Angles in the Same Segment

Angles at the circle that stand on the same arc, on the same side, are equal: spot them, including in the crossed chords bow tie, and write the reason.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 7 real past-paper questions below

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Same Arc, Same Angle

Pick two points A and B on a circle. Join them to a point C on the circle, and to another point D on the same side of AB.

Rule

Angle ACB equals angle ADB. Both stand on the same arc AB. Reason: angles in the same segment are equal.

Key idea

Look for two angles that start from the same two points and meet on the circle, on the same side. Books say both angles are “subtended by” the arc AB. It just means they stand on it.

Spot the Equal Angle

The 1-mark question asks for the angle and a reason. Find which two points the angle stands on first.

Worked example

1
$∠ A C B = 4 3^{\circ}$ — Given. Its two arms go to A and B, so it stands on the arc AB.
2
$∠ A D B also stands on A B$ — Its arms go to A and B too, and D is on the same side of AB as C. So both angles are in the same segment.
3
$∠ A D B = 4 3^{\circ}$ — Angles in the same segment are equal, so copy the 43°. Write that reason next to your answer.

Common mistake

Using a point on the other side of AB. That angle is $18 0^{\circ} - 4 3^{\circ} = 13 7^{\circ}$ , because the four points make a cyclic quadrilateral.

The Bow Tie

When two chords cross inside a circle, you get a bow tie shape. It hides two pairs of equal angles.

A, B, C and D are on a circle. Chords AC and BD cross at X. $∠ B A C = 3 5^{\circ}$ and $∠ A B D = 2 9^{\circ}$ . Find angle CXD.

Worked example

1
$∠ B D C = ∠ B A C = 3 5^{\circ}$ — Both stand on BC, and A and D are on the same side of BC. Angles in the same segment are equal.
2
$∠ A C D = ∠ A B D = 2 9^{\circ}$ — Both stand on AD, and B and C are on the same side of AD. Same reason.
3
$∠ C X D = 18 0^{\circ} - 3 5^{\circ} - 2 9^{\circ}$ — Triangle CXD has 35° at D and 29° at C. Angles in a triangle add up to 180°.
4
$∠ C X D = 11 6^{\circ}$ — 35 + 29 = 64, and 180 − 64 = 116.

Key idea

The equal angles sit at the ends of the bow tie, opposite each other across the crossing.

Copy, Then Use the Triangle

Exams often give one angle to copy across, then a triangle to finish. Copy first, then add up to 180°.

ABCD is a cyclic quadrilateral with diagonals AC and BD. $∠ A D B = 3 8^{\circ}$ and $∠ A B C = 9 7^{\circ}$ . Find angle ACB and angle BDC.

Worked example

1
$∠ A D B = 3 8^{\circ}$ — Given. It stands on AB.
2
$∠ A C B = 3 8^{\circ}$ — C is on the same side of AB as D, so ACB stands on AB too. Angles in the same segment are equal.
3
$∠ B A C = 18 0^{\circ} - 9 7^{\circ} - 3 8^{\circ}$ — Triangle ABC now has two known angles: 97° at B and 38° at C. Angles in a triangle add up to 180°.
4
$∠ B A C = 4 5^{\circ}$ — 97 + 38 = 135, and 180 − 135 = 45.
5
$∠ B D C = 4 5^{\circ}$ — BDC and BAC both stand on BC, on the same side. Angles in the same segment are equal, so copy the 45°.
6
$∠ A D C = 3 8^{\circ} + 4 5^{\circ} = 8 3^{\circ}$ — Check: 83 + 97 = 180, as opposite angles of a cyclic quadrilateral must be.

Key idea

Write the line $18 0^{\circ} - 9 7^{\circ} - 3 8^{\circ}$ even if you can do it in your head. If you slip on the arithmetic, that line still earns a mark for working.

Which Method? Read the Question

Find the two points each angle stands on.

You see	You do
Two edge angles on one arc, same side	they are equal
Two edge angles on one arc, opposite sides	Cyclic Quadrilaterals
One angle at the centre O	Angle at the Centre
Two crossing chords	bow tie: two equal pairs
Equal angles plus a triangle	copy the angle, then angles in a triangle add up to 180°

Key idea

Write the full reason: “angles in the same segment are equal”. “Angles on the same arc are equal” also scores. Some exams give the reason mark only when your angle is right, so find the angle carefully first.

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Real past-paper questions on angles in the same segment

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