GCSE Maths: Angles in the same segment (Higher)
Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Angles in the same segment
A chord cuts a circle into two segments (Section 26.1). Take any chord AB. Pick two points C and D on the circumference that are on the same side of the chord, in the same segment. The angle ACB and the angle ADB are equal. It does not matter where in the segment you put the points: the angle made at the circumference by the chord stays the same.
The reason to write is “angles in the same segment are equal”. Why is it true? Each of the two angles is half of the angle at the centre on the same arc AB (the centre theorem from 26.3). Two numbers that are both half of the same angle must be equal. The proof is worked example 26.4.5.
How to spot a pair. (1) Find the chord both angles share: it joins the other end of each arm. (2) Check the two corners are on the same side of the chord. In a picture with two chords crossing, like the one in the next examples, you can find several pairs: for each side of a quadrilateral ABCD, the two other corners see it at equal angles. If the two corners are on opposite sides of the chord, the angles are not equal: they add up to 180°, which is the subject of the next section.
Key ideas
- Angles made by the same chord, with their corners on the same side of it (the same segment), are equal.
- Reason: “angles in the same segment are equal”.
- To find a pair: name the chord, then pick another corner on the same side of it.
- Each angle is half of the angle at the centre on the same arc.
- In a cyclic quadrilateral ABCD with diagonals AC and BD: BAC = BDC, ABD = ACD, CAD = CBD, ACB = ADB.
Watch out
The two angles must be made from the same chord. Angle ACB and angle ACD share one arm but are made from different chords, so they are not equal in general.Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at A and B. So both are made from the chord AB.Angle ACB and angle ADB both stand on chord AB.
- Check the two points are on the same side of the chord. C and D are both on the same side of the line AB, so they lie in the same segment.C and D are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle ADB = angle ACB = 50°
Check. Angle at the centre on the same arc would be 2 × 50° = 100° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
- Work out the value of x.
- Work out the size of angle ACB.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.3x + 5 = 2x + 20
- Subtract 2x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.3x + 5 − 2x = 2x + 20 − 2x
x + 5 = 20 - Subtract 5 from both sides. We want the number terms on the other side.x = 20 − 5
x = 15 - Read off x. x = 15
- Find the angle. Put x into the expression.Angle ACB = 3 × 15 + 5 = 50°
Check. Other angle: 2 × 15 + 20 = 50°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Use the triangle angle sum first to find an angle that is in the same segment as the one we want.
- Pick out triangle ABE. The chords cross at E. Triangle ABE contains the angle BAC and the angle AEB that we are given.triangle ABE: angle BAE = 35°, angle AEB = 105°
- Find angle ABE (that is, angle ABD). Triangle ABE has angle sum 180°. We know two of its angles.Angle ABD = 180° − 35° − 105° = 40° (the angles in a triangle add up to 180°)
- Match it with an angle in the same segment. Angles ABD and ACD are both made from the chord AD, with B and C on the same side of it.Angle ACD = angle ABD = 40° (angles in the same segment are equal)
Check. In triangle DEC the angles are 35° (angle BDC, same segment as BAC), 40° and 105° (vertically opposite AEB): 35 + 40 + 105 = 180. ✓
Watch out. A "same segment" pair needs the same chord. Name the chord before you say the angles are equal.
Method. Use the same-segment theorem to find a second angle in triangle DEC, then the angle sum and vertically opposite angles.
- Find angle BDC. Angles BAC and BDC are both made from the chord BC, with A and D on the same side of it.Angle BDC = angle BAC = 40° (angles in the same segment are equal)
- Use triangle DEC. Its angles at D and C are now known: angle EDC is angle BDC and angle ECD is angle ACD. The angles add up to 180°.Angle DEC = 180° − 40° − 45° = 95° (the angles in a triangle add up to 180°)
- Use vertically opposite angles. The chords cross at E, so angle AEB is opposite angle DEC.Angle AEB = angle DEC = 95° (vertically opposite angles are equal)
Check. Second route: angle ABD = angle ACD = 45° (same segment, chord AD). In triangle ABE: 180° − 40° − 45° = 95°. ✓
Watch out. Do not forget vertically opposite angles at the point where two chords cross.
Method. Use the centre theorem to show that each angle is half of the same centre angle.
- Draw the angle at the centre. Join A and B to the centre O. Angle AOB stands on the same arc AB as both angles.Let angle AOB = 2x
- Use the centre theorem on C. The angle at the centre is twice the angle at the circumference, so the angle at the circumference is half the angle at the centre.angle ACB = 2x ÷ 2 = x
- Use the centre theorem on D. D is also on the circumference and its angle stands on the same arc AB.angle ADB = 2x ÷ 2 = x
- Compare. Both angles equal x.angle ACB = angle ADB. Angles in the same segment are equal. ✓
Check. If angle AOB = 100° then both angles are 50°, wherever C and D are on the major arc. ✓
Watch out. This proof assumes C and D are on the major arc. For the minor arc, both angles are half of the reflex angle at the centre.
Exercise 26D Angles in the same segment
Non-calculatorKey words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at A and B. So both are made from the chord AB.Angle ACB and angle ADB both stand on chord AB.
- Check the two points are on the same side of the chord. C and D are both on the same side of the line AB, so they lie in the same segment.C and D are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle ADB = angle ACB = 40°
Check. Angle at the centre on the same arc would be 2 × 40° = 80° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at B and C. So both are made from the chord BC.Angle BDC and angle BAC both stand on chord BC.
- Check the two points are on the same side of the chord. D and A are both on the same side of the line BC, so they lie in the same segment.D and A are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle BAC = angle BDC = 60°
Check. Angle at the centre on the same arc would be 2 × 60° = 120° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at C and D. So both are made from the chord CD.Angle CAD and angle CBD both stand on chord CD.
- Check the two points are on the same side of the chord. A and B are both on the same side of the line CD, so they lie in the same segment.A and B are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle CBD = angle CAD = 60°
Check. Angle at the centre on the same arc would be 2 × 60° = 120° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at D and A. So both are made from the chord DA.Angle DBA and angle DCA both stand on chord DA.
- Check the two points are on the same side of the chord. B and C are both on the same side of the line DA, so they lie in the same segment.B and C are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle DCA = angle DBA = 40°
Check. Angle at the centre on the same arc would be 2 × 40° = 80° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at A and B. So both are made from the chord AB.Angle ACB and angle ADB both stand on chord AB.
- Check the two points are on the same side of the chord. C and D are both on the same side of the line AB, so they lie in the same segment.C and D are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle ADB = angle ACB = 30°
Check. Angle at the centre on the same arc would be 2 × 30° = 60° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Key words. A chord cuts a circle into two segments. Two angles are in the same segment if their corners are on the same side of the chord.
Method. Two angles made from the same chord, with their corners on the same side of it, are in the same segment and are equal.
- Find the chord that both angles are drawn from. Both angles have their arms ending at B and C. So both are made from the chord BC.Angle BDC and angle BAC both stand on chord BC.
- Check the two points are on the same side of the chord. D and A are both on the same side of the line BC, so they lie in the same segment.D and A are in the same segment of the circle.
- Use the theorem. Angles in the same segment are equal.Angle BAC = angle BDC = 25°
Check. Angle at the centre on the same arc would be 2 × 25° = 50° for either point. ✓
Watch out. Make sure both corners are on the same side of the chord. Angles on opposite sides of the chord add up to 180°, not equal.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.3x + 15 = x + 35
- Subtract x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.3x + 15 − x = x + 35 − x
2x + 15 = 35 - Subtract 15 from both sides. We want the number terms on the other side.2x = 35 − 15
2x = 20 - Divide both sides by 2. 2x means 2 × x, so dividing by 2 leaves x on its own.x = 20 ÷ 2 = 10
- Find the angle. Put x into the expression.Angle ACB = 3 × 10 + 15 = 45°
Check. Other angle: 1 × 10 + 35 = 45°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.3x + 15 = x + 35
- Subtract x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.3x + 15 − x = x + 35 − x
2x + 15 = 35 - Subtract 15 from both sides. We want the number terms on the other side.2x = 35 − 15
2x = 20 - Divide both sides by 2. 2x means 2 × x, so dividing by 2 leaves x on its own.x = 20 ÷ 2 = 10
- Find the angle. Put x into the expression.Angle ACB = 3 × 10 + 15 = 45°
Check. Other angle: 1 × 10 + 35 = 45°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.4x − 8 = 2x + 16
- Subtract 2x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.4x − 8 − 2x = 2x + 16 − 2x
2x − 8 = 16 - Add 8 to both sides. We want the number terms on the other side.2x = 16 + 8
2x = 24 - Divide both sides by 2. 2x means 2 × x, so dividing by 2 leaves x on its own.x = 24 ÷ 2 = 12
- Find the angle. Put x into the expression.Angle ACB = 4 × 12 − 8 = 40°
Check. Other angle: 2 × 12 + 16 = 40°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.4x − 8 = 2x + 16
- Subtract 2x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.4x − 8 − 2x = 2x + 16 − 2x
2x − 8 = 16 - Add 8 to both sides. We want the number terms on the other side.2x = 16 + 8
2x = 24 - Divide both sides by 2. 2x means 2 × x, so dividing by 2 leaves x on its own.x = 24 ÷ 2 = 12
- Find the angle. Put x into the expression.Angle ACB = 4 × 12 − 8 = 40°
Check. Other angle: 2 × 12 + 16 = 40°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.5x + 5 = x + 25
- Subtract x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.5x + 5 − x = x + 25 − x
4x + 5 = 25 - Subtract 5 from both sides. We want the number terms on the other side.4x = 25 − 5
4x = 20 - Divide both sides by 4. 4x means 4 × x, so dividing by 4 leaves x on its own.x = 20 ÷ 4 = 5
- Find the angle. Put x into the expression.Angle ACB = 5 × 5 + 5 = 30°
Check. Other angle: 1 × 5 + 25 = 30°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles in the same segment are equal, so the two expressions are equal. Solve for x, then substitute.
- Spot that the two angles are equal. Both angles are made from the chord AB with the corners C and D on the same side of it. They are in the same segment.Angle ACB = angle ADB (angles in the same segment are equal)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.5x + 5 = x + 25
- Subtract x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.5x + 5 − x = x + 25 − x
4x + 5 = 25 - Subtract 5 from both sides. We want the number terms on the other side.4x = 25 − 5
4x = 20 - Divide both sides by 4. 4x means 4 × x, so dividing by 4 leaves x on its own.x = 20 ÷ 4 = 5
- Find the angle. Put x into the expression.Angle ACB = 5 × 5 + 5 = 30°
Check. Other angle: 1 × 5 + 25 = 30°. They match. ✓
Watch out. Equal angles means the expressions are equal (set one equal to the other), not that they add up to 180°.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at A has its arms going to B and C. So it is made from the chord BC.Angle BAC is made from chord BC.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (B and C). The other two are both on the same side of it, in the same segment.The corners left over are A and D.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle BAC = angle BDC (angles in the same segment are equal)
Check. Both angles go to the ends of chord BC. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at D has its arms going to A and B. So it is made from the chord AB.Angle ADB is made from chord AB.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (A and B). The other two are both on the same side of it, in the same segment.The corners left over are D and C.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle ADB = angle ACB (angles in the same segment are equal)
Check. Both angles go to the ends of chord AB. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at A has its arms going to C and D. So it is made from the chord CD.Angle CAD is made from chord CD.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (C and D). The other two are both on the same side of it, in the same segment.The corners left over are A and B.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle CAD = angle CBD (angles in the same segment are equal)
Check. Both angles go to the ends of chord CD. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at C has its arms going to A and D. So it is made from the chord AD.Angle ACD is made from chord AD.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (A and D). The other two are both on the same side of it, in the same segment.The corners left over are C and B.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle ACD = angle ABD (angles in the same segment are equal)
Check. Both angles go to the ends of chord AD. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at A has its arms going to B and C. So it is made from the chord BC.Angle BAC is made from chord BC.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (B and C). The other two are both on the same side of it, in the same segment.The corners left over are A and D.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle BAC = angle BDC (angles in the same segment are equal)
Check. Both angles go to the ends of chord BC. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at D has its arms going to A and B. So it is made from the chord AB.Angle ADB is made from chord AB.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (A and B). The other two are both on the same side of it, in the same segment.The corners left over are D and C.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle ADB = angle ACB (angles in the same segment are equal)
Check. Both angles go to the ends of chord AB. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at A has its arms going to C and D. So it is made from the chord CD.Angle CAD is made from chord CD.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (C and D). The other two are both on the same side of it, in the same segment.The corners left over are A and B.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle CAD = angle CBD (angles in the same segment are equal)
Check. Both angles go to the ends of chord CD. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Angles made from the same chord, with their corners on the same side of it (in the same segment), are equal.
- Find the chord that the angle is drawn from. The angle at C has its arms going to A and D. So it is made from the chord AD.Angle ACD is made from chord AD.
- Find the other corner on the same side of that chord. The four points are A, B, C and D. Two of them are the ends of the chord (A and D). The other two are both on the same side of it, in the same segment.The corners left over are C and B.
- Write the equal angle. Angles in the same segment are equal, so use the other corner with the same two arms.Angle ACD = angle ABD (angles in the same segment are equal)
Check. Both angles go to the ends of chord AD. ✓
Watch out. Check that the two corners really are on the same side of the chord. For example, angle BAC and angle BDC are both made from chord BC.
Method. Use the triangle angle sum first to find an angle that is in the same segment as the one we want.
- Pick out triangle ABE. The chords cross at E. Triangle ABE contains the angle BAC and the angle AEB that we are given.triangle ABE: angle BAE = 30°, angle AEB = 110°
- Find angle ABE (that is, angle ABD). Triangle ABE has angle sum 180°. We know two of its angles.Angle ABD = 180° − 30° − 110° = 40° (the angles in a triangle add up to 180°)
- Match it with an angle in the same segment. Angles ABD and ACD are both made from the chord AD, with B and C on the same side of it.Angle ACD = angle ABD = 40° (angles in the same segment are equal)
Check. In triangle DEC the angles are 30° (angle BDC, same segment as BAC), 40° and 110° (vertically opposite AEB): 30 + 40 + 110 = 180. ✓
Watch out. A "same segment" pair needs the same chord. Name the chord before you say the angles are equal.
Method. Use the triangle angle sum first to find an angle that is in the same segment as the one we want.
- Pick out triangle ABE. The chords cross at E. Triangle ABE contains the angle BAC and the angle AEB that we are given.triangle ABE: angle BAE = 40°, angle AEB = 95°
- Find angle ABE (that is, angle ABD). Triangle ABE has angle sum 180°. We know two of its angles.Angle ABD = 180° − 40° − 95° = 45° (the angles in a triangle add up to 180°)
- Match it with an angle in the same segment. Angles ABD and ACD are both made from the chord AD, with B and C on the same side of it.Angle ACD = angle ABD = 45° (angles in the same segment are equal)
Check. In triangle DEC the angles are 40° (angle BDC, same segment as BAC), 45° and 95° (vertically opposite AEB): 40 + 45 + 95 = 180. ✓
Watch out. A "same segment" pair needs the same chord. Name the chord before you say the angles are equal.
Method. Use the triangle angle sum first to find an angle that is in the same segment as the one we want.
- Pick out triangle ABE. The chords cross at E. Triangle ABE contains the angle BAC and the angle AEB that we are given.triangle ABE: angle BAE = 30°, angle AEB = 115°
- Find angle ABE (that is, angle ABD). Triangle ABE has angle sum 180°. We know two of its angles.Angle ABD = 180° − 30° − 115° = 35° (the angles in a triangle add up to 180°)
- Match it with an angle in the same segment. Angles ABD and ACD are both made from the chord AD, with B and C on the same side of it.Angle ACD = angle ABD = 35° (angles in the same segment are equal)
Check. In triangle DEC the angles are 30° (angle BDC, same segment as BAC), 35° and 115° (vertically opposite AEB): 30 + 35 + 115 = 180. ✓
Watch out. A "same segment" pair needs the same chord. Name the chord before you say the angles are equal.
Method. Use the same-segment theorem to find a second angle in triangle DEC, then the angle sum and vertically opposite angles.
- Find angle BDC. Angles BAC and BDC are both made from the chord BC, with A and D on the same side of it.Angle BDC = angle BAC = 40° (angles in the same segment are equal)
- Use triangle DEC. Its angles at D and C are now known: angle EDC is angle BDC and angle ECD is angle ACD. The angles add up to 180°.Angle DEC = 180° − 40° − 45° = 95° (the angles in a triangle add up to 180°)
- Use vertically opposite angles. The chords cross at E, so angle AEB is opposite angle DEC.Angle AEB = angle DEC = 95° (vertically opposite angles are equal)
Check. Second route: angle ABD = angle ACD = 45° (same segment, chord AD). In triangle ABE: 180° − 40° − 45° = 95°. ✓
Watch out. Do not forget vertically opposite angles at the point where two chords cross.
Method. Use the same-segment theorem to find a second angle in triangle DEC, then the angle sum and vertically opposite angles.
- Find angle BDC. Angles BAC and BDC are both made from the chord BC, with A and D on the same side of it.Angle BDC = angle BAC = 35° (angles in the same segment are equal)
- Use triangle DEC. Its angles at D and C are now known: angle EDC is angle BDC and angle ECD is angle ACD. The angles add up to 180°.Angle DEC = 180° − 35° − 45° = 100° (the angles in a triangle add up to 180°)
- Use vertically opposite angles. The chords cross at E, so angle AEB is opposite angle DEC.Angle AEB = angle DEC = 100° (vertically opposite angles are equal)
Check. Second route: angle ABD = angle ACD = 45° (same segment, chord AD). In triangle ABE: 180° − 35° − 45° = 100°. ✓
Watch out. Do not forget vertically opposite angles at the point where two chords cross.
Method. Use the same-segment theorem to find a second angle in triangle DEC, then the angle sum and vertically opposite angles.
- Find angle BDC. Angles BAC and BDC are both made from the chord BC, with A and D on the same side of it.Angle BDC = angle BAC = 30° (angles in the same segment are equal)
- Use triangle DEC. Its angles at D and C are now known: angle EDC is angle BDC and angle ECD is angle ACD. The angles add up to 180°.Angle DEC = 180° − 30° − 40° = 110° (the angles in a triangle add up to 180°)
- Use vertically opposite angles. The chords cross at E, so angle AEB is opposite angle DEC.Angle AEB = angle DEC = 110° (vertically opposite angles are equal)
Check. Second route: angle ABD = angle ACD = 40° (same segment, chord AD). In triangle ABE: 180° − 30° − 40° = 110°. ✓
Watch out. Do not forget vertically opposite angles at the point where two chords cross.
Method. Check whether the two corners are on the same side of the chord BD.
- Find which side each corner is on. A is on one side of the line BD and C is on the other side. So they are in different segments.The two angles are not in the same segment.
- Use the cyclic quadrilateral rule. ABCD is a cyclic quadrilateral, so opposite angles add up to 180°.angle BCD = 180° − 80° = 100°
- State the conclusion. The angles are not equal, so Ali is wrong.80° ≠ 100°. Ali is wrong.
Check. Angle BCD stands on the major arc side, so it is bigger than angle BAD on the minor side. 80° + 100° = 180°. ✓
Watch out. Equal angles need corners on the same side. Opposite sides give a sum of 180°.
Method. Use the centre theorem for each angle separately.
- Name the centre angle. Both angles stand on arc AB, so they both link to angle AOB at the centre.angle AOB = 2 × angle ACB
- Do the same for D. The same centre angle works for D.angle AOB = 2 × angle ADB
- Compare. Both angles are equal to angle AOB ÷ 2.So angle ACB = angle ADB. ✓
Check. The position of C and D does not appear in the working, so the result is the same wherever they are in the segment. ✓
Watch out. This is a proof, so write the reason beside each line.