square root of the result.b = √64 = 8
- Write the answer. 8
Check. Check: 102 = 62 + (8)2, which is 100 = 36 + 64.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 13 and 12, then subtract.132 − 122 = 169 − 144 = 25
- Take the square root. b is the square root of the result.b = √25 = 5
- Write the answer. 5
Check. Check: 132 = 122 + (5)2, which is 169 = 144 + 25.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 6 and 4, then subtract.62 − 42 = 36 − 16 = 20
- Take the square root. b is the square root of the result.b = √20 = 2√5
- Write the answer. 2√5
Check. Check: 62 = 42 + (2√5)2, which is 36 = 16 + 20.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Check both triangles with Pythagoras’ theorem.
- Check Ali. 32 + 42 = 9 + 16 = 25, and √25 = 5.c = 5
- Check Sam. 52 − 32 = 25 − 9 = 16, and √16 = 4.b = 4
- Write the answer. They use the same triangle, so both are right.Both are right.
Check. 3, 4 and 5 are a Pythagorean triple.
Method. The hypotenuse must be the longest side, so it must be bigger than the other sides.
- Look at the lengths. The hypotenuse is 5 cm and the shorter side is 5 cm.5 = 5
- Subtract the squares. 52 − 52 = 0, so the other side would be 0.0
- Write the answer. A triangle cannot have a side of length 0, so this is not possible.No
Check. The hypotenuse must be strictly longer than each shorter side.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 17 and 8, then subtract.172 − 82 = 289 − 64 = 225
- Take the square root. b is the square root of the result.b = √225 = 15
- Write the answer. 15
Check. Check: 172 = 82 + (15)2, which is 289 = 64 + 225.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 7 and 3, then subtract.72 − 32 = 49 − 9 = 40
- Take the square root. b is the square root of the result.b = √40 = 2√10
- Write the answer. 2√10
Check. Check: 72 = 32 + (2√10)2, which is 49 = 9 + 40.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 15 and 9, then subtract.152 − 92 = 225 − 81 = 144
- Take the square root. b is the square root of the result.b = √144 = 12
- Write the answer. 12
Check. Check: 152 = 92 + (12)2, which is 225 = 81 + 144.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Angles in triangles, quadrilaterals and at a point
Some angle facts are always true:
- the angles in a triangle add up to 180°;
- the angles in a quadrilateral add up to 360°;
- angles on a straight line add up to 180°;
- angles at a point add up to 360°;
- vertically opposite angles are equal.
Key ideas
- Triangle: the angles add up to 180°.
- Quadrilateral: the angles add up to 360°.
- Straight line: 180°. Full turn at a point: 360°.
- Vertically opposite angles are equal.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 62° + 48° = 110°.62 + 48 = 110
- Subtract from 180°. The third angle is what is left.180 − 110 = 70
- Write the answer. 70°
Check. 62 + 48 + 70 = 180.
Method. The angles in a quadrilateral add up to 360°.
- Add the three known angles. 85° + 110° + 95° = 290°.85 + 110 + 95 = 290
- Subtract from 360°. The fourth angle is what is left.360 − 290 = 70
- Write the answer. 70°
Check. 85 + 110 + 95 + 70 = 360.
Method. Vertically opposite angles are equal.
- Look for the opposite angle. The angle x is opposite the angle given, at the same point.x is opposite 38°
- Check the angles on a straight line. The two angles next to x each make a straight line with a, so they are both 180 − a.180 − 38 = 142
- Write the answer. Vertically opposite angles are equal.38°
Check. The two lines cross, and the opposite angles are the same.
Watch out
Watch out: a triangle has 180° and a quadrilateral has 360°. Do not use the same total for both.
Exam tip
Exam tip. Give a reason with each angle: “angles in a triangle add to 180°”. This earns the reason mark.
Exercise 9C Angles in triangles, quadrilaterals and at a point
Non-calculatorMethod. The angles in a triangle add up to 180°.
- Add the two known angles. 62° + 48° = 110°.62 + 48 = 110
- Subtract from 180°. The third angle is what is left.180 − 110 = 70
- Write the answer. 70°
Check. 62 + 48 + 70 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 35° + 85° = 120°.35 + 85 = 120
- Subtract from 180°. The third angle is what is left.180 − 120 = 60
- Write the answer. 60°
Check. 35 + 85 + 60 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 90° + 30° = 120°.90 + 30 = 120
- Subtract from 180°. The third angle is what is left.180 − 120 = 60
- Write the answer. 60°
Check. 90 + 30 + 60 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 40° + 75° = 115°.40 + 75 = 115
- Subtract from 180°. The third angle is what is left.180 − 115 = 65
- Write the answer. 65°
Check. 40 + 75 + 65 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 25° + 25° = 50°.25 + 25 = 50
- Subtract from 180°. The third angle is what is left.180 − 50 = 130
- Write the answer. 130°
Check. 25 + 25 + 130 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 70° + 50° = 120°.70 + 50 = 120
- Subtract from 180°. The third angle is what is left.180 − 120 = 60
- Write the answer. 60°
Check. 70 + 50 + 60 = 180.
Method. The angles in a quadrilateral add up to 360°.
- Add the three known angles. 85° + 110° + 95° = 290°.85 + 110 + 95 = 290
- Subtract from 360°. The fourth angle is what is left.360 − 290 = 70
- Write the answer. 70°
Check. 85 + 110 + 95 + 70 = 360.
Method. The angles in a quadrilateral add up to 360°.
- Add the three known angles. 90° + 90° + 60° = 240°.90 + 90 + 60 = 240
- Subtract from 360°. The fourth angle is what is left.360 − 240 = 120
- Write the answer. 120°
Check. 90 + 90 + 60 + 120 = 360.
Method. The angles in a quadrilateral add up to 360°.
- Add the three known angles. 70° + 80° + 100° = 250°.70 + 80 + 100 = 250
- Subtract from 360°. The fourth angle is what is left.360 − 250 = 110
- Write the answer. 110°
Check. 70 + 80 + 100 + 110 = 360.
Method. Angles on a straight line add up to 180°.
- Write the rule. The two angles make a straight line.130 + x = 180
- Subtract. Take 130° from 180°.180 − 130 = 50
- Write the answer. 50°
Check. 130 + 50 = 180.
Method. Angles at a point add up to 360°.
- Add the known angles. Add the angles around the point.90 + 120 = 210
- Subtract from 360°. The unknown angle is what is left.360 − 210 = 150
- Write the answer. 150°
Check. All the angles add to 360°.
Method. Angles at a point add up to 360°.
- Add the known angles. Add the angles around the point.100 + 80 + 70 = 250
- Subtract from 360°. The unknown angle is what is left.360 − 250 = 110
- Write the answer. 110°
Check. All the angles add to 360°.
Method. Vertically opposite angles are equal.
- Look for the opposite angle. The angle x is opposite the angle given, at the same point.x is opposite 38°
- Check the angles on a straight line. The two angles next to x each make a straight line with a, so they are both 180 − a.180 − 38 = 142
- Write the answer. Vertically opposite angles are equal.38°
Check. The two lines cross, and the opposite angles are the same.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 55° + 55° = 110°.55 + 55 = 110
- Subtract from 180°. The third angle is what is left.180 − 110 = 70
- Write the answer. 70°
Check. 55 + 55 + 70 = 180.
Method. The angles in a triangle add up to 180°, so x + 2x + 3x = 180.
- Write the equation. Add the three angles and set them equal to 180.6x = 180
- Divide by 6. Divide both sides by 6.x = 30
- Write the answer. 30°
Check. 30 + 60 + 90 = 180.
Method. The angles of a quadrilateral add up to 360°.
- Write the equation. Add the four angles and set them equal to 360.2x + 60 + 120 = 360
- Simplify. Add 60 and 120 to get 180.2x + 180 = 360
- Solve. Subtract 180, then divide by 2.x = 90
- Write the answer. 90°
Check. 90 + 90 + 60 + 120 = 360.
Method. Look at two lines that cross each other.
- Look at the angles. Two lines that cross make four angles. The opposite angles are equal.opposite = equal
- Check the total. An angle and the next one along add up to 180°, but opposite ones do not.adjacent add to 180
- Write the answer. Ali is not right. Vertically opposite angles are equal.No
Check. If one angle is 40°, the opposite angle is also 40°.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 90° + 35° = 125°.90 + 35 = 125
- Subtract from 180°. The third angle is what is left.180 − 125 = 55
- Write the answer. 55°
Check. 90 + 35 + 55 = 180.
Angles in parallel lines
When a straight line (a transversal) crosses two parallel lines, it makes several angles that are related:
- corresponding angles are in the same position, and they are equal (an F shape);
- alternate angles are on opposite sides of the transversal, and they are equal (a Z shape);
- co-interior angles are between the parallel lines on the same side, and they add up to 180° (a C shape).
Key ideas
- Corresponding angles (F shape) are equal.
- Alternate angles (Z shape) are equal.
- Co-interior angles (C shape) add up to 180°.
- The lines must be parallel for these rules to work.
Method. Corresponding angles are equal when the lines are parallel.
- Name the rule. Corresponding angles are equal.x = 65
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 65°
Check. The corresponding angle has the same size, because the lines are parallel.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 40
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 40°
Check. The alternate angle has the same size, because the lines are parallel.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.110 + x = 180
- Subtract. Take 110° from 180°.180 − 110 = 70
- Write the answer. 70°
Check. 110 + 70 = 180.
Watch out
Watch out: co-interior angles add up to 180°, but corresponding and alternate angles are equal. Check the letter shape.
Exam tip
Exam tip. Write the name of the rule, such as “alternate angles are equal”. A reason is needed for full marks.
Exercise 9D Parallel lines
Non-calculatorMethod. Corresponding angles are equal when the lines are parallel.
- Name the rule. Corresponding angles are equal.x = 65
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 65°
Check. The corresponding angle has the same size, because the lines are parallel.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 40
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 40°
Check. The alternate angle has the same size, because the lines are parallel.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.110 + x = 180
- Subtract. Take 110° from 180°.180 − 110 = 70
- Write the answer. 70°
Check. 110 + 70 = 180.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.75 + x = 180
- Subtract. Take 75° from 180°.180 − 75 = 105
- Write the answer. 105°
Check. 75 + 105 = 180.
Method. Corresponding angles are equal when the lines are parallel.
- Name the rule. Corresponding angles are equal.x = 120
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 120°
Check. The corresponding angle has the same size, because the lines are parallel.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 55
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 55°
Check. The alternate angle has the same size, because the lines are parallel.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.60 + x = 180
- Subtract. Take 60° from 180°.180 − 60 = 120
- Write the answer. 120°
Check. 60 + 120 = 180.
Method. Corresponding angles are equal when the lines are parallel.
- Name the rule. Corresponding angles are equal.x = 98
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 98°
Check. The corresponding angle has the same size, because the lines are parallel.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 72
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 72°
Check. The alternate angle has the same size, because the lines are parallel.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.105 + x = 180
- Subtract. Take 105° from 180°.180 − 105 = 75
- Write the answer. 75°
Check. 105 + 75 = 180.
Method. Look at the rule for co-interior angles.
- Recall the rule. Co-interior angles are on the same side of the transversal, between the lines.same side
- Check the total. Co-interior angles add up to 180°, so they are equal only when both are 90°.sum = 180
- Write the answer. Ali is not right in general.No
Check. Co-interior angles of 110° and 70° are not equal.
Method. Alternate angles are equal, so 2x = 80.
- Write the rule. Alternate angles are equal.2x = 80
- Divide by 2. Divide both sides by 2.x = 40
- Write the answer. 40°
Check. 2 × 40 = 80.
Method. Co-interior angles add up to 180°, so x + 3x = 180.
- Write the equation. The two angles add to 180.4x = 180
- Divide by 4. Divide both sides by 4.x = 45
- Write the answer. 45°
Check. 45 + 135 = 180.
Method. Corresponding angles are equal, so 2x + 10 = 70.
- Write the equation. The corresponding angles are equal.2x + 10 = 70
- Subtract 10. Take 10 from both sides.2x = 60
- Divide by 2. x = 30
- Write the answer. 30
Check. 2 × 30 + 10 = 70.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.115 + x = 180
- Subtract. Take 115° from 180°.180 − 115 = 65
- Write the answer. 65°
Check. 115 + 65 = 180.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 60
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 60°
Check. The alternate angle has the same size, because the lines are parallel.
Interior and exterior angles of polygons
A polygon is a closed shape with straight sides. A regular polygon has all sides and all angles equal.
- The interior angles of a polygon with n sides add up to (n − 2) × 180°.
- The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360° ÷ n.
- An interior angle and its exterior angle add up to 180°.
Key ideas
- Sum of the interior angles: (n − 2) × 180°.
- Sum of the exterior angles: 360° (for every polygon).
- Regular polygon: each exterior angle = 360° ÷ n.
- Interior angle = 180° − exterior angle.
Method. Use the rule for the sum of the interior angles: (n − 2) × 180°.
- Write the rule. For a polygon with n sides.(n − 2) × 180
- Put in n. The polygon has 7 sides.(7 − 2) × 180
- Work it out. Subtract 2 and multiply by 180.5 × 180 = 900
- Write the answer. 900°
Check. Split the polygon into 5 triangles, each with 180°.
Method. The interior angles of a polygon with n sides add up to (n − 2) × 180°. A regular polygon has equal angles.
- Find the sum of the interior angles. Use (n − 2) × 180 with n = 5.(5 − 2) × 180 = 540
- Share the sum equally (regular polygon). There are 5 equal angles.540 ÷ 5 = 108
- Write the answer. 108°
Check. The exterior angle is 180 − 108 = 72, and 5 × 72 = 360.
Method. Use the exterior angle: n = 360 ÷ (exterior angle).
- Write the rule. The number of sides is 360 divided by one exterior angle.n = 360 ÷ 40
- Divide. Work out how many exterior angles fit into 360°.360 ÷ 40 = 9
- Write the answer. 9 sides
Check. 9 × 40 = 360.
Watch out
Watch out: the exterior angles of ANY polygon add up to 360°. The interior angles depend on the number of sides.
Exam tip
Exam tip. Write the rule first: (n − 2) × 180 or 360 ÷ n. Put in the number of sides last.
Exercise 9E Interior and exterior angles of polygons
Non-calculatorMethod. The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360 ÷ n.
- Write the rule. A regular polygon with 5 sides has 5 equal exterior angles.exterior angle = 360 ÷ 5
- Divide. Share 360° equally between the sides.360 ÷ 5 = 72
- Write the answer. 72°
Check. 5 × 72 = 360.
Method. The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360 ÷ n.
- Write the rule. A regular polygon with 6 sides has 6 equal exterior angles.exterior angle = 360 ÷ 6
- Divide. Share 360° equally between the sides.360 ÷ 6 = 60
- Write the answer. 60°
Check. 6 × 60 = 360.
Method. The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360 ÷ n.
- Write the rule. A regular polygon with 8 sides has 8 equal exterior angles.exterior angle = 360 ÷ 8
- Divide. Share 360° equally between the sides.360 ÷ 8 = 45
- Write the answer. 45°
Check. 8 × 45 = 360.
Method. The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360 ÷ n.
- Write the rule. A regular polygon with 10 sides has 10 equal exterior angles.exterior angle = 360 ÷ 10
- Divide. Share 360° equally between the sides.360 ÷ 10 = 36
- Write the answer. 36°
Check. 10 × 36 = 360.
Method. The interior angles of a polygon with n sides add up to (n − 2) × 180°. A regular polygon has equal angles.
- Find the sum of the interior angles. Use (n − 2) × 180 with n = 5.(5 − 2) × 180 = 540
- Share the sum equally (regular polygon). There are 5 equal angles.540 ÷ 5 = 108
- Write the answer. 108°
Check. The exterior angle is 180 − 108 = 72, and 5 × 72 = 360.
Method. The interior angles of a polygon with n sides add up to (n − 2) × 180°. A regular polygon has equal angles.
- Find the sum of the interior angles. Use (n − 2) × 180 with n = 6.(6 − 2) × 180 = 720
- Share the sum equally (regular polygon). There are 6 equal angles.720 ÷ 6 = 120
- Write the answer. 120°
Check. The exterior angle is 180 − 120 = 60, and 6 × 60 = 360.
Method. The interior angles of a polygon with n sides add up to (n − 2) × 180°. A regular polygon has equal angles.
- Find the sum of the interior angles. Use (n − 2) × 180 with n = 8.(8 − 2) × 180 = 1080
- Share the sum equally (regular polygon). There are 8 equal angles.1080 ÷ 8 = 135
- Write the answer. 135°
Check. The exterior angle is 180 − 135 = 45, and 8 × 45 = 360.
Method. Use the rule for the sum of the interior angles: (n − 2) × 180°.
- Write the rule. For a polygon with n sides.(n − 2) × 180
- Put in n. The polygon has 7 sides.(7 − 2) × 180
- Work it out. Subtract 2 and multiply by 180.5 × 180 = 900
- Write the answer. 900°
Check. Split the polygon into 5 triangles, each with 180°.
Method. Use the rule for the sum of the interior angles: (n − 2) × 180°.
- Write the rule. For a polygon with n sides.(n − 2) × 180
- Put in n. The polygon has 9 sides.(9 − 2) × 180
- Work it out. Subtract 2 and multiply by 180.7 × 180 = 1260
- Write the answer. 1260°
Check. Split the polygon into 7 triangles, each with 180°.
Method. Use the exterior angle: n = 360 ÷ (exterior angle).
- Write the rule. The number of sides is 360 divided by one exterior angle.n = 360 ÷ 40
- Divide. Work out how many exterior angles fit into 360°.360 ÷ 40 = 9
- Write the answer. 9 sides
Check. 9 × 40 = 360.
Method. Use the exterior angle: n = 360 ÷ (exterior angle).
- Write the rule. The number of sides is 360 divided by one exterior angle.n = 360 ÷ 30
- Divide. Work out how many exterior angles fit into 360°.360 ÷ 30 = 12
- Write the answer. 12 sides
Check. 12 × 30 = 360.
Method. Find the exterior angle first, then use it to find n.
- Find the exterior angle. An interior angle and its exterior angle add up to 180°, so 180 − 150 = 30.30°
- Use the exterior angle. n = 360 ÷ 30.n = 12
- Write the answer. 12 sides
Check. 12 × 30 = 360.
Method. Find the exterior angle, then the number of sides.
- Find the exterior angle. 180 − 135 = 45.45°
- Find n. n = 360 ÷ 45 = 8.n = 8
- Write the answer. 8 sides
Check. 8 × 45 = 360.
Method. Share the sum by the number of angles.
- Use the sum. The sum of the interior angles of a hexagon is 720°.720
- Share equally. Each angle is 720 ÷ 6 = 120° if it is regular.120
- Write the answer. Ali is right, if the hexagon is regular.Yes
Check. 6 × 120 = 720.
Method. Imagine walking around the outside of the shape.
- Walk round the shape. At each corner you turn through the exterior angle.turn at each corner
- Count the turns. When you return to the start, you have made one full turn.360°
- Write the answer. 360°
Check. The total turn is always one full turn, for any polygon.
Chapter summary
| Skill | Method |
|---|---|
| Pythagoras (hypotenuse) | c2 = a2 + b2: add the squares, then square root |
| Pythagoras (shorter side) | b2 = c2 − a2: subtract the squares, then square root |
| Triangle | Angles add up to 180° |
| Quadrilateral | Angles add up to 360° |
| Straight line and point | 180° on a straight line; 360° around a point |
| Parallel lines | Corresponding and alternate angles are equal; co-interior angles add to 180° |
| Polygons | Interior sum (n − 2) × 180°; exterior 360°; regular exterior = 360° ÷ n |
Can you…?
- I can find the hypotenuse of a right-angled triangle, and give an exact answer
- I can find a shorter side, by subtracting the squares
- I can find missing angles in triangles, quadrilaterals and at a point
- I can name corresponding, alternate and co-interior angles, and use them
- I can find the interior and exterior angles of a regular polygon
Key terms
- hypotenuse
- the longest side of a right-angled triangle, opposite the right angle
- transversal
- a line that crosses two or more other lines
- corresponding angles
- angles in the same position at each crossing; they are equal for parallel lines
- alternate angles
- angles on opposite sides of the transversal; they are equal for parallel lines
- co-interior angles
- angles on the same side of the transversal, between parallel lines; they add to 180°
- polygon
- a closed shape with straight sides
- regular polygon
- a polygon with all sides and all angles equal
Exercise 9Z Check-up
Non-calculatorMethod. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 5 and 12.52 + 122 = 25 + 144
- Add. Add the two squares to find c2.c2 = 169
- Take the square root. c is the square root of the total.c = √169 = 13
- Write the answer. 13
Check. Check: 13 squared is 169, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 1 and 2.12 + 22 = 1 + 4
- Add. Add the two squares to find c2.c2 = 5
- Take the square root. c is the square root of the total.c = √5 = √5
- Write the answer. √5
Check. Check: (√5)2 = 5, and 12 + 22 = 5.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 10 and 8, then subtract.102 − 82 = 100 − 64 = 36
- Take the square root. b is the square root of the result.b = √36 = 6
- Write the answer. 6
Check. Check: 102 = 82 + (6)2, which is 100 = 64 + 36.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 9 and 4, then subtract.92 − 42 = 81 − 16 = 65
- Take the square root. b is the square root of the result.b = √65 = √65
- Write the answer. √65
Check. Check: 92 = 42 + (√65)2, which is 81 = 16 + 65.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 74° + 56° = 130°.74 + 56 = 130
- Subtract from 180°. The third angle is what is left.180 − 130 = 50
- Write the answer. 50°
Check. 74 + 56 + 50 = 180.
Method. Angles on a straight line add up to 180°.
- Write the rule. The two angles make a straight line.115 + x = 180
- Subtract. Take 115° from 180°.180 − 115 = 65
- Write the answer. 65°
Check. 115 + 65 = 180.
Method. Angles at a point add up to 360°.
- Add the known angles. Add the angles around the point.110 + 95 = 205
- Subtract from 360°. The unknown angle is what is left.360 − 205 = 155
- Write the answer. 155°
Check. All the angles add to 360°.
Method. Alternate angles are equal when the lines are parallel.
- Name the rule. Alternate angles are equal.x = 82
- Check the position. The angles are in matching positions (an F shape or a Z shape) on the two parallel lines.F or Z shape
- Write the answer. 82°
Check. The alternate angle has the same size, because the lines are parallel.
Method. Use the rule for the sum of the interior angles: (n − 2) × 180°.
- Write the rule. For a polygon with n sides.(n − 2) × 180
- Put in n. The polygon has 8 sides.(8 − 2) × 180
- Work it out. Subtract 2 and multiply by 180.6 × 180 = 1080
- Write the answer. 1080°
Check. Split the polygon into 6 triangles, each with 180°.
Method. Use the exterior angle: n = 360 ÷ (exterior angle).
- Write the rule. The number of sides is 360 divided by one exterior angle.n = 360 ÷ 45
- Divide. Work out how many exterior angles fit into 360°.360 ÷ 45 = 8
- Write the answer. 8 sides
Check. 8 × 45 = 360.
Total 28 marks. Allow about 28 minutes.
Exercise 9X Exam-style questions
Non-calculatorMethod. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 3 and 6.32 + 62 = 9 + 36
- Add. Add the two squares to find c2.c2 = 45
- Take the square root. c is the square root of the total.c = √45 = 3√5
- Write the answer. 3√5
Check. Check: 3√5 squared is 9 × 5 = 45, which is correct.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.
Method. Rearrange Pythagoras' theorem to find the shorter side: b2 = c2 − a2.
- Write the rule. The hypotenuse c is the longest side, so subtract the known shorter side squared.b2 = c2 − a2
- Square and subtract. Square 15 and 12, then subtract.152 − 122 = 225 − 144 = 81
- Take the square root. b is the square root of the result.b = √81 = 9
- Write the answer. 9
Check. Check: 152 = 122 + (9)2, which is 225 = 144 + 81.
Watch out. Subtract the squares: the hypotenuse is always the biggest. Check your answer is smaller than c.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 40° + 65° = 105°.40 + 65 = 105
- Subtract from 180°. The third angle is what is left.180 − 105 = 75
- Write the answer. 75°
Check. 40 + 65 + 75 = 180.
Method. Angles on a straight line add up to 180°.
- Write the equation. x + 3x = 180.4x = 180
- Divide by 4. Divide both sides by 4.x = 45
- Write the answer. 45°
Check. 45 + 135 = 180.
Method. The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360 ÷ n.
- Write the rule. A regular polygon with 8 sides has 8 equal exterior angles.exterior angle = 360 ÷ 8
- Divide. Share 360° equally between the sides.360 ÷ 8 = 45
- Write the answer. 45°
Check. 8 × 45 = 360.
Method. Find the exterior angle, then divide 360 by it.
- Find the exterior angle. 180 − 140 = 40.40°
- Find n. n = 360 ÷ 40 = 9.n = 9
- Write the answer. 9 sides
Check. 9 × 40 = 360.
Method. Co-interior angles (between the parallel lines, on the same side of the transversal) add up to 180°.
- Name the rule. Co-interior angles add up to 180°.118 + x = 180
- Subtract. Take 118° from 180°.180 − 118 = 62
- Write the answer. 62°
Check. 118 + 62 = 180.
Method. The angles in a triangle add up to 180°.
- Add the two known angles. 90° + 52° = 142°.90 + 52 = 142
- Subtract from 180°. The third angle is what is left.180 − 142 = 38
- Write the answer. 38°
Check. 90 + 52 + 38 = 180.
Method. The angles in a quadrilateral add up to 360°.
- Add the three known angles. 80° + 100° + 110° = 290°.80 + 100 + 110 = 290
- Subtract from 360°. The fourth angle is what is left.360 − 290 = 70
- Write the answer. 70°
Check. 80 + 100 + 110 + 70 = 360.
Method. The interior angles of a polygon with n sides add up to (n − 2) × 180°. A regular polygon has equal angles.
- Find the sum of the interior angles. Use (n − 2) × 180 with n = 10.(10 − 2) × 180 = 1440
- Share the sum equally (regular polygon). There are 10 equal angles.1440 ÷ 10 = 144
- Write the answer. 144°
Check. The exterior angle is 180 − 144 = 36, and 10 × 36 = 360.
Method. Use Pythagoras' theorem: c2 = a2 + b2. Add the squares, then take the square root.
- Write the rule. The hypotenuse is the longest side, so c is the side opposite the right angle.c2 = a2 + b2
- Square the two shorter sides. Square 6 and 8.62 + 82 = 36 + 64
- Add. Add the two squares to find c2.c2 = 100
- Take the square root. c is the square root of the total.c = √100 = 10
- Write the answer. 10
Check. Check: 10 squared is 100, which is the sum of the squares of the two shorter sides.
Watch out. Do not add the lengths. Square them first, add the squares, then take the square root.