Year 6: Angles in triangles and quadrilaterals
Part of the Year 6 angles, shapes and circles chapter. Chapter overview · whole chapter on one page
Angles in triangles and quadrilaterals
The angles inside a triangle always add up to 180°. The angles inside a quadrilateral (a four-sided shape) always add up to 360°. A diagonal splits a quadrilateral into two triangles, so 360 = 2 × 180.
| Shape | The angles inside add up to |
|---|---|
| Triangle | 180° |
| Quadrilateral | 360° |
We can classify a shape by its angles and sides. An isosceles triangle has two equal sides, and its two base angles are equal. An equilateral triangle has three equal sides and three angles of 60°. In a parallelogram the opposite angles are equal, and the two angles next to each other add up to 180°.
Key ideas
- A triangle has angles that add up to 180°.
- A quadrilateral has angles that add up to 360°.
- In an isosceles triangle, the two base angles are equal.
- In a parallelogram, opposite angles are equal and adjacent angles add up to 180°.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.47° + 68° = 115°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 115 = 65
- Write the answer. The missing angle is 65 degrees.65
Check. 65 + 115 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 75° + 110° = 275°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 275 = 85
- Write the answer. The missing angle is 85 degrees.85
Check. 85 + 275 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 36 = 144
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.144 ÷ 2 = 72
- Write the answer. Each base angle is this size.72
Check. 72 + 72 + 36 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Exercise 10B Practice: triangles, quadrilaterals and special shapes
Non-calculatorKey words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.50° + 70° = 120°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 120 = 60
- Write the answer. The missing angle is 60 degrees.60
Check. 60 + 120 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.38° + 92° = 130°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 130 = 50
- Write the answer. The missing angle is 50 degrees.50
Check. 50 + 130 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.90° + 27° = 117°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 117 = 63
- Write the answer. The missing angle is 63 degrees.63
Check. 63 + 117 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.115° + 28° = 143°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 143 = 37
- Write the answer. The missing angle is 37 degrees.37
Check. 37 + 143 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 40 = 140
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.140 ÷ 2 = 70
- Write the answer. Each base angle is this size.70
Check. 70 + 70 + 40 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.85° + 110° + 95° = 290°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 290 = 70
- Write the answer. The missing angle is 70 degrees.70
Check. 70 + 290 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 90° + 70° = 250°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 250 = 110
- Write the answer. The missing angle is 110 degrees.110
Check. 110 + 250 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Adjacent angles in a parallelogram add up to 180 degrees, so subtract the given angle from 180.
- Know the rule for a parallelogram. The two angles next to each other on the same side add up to 180 degrees.64° + ? = 180°
- Take the given angle away from 180. The missing angle is what is left over.180 - 64 = 116
- Write the answer. The angle is 116 degrees.116
Check. 64 + 116 = 180. ✓
Watch out. Next to each other means on the same side. Do not copy the angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Opposite angles in a parallelogram are equal, so copy the angle.
- Know the rule for a parallelogram. Opposite angles in a parallelogram are equal.Opposite angle = 64°
- Write the answer. The opposite angle is the same size.64
Check. The four angles are 64, 116, 64 and 116, and 64 + 116 + 64 + 116 = 360. ✓
Watch out. Opposite angles are equal. Angles next to each other add up to 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Subtract the known angles from the total, then share the rest equally.
- Add the angles you know. Put them together first.100° + 100° = 200°
- Take the total away. What is left is shared by the 2 equal angles.360 - 200 = 160
- Share the rest equally. Each of the equal angles is the same size.160 ÷ 2 = 80
- Write the answer. Each equal angle is this size.80
Check. 80 × 2 = 160, and 160 + 200 = 360. ✓
Watch out. Share only the angles that are left over. Do not share the total.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Share the total into equal parts using the ratio, then work out each angle.
- Add the parts of the ratio. Each part of the ratio is one share of the total.1 : 2 : 3: 6 parts in all
- Find the size of one part. Share the total of 180 degrees into 6 equal parts.180 ÷ 6 = 30
- Work out each angle. Multiply each ratio number by the size of one part.1 × 30 = 30, 2 × 30 = 60, 3 × 30 = 90
- Write the answer. The largest angle is 90 degrees.90
Check. The angles add up to 30 + 60 + 90 = 180. ✓
Watch out. Share the total by the number of parts, not by the number of angles.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Use the rule that the angles of a triangle add up to 180 degrees.
- Add the two right angles. Each right angle is 90 degrees.90 + 90 = 180
- Look at what is left for the third angle. The three angles must add up to 180, so nothing is left.180 - 180 = 0
- Write the answer. A triangle cannot have an angle of 0 degrees, so Sam is not right.No
Check. Two right angles would already make the full 180 degrees of a triangle. ✓
Watch out. A triangle can have at most one right angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Write the three angles in terms of x, add them, and solve.
- Write the three angles and add them up. The angles of a triangle add up to 180 degrees.x + (x + 20) + (x + 40) = 180
- Collect the x terms and the numbers. x + x + x is 3x, and 20 + 40 is 60.3x + 60 = 180
- Take 60 from 180. What is left is 3 lots of x.180 - 60 = 120, so 3x = 120
- Divide by 3 to find x. This gives the size of x.120 ÷ 3 = 40, so x = 40
- Find the largest angle. The three angles are 40, 60 and 80, so the largest is x + 40.x + 40 = 40 + 40 = 80
- Write the answer. The largest angle is 80 degrees.80
Check. The three angles are 40, 60 and 80, and 40 + 60 + 80 = 180. ✓
Watch out. Find x first. Then work out each angle before you choose the largest.
Watch out
Common mistake: using 180 for a quadrilateral. A four-sided shape has 360 degrees inside it. Also check that a missing angle is positive.
Exam tip
If a question gives a diagram that is not drawn to scale, use the numbers on it, not the size of the drawn angles.