When a point P on a circle looks at two other points A and B on the circle, the angle ∠APB is called the inscribed angle on arc AB.
An inscribed angle is exactly half the central angle ∠AOB on the same arc, and it does not change wherever P moves along the same side of the circle.
So an inscribed angle on a diameter is half of 180°, that is 90°, and the opposite angles of a quadrilateral inscribed in a circle add up to 180°.
Moving P around the circle in the first picture changes how the segments lean, but the angle marked at P keeps the same size.
Central and inscribed angles
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The central angle ∠AOB on arc AB and the inscribed angle ∠APB seen from a point P on the circle
80
30
👀 How inscribed and central angles relate
①An inscribed angle on an arc is half the central angle on that arc.
②Conversely, the central angle is twice the inscribed angle.
③As long as P stays on the same side, the inscribed angle is the same wherever P is. With a central angle of 80° in the picture, the inscribed angle stays at 40° however P moves.
Inscribed angles on the same arc are equal
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The inscribed angles at three points P₁, P₂, P₃ looking at the same arc AB
100
🔑 Property of inscribed angles
①In one circle, inscribed angles on the same arc (or on equal arcs) are all equal.
②An inscribed angle on a semicircle, that is on a diameter, is 90°.
③Changing the size of the arc changes all three inscribed angles together, and they always share the same value.
The inscribed angle on a diameter
Inscribed angle on a semicircle
inscribed angle on diameter AB: ∠APB = 90°
The central angle on a diameter is 180°, so the inscribed angle is half of it, 90°.
📐 Diameters and right angles
①Joining a point on a circle to both ends of a diameter always makes a right angle at that point.
②Conversely, if an inscribed angle is 90°, the chord it looks at is a diameter.
③So the hypotenuse of a right triangle is a diameter of the circle through its three vertices.
Quadrilaterals inscribed in a circle
Property of a cyclic quadrilateral
sum of two opposite angles = 180°
The arcs that the two angles look at together make the whole circle (central angle 360°).
Exterior angle property
an exterior angle = the interior angle opposite the adjacent interior angle
The exterior angle at a vertex equals the interior angle at the opposite vertex.
💡 Using cyclic quadrilaterals
①In a quadrilateral ABCD inscribed in a circle, ∠A + ∠C = 180° and ∠B + ∠D = 180°.
②Conversely, a quadrilateral whose opposite angles add up to 180° can be inscribed in a circle.
③If two angles looking at one segment from the same side are equal, the four points lie on one circle.
Worked examples
Example 1
The central angle on an arc is 100°. Find the inscribed angle on the same arc.
1
An inscribed angle is half the central angle on the same arc.
inscribed = central ÷ 2
2
Divide the central angle 100° by 2.
= 100° ÷ 2 = 50°
▸ 50°
An inscribed angle is half the central angle on the same arc.
Example 2
For a diameter AB and a point P on the circle, find ∠APB.
1
The central angle on diameter AB is 180°.
central = 180°
2
The inscribed angle is half of it.
∠APB = 180° ÷ 2 = 90°
▸ 90°
The inscribed angle on a diameter is always 90°.
Exam checklist
Key facts about inscribed angles
inscribed angle = central angle ÷ 2 · same arc → same inscribed angle · diameter → 90°
Most inscribed-angle problems are solved by one of these three facts.
G9 school-exam type
If the inscribed angle on an arc is 35°, what is the central angle on the same arc?
①17.5°
②35°
③55°
④70°
⑤145°
▸ ④ 70°
1
The central angle is twice the inscribed angle.
central = inscribed × 2
2
= 35° × 2 = 70°.
= 35° × 2 = 70°
🎯 Exam points
①An inscribed angle is half the central angle on the same arc.
②Inscribed angles on the same arc are equal wherever the point is.
③The inscribed angle on a diameter is 90°.
④The opposite angles of a cyclic quadrilateral add up to 180°.
⑤Many problems open up once a diameter is drawn as an extra line to create a right angle.