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Base angles and exterior angles

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

In an isosceles triangle the base angles (Basiswinkel) are equal. An exterior angle (Außenwinkel) is as big as the two non-adjacent interior angles together.
📄 Open worksheet (PDF, German)

The video

In the video we calculate base angles, extend a side of a triangle and discover the exterior angle theorem \(\beta' = \alpha + \gamma\).

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

Contents of the video

  1. 0:00 The base angle theorem
  2. 0:27 Calculating angles
  3. 0:49 The exterior angle
  4. 1:10 The exterior angle theorem
  5. 1:42 Careful, mistake!
  6. 2:01 Try it yourself

The most important points

Base angle theorem

In an isosceles triangle the two base angles are equal: \(\alpha = \beta\). Conversely: if two angles are equal, the triangle is isosceles.

Isosceles triangle with its axis of symmetry: \alpha = \beta.
Isosceles triangle with its axis of symmetry: \(\alpha = \beta\).

Example: \(\gamma = 40^\circ\). Then \(\alpha + \beta = 140^\circ\), so \(\alpha = \beta = 70^\circ\).

Exterior angle theorem

The exterior angle \(\beta'\) is the supplementary angle of \(\beta\): \(\beta' = 180^\circ - \beta\). It equals the two non-adjacent interior angles: \(\beta' = \alpha + \gamma\).

\beta' = 120^\circ = 50^\circ + 70^\circ.
\(\beta' = 120^\circ = 50^\circ + 70^\circ\).

Careful: \(\beta' = \alpha + \beta\) is wrong. \(\beta\) lies right next to \(\beta'\).

Correct: the two opposite interior angles.
Correct: the two opposite interior angles.

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
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The worksheet – Page 3
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The worksheet – Page 4
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The worksheet – Page 5
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Show the worksheet solutions
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Try it yourself

Task

An isosceles triangle has \(\gamma = 50^\circ\) at the top. Calculate the base angles and the exterior angle at \(B\).

Show solution

\(\alpha = \beta = (180^\circ - 50^\circ) : 2 = 65^\circ\); \(\beta' = 180^\circ - 65^\circ = 115^\circ = 65^\circ + 50^\circ\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: base angle theorem and exterior angle theorem (Basiswinkelsatz, Außenwinkelsatz). 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.