Inscribed angles

key notes :

🔵 What is an Inscribed Angle?

An inscribed angle is an angle whose vertex lies on the circle, and its sides (arms) touch the circle.
👉 It “sits” on the circle’s boundary.

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🔴 Intercepted Arc 🎯

The intercepted arc is the part of the circle cut off or covered by the inscribed angle.

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🟢 Inscribed Angle Theorem 💡

The measure of an inscribed angle = ½ × measure of the intercepted arc.
📌 If the arc is 80°, the inscribed angle is 40°.

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🟣 Angles on the Same Arc Are Equal ⚖️

If two inscribed angles intercept the same arc, they have the same measure.

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🟠 Angle in a Semicircle = 90° ⭕

If the endpoints of an inscribed angle lie on the diameter, the angle is a right angle (90°).
👉 Also called Thales’ Theorem.

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🟡 Inscribed vs. Central Angles 🆚

• Central angle = vertex at the center; equals the arc.
• Inscribed angle = vertex on the circle; equals half of the arc.

📌 Example: If a central angle is 100°, the inscribed angle on the same arc is 50°.

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🔵 Useful for Solving Problems ✏️

You can use inscribed angles to find:
✔ Missing angle measures
✔ Arc lengths
✔ Right triangles inside circles
✔ Relations with central angles

Learn with an example

let’s practice!