GCSE Maths: Angles in triangles (Foundation)
Part of the GCSE Maths angles, polygons and symmetry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Angles in triangles
The three angles of any triangle add up to 180°. You can see why with a piece of paper: draw any triangle, tear off its three corners, and place them side by side with the points touching. They always fit together to make a straight line, and a straight line is 180°. (Section 17.9 explains a proper proof.)
To find a missing angle, add the two you know and subtract the total from 180°.
Isosceles triangles. An isosceles triangle has two equal sides. The angles opposite the equal sides are also equal. These are called the base angles. The ticks on the diagram show which sides are equal. If you know the angle at the top (the apex), subtract it from 180° and share the rest equally between the two base angles. If you know a base angle, the other base angle is the same, and the apex is 180° minus the two base angles.
Equilateral triangles. An equilateral triangle has three equal sides and three equal angles. Each angle is 180° ÷ 3 = 60°.
Right-angled triangles have one angle of 90°, so the other two angles add up to 90°. A triangle with all angles less than 90° is acute-angled, and one with an angle bigger than 90° is obtuse-angled. A scalene triangle has no equal sides.
Exterior angles. If you extend one side of a triangle, you make an exterior angle outside the triangle. The exterior angle and the interior angle next to it are on a straight line, so they add up to 180°. This leads to a very useful shortcut: an exterior angle of a triangle equals the sum of the two opposite interior angles (the two corners it is not next to).
Key ideas
- Angles in a triangle add up to 180°.
- Isosceles: two equal sides, and the base angles (opposite the equal sides) are equal.
- Equilateral: three equal sides and three angles of 60°.
- Right-angled: one angle is 90°, so the other two add to 90°.
- Exterior angle of a triangle = the sum of the two opposite interior angles.
- Exterior angle + its interior angle = 180° (straight line).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.52 + 71 = 123°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 123 = 57°
x = 57°
Check. Add all the angles: 52 + 71 + 57 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.48° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 48 = 132°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 132 ÷ 2 = 66°
Check. Add all three angles: 48 + 66 + 66 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 38°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.38 + 38 = 76°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 76 = 104°
x = 104°
Check. Add all three angles: 38 + 38 + 104 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 47° and 68°.x = 47 + 68 = 115°
Check. Another way: the third angle of the triangle is 180 − 47 − 68 = 65°, and on a straight line 65 + x = 180, so x = 115°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Watch out
The base angles of an isosceles triangle are the angles opposite the equal sides, which are at the ends of the third side (the base). Students often assume that the two angles next to the 50° are equal even when the ticks are on different sides. Always read the ticks.Exercise 17C Angles in triangles
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.50 + 65 = 115°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 115 = 65°
x = 65°
Check. Add all the angles: 50 + 65 + 65 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.38 + 72 = 110°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 110 = 70°
x = 70°
Check. Add all the angles: 38 + 72 + 70 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.24 + 105 = 129°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 129 = 51°
x = 51°
Check. Add all the angles: 24 + 105 + 51 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.34 + 90 = 124°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 124 = 56°
x = 56°
Check. Add all the angles: 34 + 90 + 56 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.40° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 40 = 140°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 140 ÷ 2 = 70°
Check. Add all three angles: 40 + 70 + 70 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 65°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.65 + 65 = 130°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 130 = 50°
x = 50°
Check. Add all three angles: 65 + 65 + 50 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.100° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 100 = 80°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 80 ÷ 2 = 40°
Check. Add all three angles: 100 + 40 + 40 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 35°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.35 + 35 = 70°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 70 = 110°
x = 110°
Check. Add all three angles: 35 + 35 + 110 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Method. An equilateral triangle has three equal sides, so it has three equal angles.
- State what we know. The ticks show that all three sides are equal, so all three angles are equal.Three equal angles
- Use the angle sum. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 ÷ 3 = 60°
- Write the answer. Equal angles each take a third of 180°.x = 60°
Check. 60 + 60 + 60 = 180°. ✓
Watch out. Every equilateral triangle has three 60° angles, whatever its size.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 50° and 60°.x = 50 + 60 = 110°
Check. Another way: the third angle of the triangle is 180 − 50 − 60 = 70°, and on a straight line 70 + x = 180, so x = 110°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 35° and 78°.x = 35 + 78 = 113°
Check. Another way: the third angle of the triangle is 180 − 35 − 78 = 67°, and on a straight line 67 + x = 180, so x = 113°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Method. Find the third angle, then compare each angle with 90°.
- Find angle C. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 70 − 45 = 65°
- Compare with 90°. A right-angled triangle has a 90° angle; an obtuse-angled one has an angle bigger than 90°.70°, 45° and 65° are all less than 90°
- Name the triangle. All three angles are acute.acute-angled
Check. 70 + 45 + 65 = 180°. ✓
Watch out. You must find the third angle before you can decide.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.110° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 110 = 70°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 70 ÷ 2 = 35°
Check. Add all three angles: 110 + 35 + 35 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Method. Share 180° in the ratio 1 : 2 : 3.
- Add the parts of the ratio. The ratio tells us how many equal parts the 180° is split into.1 + 2 + 3 = 6 parts
- Find one part. Divide 180° by 6.180 ÷ 6 = 30°
- Multiply to find each angle. 1 × 30 = 30°, 2 × 30 = 60°, 3 × 30 = 90°
- Name the triangle. One angle is 90°.right-angled
Check. 30 + 60 + 90 = 180°. ✓
Watch out. See Chapter 15 for sharing in a ratio.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.x° + 2x° + 3x° = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x = 180
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 180 ÷ 6 = 30
- Substitute x = 30 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 30 = 30°
2x° = 2 × 30 = 60°
3x° = 3 × 30 = 90°
Check. Add the angles: 30 + 60 + 90 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.(2x + 10°) + 3x° + (x + 20°) = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x + 30 = 180
- Get the number on the other side. Subtract 30 from both sides.6x = 150
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 150 ÷ 6 = 25
- Substitute x = 25 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x + 10° = 2 × 25 + 10 = 60°
3x° = 3 × 25 = 75°
x + 20° = 1 × 25 + 20 = 45°
Check. Add the angles: 60 + 75 + 45 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Use the fact that the angles of a triangle add up to 180°.
- Suppose a triangle has two right angles. Each right angle is 90°.90° + 90° = 180°
- Look at what is left for the third angle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180° − 180° = 0°
- Decide. A corner of 0° is not a corner at all, so the shape would not be a triangle.It is impossible.
Check. A triangle must have three angles bigger than 0°. ✓
Watch out. This is a good example of a short argument that uses the angle sum fact.
Method. Add the angles and compare with 180°.
- Recall the fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add. The total of the three angles should be 180°.70 + 60 + 60 = 190°
- Decide. 190 is bigger than 180.No, Sam cannot be correct.
Check. If the angles were 70°, 60° and 50° they would total 180°. ✓
Method. Find the interior angle at C using the straight line, then use the isosceles triangle.
- Find angle ACB. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 130 = 50°
- Say which angles are equal. AC = BC, so the angles opposite them, at B and at A, are equal.angle CAB = angle CBA
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 50 = 130° for the two base angles
- Divide by 2. The two base angles are equal.130 ÷ 2 = 65°
Check. 65 + 65 + 50 = 180°. ✓
Watch out. Another way: the exterior angle equals the sum of the two opposite angles: 65 + 65 = 130°.