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Inscribed angle theorem

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

You learn central angles (Zentriwinkel) and inscribed angles (Umfangswinkel). The central angle is twice the inscribed angle. The theorem of Thales is a special case.
📄 Open worksheet (PDF, German)

The video

In the video a point \(P\) moves along the circle. The inscribed angle stays the same. Then comes the proof with two isosceles triangles.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:18 The inscribed angle
  3. 0:33 The theorem
  4. 1:02 Why?
  5. 2:02 Careful, mistake!
  6. 2:23 Try it yourself

The most important points

Central angle and inscribed angle

The angle \(\angle AMB\) at the centre is the central angle (der Zentriwinkel), the angle \(\angle APB\) at the edge is the inscribed angle (der Umfangswinkel).

The inscribed angle stays 40^\circ when P moves along the circle.
The inscribed angle stays \(40^\circ\) when \(P\) moves along the circle.

Inscribed angle theorem (der Kreiswinkelsatz)

The central angle is twice as big as the inscribed angle (over the same chord, \(P\) on the long arc). For a diameter: \(180^\circ = 2 \cdot 90^\circ\) (theorem of Thales).

Proof: 20^\circ + 20^\circ = 40^\circ and 360^\circ - 140^\circ - 140^\circ = 80^\circ.
Proof: \(20^\circ + 20^\circ = 40^\circ\) and \(360^\circ - 140^\circ - 140^\circ = 80^\circ\).
Mistake: P lies on the other arc. There the inscribed angle is 180^\circ - 40^\circ = 140^\circ.
Mistake: \(P\) lies on the other arc. There the inscribed angle is \(180^\circ - 40^\circ = 140^\circ\).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
Page 2/3
Show the worksheet solutions
The worksheet – Page 3
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Try it yourself

Task

The central angle \(\angle AMB\) is \(70^\circ\). How big is the inscribed angle \(\angle APB\) (\(P\) on the long arc)?

Show solution

\(70^\circ : 2 = 35^\circ\)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Kreiswinkelsatz (Zentriwinkel, Umfangswinkel, Sehne, Satz des Thales als Sonderfall). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Use simple German sentences.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.