GCSE Maths: Cyclic quadrilaterals (Higher)
Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Cyclic quadrilaterals
A cyclic quadrilateral is a four-sided shape with all four corners on a circle. The word “cyclic” comes from the Greek word for circle. The theorem says: opposite angles of a cyclic quadrilateral add up to 180°. Opposite means across the shape, not next to each other.
The reason to write is “opposite angles of a cyclic quadrilateral add up to 180°”. Why is it true? Join B and D to the centre O. Let angle BAD = a. It stands on the arc BCD, so the angle at the centre on that arc is 2a (centre theorem). The angle round the other side of the centre is 360° − 2a (angles round a point). Angle BCD stands on that other arc, so it is half of that: (360° − 2a) ÷ 2 = 180° − a. Add: a + 180° − a = 180°. The full proof is worked example 26.5.5.
Exterior angles. If you extend one side of a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle. This is a fast shortcut: the exterior angle and its neighbouring interior angle make a straight line (180°), and the interior angle plus its opposite angle make 180° too. So the exterior angle equals the opposite one. You can use the shortcut in the exam if you give the reason “exterior angle of a cyclic quadrilateral is equal to the interior opposite angle”, but it is safer to show the two steps.
Key ideas
- Opposite angles of a cyclic quadrilateral add up to 180°.
- Reason: “opposite angles of a cyclic quadrilateral add up to 180°”.
- Opposite means A with C, and B with D. Next-door angles do not add to 180° in general.
- The exterior angle (made by extending a side) equals the interior opposite angle.
- The angles of every quadrilateral add up to 360°; a cyclic quadrilateral has the extra property above.
Watch out
Only opposite angles add up to 180°. And the theorem needs all four corners on the circle. A parallelogram that is not a rectangle has opposite angles that add to more or less than 180°, so it cannot be cyclic.Key words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle DAB and angle BCD are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle DAB + angle BCD = 180°
- Subtract. Take the known angle away from 180°.Angle BCD = 180° − 95° = 85°
Check. 95° + 85° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Method. Find the interior angle at the corner first (opposite angles of a cyclic quadrilateral add to 180°), then use angles on a straight line.
- Mark the exterior angle. The side BC has been extended to E. The angle ECD is outside the quadrilateral, between the extension and the next side CD.We want angle ECD.
- Find the interior angle at C. Corner C is opposite corner A in the cyclic quadrilateral, so these two angles add up to 180°.Angle BCD = 180° − 95° = 85° (opposite angles of a cyclic quadrilateral add up to 180°)
- Use the straight line. B, C and E are on a straight line, so angle BCD and angle ECD are on a straight line and add up to 180°.Angle ECD = 180° − 85° = 95° (angles on a straight line add up to 180°)
Check. The exterior angle 95° equals the opposite interior angle 95°. This is a useful shortcut: the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. ✓
Watch out. The exterior angle is equal to the interior angle opposite the corner, not the one next to it.
- Work out the value of x.
- Work out angles DAB and BCD.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. A and C are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(5x + 20) + (3x + 40) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.8x + 60 = 180
- Subtract 60 from both sides. We want the number terms on the other side.8x = 180 − 60
8x = 120 - Divide both sides by 8. 8x means 8 × x, so dividing by 8 leaves x on its own.x = 120 ÷ 8 = 15
- Work out the two angles. Put x back into each expression.Angle DAB = 5 × 15 + 20 = 95°
Angle BCD = 3 × 15 + 40 = 85°
Check. 95° + 85° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
- Work out angle ABC
- Work out angle ADC
Part (a)
Method. Angle ABC and angle AOC stand on the same arc AC (the arc through D), so use the angle-at-the-centre theorem.
- Spot the two angles. Angle AOC is at the centre, angle ABC is at the circumference, and both stand on the arc AC that contains D.Centre: 160°
Circumference: angle ABC = ? - Halve it. The angle at the centre is twice the angle at the circumference.Angle ABC = 160° ÷ 2 = 80° (the angle at the centre is twice the angle at the circumference)
Check. 2 × 80° = 160°. ✓
Watch out. Check which arc the angles stand on. B is on the other side of the circle from the arc ADC.
Part (b)
Method. ABCD is a cyclic quadrilateral, so opposite angles add up to 180°. Use the answer to part (a).
- Spot the opposite corner. D is opposite B in the quadrilateral ABCD.Angle ABC = 80° (from part a)
- Subtract from 180°. Opposite angles of a cyclic quadrilateral add up to 180°.Angle ADC = 180° − 80° = 100° (opposite angles of a cyclic quadrilateral add up to 180°)
Check. Second method: the reflex angle AOC = 360° − 160° = 200°, and half of it is 100°. ✓
Watch out. Angle ADC is obtuse here, because D is on the minor arc.
Method. Join B and D to the centre. Use the centre theorem for each of the two angles, one on each arc.
- Name the angle at A. Use a letter so the proof works for any size of angle.Let angle BAD = a
- Find the angle at the centre on the arc BCD. Angle BAD stands on arc BCD. The angle at the centre is twice the angle at the circumference.angle BOD (on the C side) = 2a
- Find the other angle at the centre. The full turn round O is 360°, so the angle on the other side is what remains.reflex angle BOD = 360° − 2a (angles round a point add up to 360°)
- Use the centre theorem again at C. Angle BCD stands on the arc BAD that goes the other way round, so it is half the reflex angle.angle BCD = (360° − 2a) ÷ 2 = 180° − a
- Add the opposite angles. angle BAD + angle BCD = a + 180° − a = 180° ✓
Check. Try a = 80°: the angle at the centre is 160°, the reflex angle is 200°, angle BCD = 100°. 80° + 100° = 180°. ✓
Watch out. The same argument works for the other pair of opposite angles, B and D.
Exercise 26E Cyclic quadrilaterals
Non-calculatorKey words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle DAB and angle BCD are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle DAB + angle BCD = 180°
- Subtract. Take the known angle away from 180°.Angle BCD = 180° − 85° = 95°
Check. 85° + 95° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Key words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle ABC and angle CDA are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle ABC + angle CDA = 180°
- Subtract. Take the known angle away from 180°.Angle CDA = 180° − 85° = 95°
Check. 85° + 95° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Key words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle BCD and angle DAB are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle BCD + angle DAB = 180°
- Subtract. Take the known angle away from 180°.Angle DAB = 180° − 110° = 70°
Check. 110° + 70° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Key words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle CDA and angle ABC are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle CDA + angle ABC = 180°
- Subtract. Take the known angle away from 180°.Angle ABC = 180° − 80° = 100°
Check. 80° + 100° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Key words. A cyclic quadrilateral is a four-sided shape with all four corners on a circle. Opposite angles are at corners that are not next to each other.
Method. In a cyclic quadrilateral, opposite angles add up to 180°. Subtract the known angle from 180°.
- Check the angles are opposite. In a quadrilateral ABCD, A is opposite C and B is opposite D. Opposite corners are not next to each other.Angle DAB and angle BCD are opposite corners.
- Write the cyclic quadrilateral fact. All four corners are on the circle, so opposite angles add up to 180°.Angle DAB + angle BCD = 180°
- Subtract. Take the known angle away from 180°.Angle BCD = 180° − 80° = 100°
Check. 80° + 100° = 180°. ✓
Watch out. Opposite means the corner across the quadrilateral. Neighbouring angles do not add up to 180°.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Find the angle opposite DAB. Corner A is opposite corner C.Angle DAB + angle BCD = 180° (opposite angles of a cyclic quadrilateral add up to 180°)
- Subtract. Take 95° from 180°.Angle BCD = 180° − 95° = 85°
Check. 95° + 85° = 180°. ✓
Watch out. Pair each angle with the one opposite, not with its neighbour.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Find the angle opposite ABC. Corner B is opposite corner D.Angle ABC + angle CDA = 180° (opposite angles of a cyclic quadrilateral add up to 180°)
- Subtract. Take 90° from 180°.Angle CDA = 180° − 90° = 90°
Check. 90° + 90° = 180°. ✓ Now all four angles add up to 360°, so they make a full 360°.
Watch out. Pair each angle with the one opposite, not with its neighbour.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Find the angle opposite BCD. Corner C is opposite corner A.Angle BCD + angle DAB = 180° (opposite angles of a cyclic quadrilateral add up to 180°)
- Subtract. Take 110° from 180°.Angle DAB = 180° − 110° = 70°
Check. 110° + 70° = 180°. ✓
Watch out. Pair each angle with the one opposite, not with its neighbour.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Find the angle opposite CDA. Corner D is opposite corner B.Angle CDA + angle ABC = 180° (opposite angles of a cyclic quadrilateral add up to 180°)
- Subtract. Take 90° from 180°.Angle ABC = 180° − 90° = 90°
Check. 90° + 90° = 180°. ✓ Now all four angles add up to 360°, so they make a full 360°.
Watch out. Pair each angle with the one opposite, not with its neighbour.
Method. Find the interior angle at the corner first (opposite angles of a cyclic quadrilateral add to 180°), then use angles on a straight line.
- Mark the exterior angle. The side BC has been extended to E. The angle ECD is outside the quadrilateral, between the extension and the next side CD.We want angle ECD.
- Find the interior angle at C. Corner C is opposite corner A in the cyclic quadrilateral, so these two angles add up to 180°.Angle BCD = 180° − 80° = 100° (opposite angles of a cyclic quadrilateral add up to 180°)
- Use the straight line. B, C and E are on a straight line, so angle BCD and angle ECD are on a straight line and add up to 180°.Angle ECD = 180° − 100° = 80° (angles on a straight line add up to 180°)
Check. The exterior angle 80° equals the opposite interior angle 80°. This is a useful shortcut: the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. ✓
Watch out. The exterior angle is equal to the interior angle opposite the corner, not the one next to it.
Method. Find the interior angle at the corner first (opposite angles of a cyclic quadrilateral add to 180°), then use angles on a straight line.
- Mark the exterior angle. The side AB has been extended to E. The angle EBC is outside the quadrilateral, between the extension and the next side BC.We want angle EBC.
- Find the interior angle at B. Corner B is opposite corner D in the cyclic quadrilateral, so these two angles add up to 180°.Angle ABC = 180° − 95° = 85° (opposite angles of a cyclic quadrilateral add up to 180°)
- Use the straight line. A, B and E are on a straight line, so angle ABC and angle EBC are on a straight line and add up to 180°.Angle EBC = 180° − 85° = 95° (angles on a straight line add up to 180°)
Check. The exterior angle 95° equals the opposite interior angle 95°. This is a useful shortcut: the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. ✓
Watch out. The exterior angle is equal to the interior angle opposite the corner, not the one next to it.
Method. Find the interior angle at the corner first (opposite angles of a cyclic quadrilateral add to 180°), then use angles on a straight line.
- Mark the exterior angle. The side CD has been extended to E. The angle EDA is outside the quadrilateral, between the extension and the next side DA.We want angle EDA.
- Find the interior angle at D. Corner D is opposite corner B in the cyclic quadrilateral, so these two angles add up to 180°.Angle CDA = 180° − 100° = 80° (opposite angles of a cyclic quadrilateral add up to 180°)
- Use the straight line. C, D and E are on a straight line, so angle CDA and angle EDA are on a straight line and add up to 180°.Angle EDA = 180° − 80° = 100° (angles on a straight line add up to 180°)
Check. The exterior angle 100° equals the opposite interior angle 100°. This is a useful shortcut: the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. ✓
Watch out. The exterior angle is equal to the interior angle opposite the corner, not the one next to it.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. B and D are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(6x) + (5x + 15) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.11x + 15 = 180
- Subtract 15 from both sides. We want the number terms on the other side.11x = 180 − 15
11x = 165 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 165 ÷ 11 = 15
- Work out the two angles. Put x back into each expression.Angle ABC = 6 × 15 + 0 = 90°
Angle CDA = 5 × 15 + 15 = 90°
Check. 90° + 90° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. B and D are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(6x) + (5x + 15) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.11x + 15 = 180
- Subtract 15 from both sides. We want the number terms on the other side.11x = 180 − 15
11x = 165 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 165 ÷ 11 = 15
- Work out the two angles. Put x back into each expression.Angle ABC = 6 × 15 + 0 = 90°
Angle CDA = 5 × 15 + 15 = 90°
Check. 90° + 90° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. B and D are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(6x + 25) + (2x + 75) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.8x + 100 = 180
- Subtract 100 from both sides. We want the number terms on the other side.8x = 180 − 100
8x = 80 - Divide both sides by 8. 8x means 8 × x, so dividing by 8 leaves x on its own.x = 80 ÷ 8 = 10
- Work out the two angles. Put x back into each expression.Angle ABC = 6 × 10 + 25 = 85°
Angle CDA = 2 × 10 + 75 = 95°
Check. 85° + 95° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. B and D are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(6x + 25) + (2x + 75) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.8x + 100 = 180
- Subtract 100 from both sides. We want the number terms on the other side.8x = 180 − 100
8x = 80 - Divide both sides by 8. 8x means 8 × x, so dividing by 8 leaves x on its own.x = 80 ÷ 8 = 10
- Work out the two angles. Put x back into each expression.Angle ABC = 6 × 10 + 25 = 85°
Angle CDA = 2 × 10 + 75 = 95°
Check. 85° + 95° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. A and C are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(4x + 22) + (7x + 26) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.11x + 48 = 180
- Subtract 48 from both sides. We want the number terms on the other side.11x = 180 − 48
11x = 132 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 132 ÷ 11 = 12
- Work out the two angles. Put x back into each expression.Angle DAB = 4 × 12 + 22 = 70°
Angle BCD = 7 × 12 + 26 = 110°
Check. 70° + 110° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°. Write that as an equation, solve for x, then find the angles.
- Write the angle fact. A and C are opposite corners of a cyclic quadrilateral, so the angles add up to 180°.(4x + 22) + (7x + 26) = 180 (opposite angles of a cyclic quadrilateral add up to 180°)
- Collect like terms. Add the x terms together, then the numbers.11x + 48 = 180
- Subtract 48 from both sides. We want the number terms on the other side.11x = 180 − 48
11x = 132 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 132 ÷ 11 = 12
- Work out the two angles. Put x back into each expression.Angle DAB = 4 × 12 + 22 = 70°
Angle BCD = 7 × 12 + 26 = 110°
Check. 70° + 110° = 180°. ✓
Watch out. Put each expression in brackets and add them. Do not try to make one expression equal the other.
Method. Opposite angles of a cyclic quadrilateral add up to 180°, so there are two separate equations: one with x and one with y.
- Pair up the opposite angles. A is opposite C, and B is opposite D. Each pair of opposite angles adds up to 180°, so we get two separate equations.(2x + 40) + (3x + 40) = 180
(5y + 20) + (3y + 40) = 180 (opposite angles of a cyclic quadrilateral add up to 180°) - Solve the first equation for x. Collect the x terms and numbers, then solve.5x + 80 = 180
- Subtract 80 from both sides. We want the number terms on the other side.5x = 180 − 80
5x = 100 - Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.x = 100 ÷ 5 = 20
- Solve the second equation for y. It is the same idea with y.8y + 60 = 180
- Subtract 60 from both sides. We want the number terms on the other side.8y = 180 − 60
8y = 120 - Divide both sides by 8. 8y means 8 × y, so dividing by 8 leaves y on its own.y = 120 ÷ 8 = 15
Check. The angles are 80°, 95°, 100°, 85°. 80 + 100 = 180 and 95 + 85 = 180. ✓
Watch out. Do not mix the letters. The two equations are solved separately.
Method. Opposite angles of a cyclic quadrilateral add up to 180°, so there are two separate equations: one with x and one with y.
- Pair up the opposite angles. A is opposite C, and B is opposite D. Each pair of opposite angles adds up to 180°, so we get two separate equations.(2x + 40) + (3x + 40) = 180
(5y + 20) + (3y + 40) = 180 (opposite angles of a cyclic quadrilateral add up to 180°) - Solve the first equation for x. Collect the x terms and numbers, then solve.5x + 80 = 180
- Subtract 80 from both sides. We want the number terms on the other side.5x = 180 − 80
5x = 100 - Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.x = 100 ÷ 5 = 20
- Solve the second equation for y. It is the same idea with y.8y + 60 = 180
- Subtract 60 from both sides. We want the number terms on the other side.8y = 180 − 60
8y = 120 - Divide both sides by 8. 8y means 8 × y, so dividing by 8 leaves y on its own.y = 120 ÷ 8 = 15
Check. The angles are 80°, 95°, 100°, 85°. 80 + 100 = 180 and 95 + 85 = 180. ✓
Watch out. Do not mix the letters. The two equations are solved separately.
Method. Opposite angles of a cyclic quadrilateral add up to 180°, so there are two separate equations: one with x and one with y.
- Pair up the opposite angles. A is opposite C, and B is opposite D. Each pair of opposite angles adds up to 180°, so we get two separate equations.(4x + 30) + (7x + 40) = 180
(6y + 18) + (3y + 54) = 180 (opposite angles of a cyclic quadrilateral add up to 180°) - Solve the first equation for x. Collect the x terms and numbers, then solve.11x + 70 = 180
- Subtract 70 from both sides. We want the number terms on the other side.11x = 180 − 70
11x = 110 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 110 ÷ 11 = 10
- Solve the second equation for y. It is the same idea with y.9y + 72 = 180
- Subtract 72 from both sides. We want the number terms on the other side.9y = 180 − 72
9y = 108 - Divide both sides by 9. 9y means 9 × y, so dividing by 9 leaves y on its own.y = 108 ÷ 9 = 12
Check. The angles are 70°, 90°, 110°, 90°. 70 + 110 = 180 and 90 + 90 = 180. ✓
Watch out. Do not mix the letters. The two equations are solved separately.
Method. Opposite angles of a cyclic quadrilateral add up to 180°, so there are two separate equations: one with x and one with y.
- Pair up the opposite angles. A is opposite C, and B is opposite D. Each pair of opposite angles adds up to 180°, so we get two separate equations.(4x + 30) + (7x + 40) = 180
(6y + 18) + (3y + 54) = 180 (opposite angles of a cyclic quadrilateral add up to 180°) - Solve the first equation for x. Collect the x terms and numbers, then solve.11x + 70 = 180
- Subtract 70 from both sides. We want the number terms on the other side.11x = 180 − 70
11x = 110 - Divide both sides by 11. 11x means 11 × x, so dividing by 11 leaves x on its own.x = 110 ÷ 11 = 10
- Solve the second equation for y. It is the same idea with y.9y + 72 = 180
- Subtract 72 from both sides. We want the number terms on the other side.9y = 180 − 72
9y = 108 - Divide both sides by 9. 9y means 9 × y, so dividing by 9 leaves y on its own.y = 108 ÷ 9 = 12
Check. The angles are 70°, 90°, 110°, 90°. 70 + 110 = 180 and 90 + 90 = 180. ✓
Watch out. Do not mix the letters. The two equations are solved separately.
Method. Angle ABC and angle AOC stand on the same arc AC (the arc through D), so use the angle-at-the-centre theorem.
- Spot the two angles. Angle AOC is at the centre, angle ABC is at the circumference, and both stand on the arc AC that contains D.Centre: 160°
Circumference: angle ABC = ? - Halve it. The angle at the centre is twice the angle at the circumference.Angle ABC = 160° ÷ 2 = 80° (the angle at the centre is twice the angle at the circumference)
Check. 2 × 80° = 160°. ✓
Watch out. Check which arc the angles stand on. B is on the other side of the circle from the arc ADC.
Method. ABCD is a cyclic quadrilateral, so opposite angles add up to 180°. Use the answer to part (a).
- Spot the opposite corner. D is opposite B in the quadrilateral ABCD.Angle ABC = 80° (from part a)
- Subtract from 180°. Opposite angles of a cyclic quadrilateral add up to 180°.Angle ADC = 180° − 80° = 100° (opposite angles of a cyclic quadrilateral add up to 180°)
Check. Second method: the reflex angle AOC = 360° − 160° = 200°, and half of it is 100°. ✓
Watch out. Angle ADC is obtuse here, because D is on the minor arc.
Method. Angle ABC and angle AOC stand on the same arc AC (the arc through D), so use the angle-at-the-centre theorem.
- Spot the two angles. Angle AOC is at the centre, angle ABC is at the circumference, and both stand on the arc AC that contains D.Centre: 170°
Circumference: angle ABC = ? - Halve it. The angle at the centre is twice the angle at the circumference.Angle ABC = 170° ÷ 2 = 85° (the angle at the centre is twice the angle at the circumference)
Check. 2 × 85° = 170°. ✓
Watch out. Check which arc the angles stand on. B is on the other side of the circle from the arc ADC.
Method. ABCD is a cyclic quadrilateral, so opposite angles add up to 180°. Use the answer to part (a).
- Spot the opposite corner. D is opposite B in the quadrilateral ABCD.Angle ABC = 85° (from part a)
- Subtract from 180°. Opposite angles of a cyclic quadrilateral add up to 180°.Angle ADC = 180° − 85° = 95° (opposite angles of a cyclic quadrilateral add up to 180°)
Check. Second method: the reflex angle AOC = 360° − 170° = 190°, and half of it is 95°. ✓
Watch out. Angle ADC is obtuse here, because D is on the minor arc.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Spot the opposite corners. B is opposite D in the quadrilateral.Angle ADC = 90°
- Subtract from 180°. Opposite angles of a cyclic quadrilateral add up to 180°.Angle ABC = 180° − 90° = 90°
Check. 90° + 90° = 180°. ✓
Watch out. Opposite corners, not neighbours.
Method. AB and DC are parallel, so the angles BAD and ADC are co-interior angles (they sit between the parallel lines on the same side of the line AD).
- Spot the co-interior angles. Angle BAD and angle ADC are between the parallel lines AB and DC, on the same side of the transversal AD.Angle ADC = 90°
- Subtract from 180°. Co-interior angles between parallel lines add up to 180°.Angle BAD = 180° − 90° = 90° (co-interior angles between parallel lines add up to 180°)
Check. Opposite angle check: angle BCD = 180° − 90° = 90°, and BCD and ADC are equal as they should be in this isosceles trapezium. ✓
Watch out. Parallel lines give co-interior angles that add to 180°, not equal angles.
Method. Opposite angles of a cyclic quadrilateral add up to 180°.
- Spot the opposite corners. B is opposite D in the quadrilateral.Angle ADC = 95°
- Subtract from 180°. Opposite angles of a cyclic quadrilateral add up to 180°.Angle ABC = 180° − 95° = 85°
Check. 95° + 85° = 180°. ✓
Watch out. Opposite corners, not neighbours.
Method. AB and DC are parallel, so the angles BAD and ADC are co-interior angles (they sit between the parallel lines on the same side of the line AD).
- Spot the co-interior angles. Angle BAD and angle ADC are between the parallel lines AB and DC, on the same side of the transversal AD.Angle ADC = 95°
- Subtract from 180°. Co-interior angles between parallel lines add up to 180°.Angle BAD = 180° − 95° = 85° (co-interior angles between parallel lines add up to 180°)
Check. Opposite angle check: angle BCD = 180° − 85° = 95°, and BCD and ADC are equal as they should be in this isosceles trapezium. ✓
Watch out. Parallel lines give co-interior angles that add to 180°, not equal angles.
Method. Test the cyclic quadrilateral rule for the opposite pair A and C.
- Write the rule. If ABCD were cyclic, then opposite angles would add up to 180°.angle A + angle C should be 180°
- Add the given angles. A and C are opposite corners.85° + 100° = 185°
- Compare and conclude. 185° is not 180°.So ABCD cannot be a cyclic quadrilateral. ✓
Check. If angle C were 95° the angles would add to 180°. ✓
Watch out. Check the opposite pairs only. Next-door angles can add to anything.
Method. Combine the parallelogram property with the cyclic quadrilateral rule.
- Parallelogram property. Opposite angles of a parallelogram are equal.angle A = angle C = x
- Cyclic quadrilateral rule. Opposite angles of a cyclic quadrilateral add up to 180°.x + x = 180°
- Solve. Divide by 2.x = 90°
- Say what shape it is. The other pair of angles is equal and also adds to 180°, so every angle is 90°. A parallelogram with right angles is a rectangle.So ABCD is a rectangle. ✓
Check. A rectangle does have a circle through all four corners (centre at the middle). ✓
Watch out. This explains why a non-rectangle parallelogram cannot be cyclic.
Method. Write angle DCE and angle BAD in terms of angle BCD.
- Use the straight line BCE. B, C and E are in a straight line, so angle BCD and angle DCE add up to 180°.angle DCE = 180° − angle BCD
- Use the cyclic quadrilateral. A and C are opposite corners, so angle BAD and angle BCD add up to 180°.angle BAD = 180° − angle BCD
- Compare. Both equal 180° − angle BCD.angle DCE = angle BAD. ✓
Check. If angle BCD = 100° both angles are 80°. ✓
Watch out. This is a proof, so write each reason next to the line it supports.