Angles in inscribed right triangles

Key Notes :

🔹 Inscribed Triangle Definition

An inscribed triangle is a triangle whose all vertices lie on a circle.
👉 This circle is called the circumcircle.

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🔹 Right Triangle in a Circle

A right triangle can always be inscribed in a circle.
✨ The hypotenuse becomes the diameter of the circle.

🟢 Important Rule:
➡️ If a triangle is inscribed in a circle and one side is the diameter,
then the angle opposite the diameter is a right angle (90°).

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🔹 Thales’ Theorem 🎯

This amazing theorem states:
👉 Any angle inscribed in a semicircle is a right angle (90°).
So, if you see a triangle inside a circle with one side as diameter → 90° for sure! ✔️

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🔹 Angle Relationships

In an inscribed right triangle:

• The right angle is opposite the diameter
• The other two angles are acute angles
• These two acute angles add up to 90°

🧩 Example: If one angle is 30°, the other must be 60°.

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🔹 Central Angle vs Inscribed Angle 🔵📐

An inscribed angle is half of the central angle that subtends the same arc.
👉 In a right triangle, the arc corresponding to the right angle is a semicircle (180°).
So, inscribed angle = 180° ÷ 2 = 90° 👍

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🔹 Identifying Right Angles in Diagrams 👀

If you spot:

• A triangle inside a circle
• One side as the diameter

🎉 Then you instantly know the angle opposite is 90°!

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🔹 Real-World Uses 🌍

Inscribed right triangles help in:

• Finding missing angles
• Solving circle geometry problems
• Understanding trigonometry basics
• Proving theorems in coordinate geometry

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🌈 Summary

📌 The hypotenuse of an inscribed right triangle is always the diameter.
📌 The angle opposite that diameter is always 90°.
📌 The other two angles are acute and complementary.
📌 This is based on Thales’ Theorem.

Learn with an example

let’s practice!