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Congruence theorems

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

Two triangles are congruent (kongruent) if they fit exactly on top of each other. The congruence theorems say which three parts are enough.
📄 Open worksheet (PDF, German)

The video

In the video we slide and turn a triangle, learn SSS, SAS, ASA and SsA and see why three angles are not enough.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:19 The congruence theorems
  3. 1:00 Three angles?
  4. 1:21 Careful, mistake!
  5. 1:54 Try it yourself

The most important points

Congruent

Congruent means identical in shape and size (deckungsgleich): you can slide, turn (or reflect) one triangle exactly onto the other.

The second triangle fits exactly on the first.
The second triangle fits exactly on the first.

The congruence theorems

SSS: three sides. SAS (SWS): two sides and the angle between them. ASA (WSW): one side and the two angles on it. SsA (SsW): two sides and the angle opposite the longer side.

Three equal angles, but different sizes: only similar.
Three equal angles, but different sizes: only similar.

Careful: If the angle is opposite the shorter side, there can be two different triangles.

The circle around B cuts the ray twice.
The circle around \(B\) cuts the ray twice.

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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Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Surely congruent? a) both \(5\), \(6\), \(7\,\text{cm}\); b) both \(40^\circ\), \(60^\circ\), \(80^\circ\); c) both \(b = 4\,\text{cm}\), \(c = 6\,\text{cm}\), \(\alpha = 50^\circ\).

Show solution

a) yes (SSS); b) no (only angles); c) yes (SAS).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: congruence theorems (Kongruenzsätze: SSS, SWS, WSW, SsW). 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.