Inscribed Angle

An inscribed angle is what you see looking at two points from the circle. Every inscribed angle on the same arc is equal wherever you stand, and it is exactly half the central angle. An angle on a diameter is always 90°, so right triangles are easy to spot in a circle. Move the point around and see why the inscribed angle never changes.

Central Angle vs Inscribed Angle

Central angle (°)80

Position of P30°

👀 Key Relationship

①Inscribed angle on an arc = half the central angle

②Central = inscribed × 2

③The inscribed angle stays the same wherever P sits on that arc!

All Inscribed Angles on the Same Arc Are Equal

Arc size (°)100

🔑 Inscribed Angle Property

①All inscribed angles on the same arc (or chord) are equal

②Inscribed angle on a semicircle = 90° (Thales' theorem)

③This is the most powerful property!

Inscribed Angle on a Semicircle

Semicircle Inscribed Angle

For diameter AB, ∠APB = 90°

An inscribed angle on a diameter is always a right angle

📐 Using the Semicircle Theorem

①Diameter → inscribed angle 90° (Thales)

②Converse: if inscribed angle is 90°, the chord is a diameter

③Hypotenuse of a right triangle = diameter of its circumcircle

Cyclic Quadrilateral

Opposite angles sum to 180°

In a cyclic quadrilateral, opposite angles add up

Exterior-Interior Property

Exterior angle = opposite interior angle

An exterior angle at one vertex equals the opposite interior

💡 Using Cyclic Quadrilaterals

①∠A + ∠C = 180°, ∠B + ∠D = 180°

②Converse: if opposite angles sum to 180°, it is cyclic

③Equal inscribed angles on a chord ⇒ points are concyclic

Work It Out

Example 1

The central angle on an arc is 100°. Find the inscribed angle on the same arc.

The inscribed angle is half the central angle.

inscribed = central ÷ 2

= 100° ÷ 2 = 50°.

= 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.

An angle on a diameter corresponds to a central angle of 180°.

central = 180°

inscribed = 180° ÷ 2 = 90°.

∠APB = 180° ÷ 2 = 90°

▸ 90°

An inscribed angle on a diameter is always 90° (Thales).

Exam Wrap-up

Inscribed Angle Core

inscribed = central/2, same arc → same angle, diameter → 90°

Three big properties

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°

The central angle is twice the inscribed angle.

central = inscribed × 2

= 35° × 2 = 70°.

= 35° × 2 = 70°

🎯 Exam Points

①Inscribed = central / 2 (foundation)

②Same arc → same inscribed angle, regardless of location

③Diameter → inscribed angle 90°

④Cyclic quadrilateral: opposite angles sum to 180°

⑤Strategy: draw a diameter to create a right angle

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Circle & Line

Mean & Spread

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