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Angle sum in triangles and quadrilaterals

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

You learn: the angles of a triangle add up to \(180^\circ\), the angles of a quadrilateral to \(360^\circ\). With this you calculate a missing angle.
📄 Open worksheet (PDF, German)

The video

In the video the three angles of a triangle are laid side by side: they form a straight line. Then a diagonal cuts a quadrilateral into two 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. 0:17 Angle sum in a triangle
  3. 0:46 Proof with parallels
  4. 1:51 Angle sum in a quadrilateral
  5. 2:24 Calculating a missing angle
  6. 2:46 Careful, mistake!
  7. 3:21 Try it yourself

The most important points

Angle sum in a triangle

In every triangle: \(\alpha + \beta + \gamma = 180^\circ\). If you put the three angles side by side, they form a straight angle.

The three angles side by side form a straight line.
The three angles side by side form a straight line.

Proof with parallels

Extend \(\overline{AB}\) beyond \(B\) and draw the parallel to \(\overline{AC}\) through \(B\). \(\alpha\) at \(A\) and at \(B\) are corresponding angles (die Stufenwinkel), \(\gamma\) at \(C\) and at \(B\) are alternate angles (die Wechselwinkel). At \(B\), \(\alpha\), \(\gamma\) and \(\beta\) lie on one straight line: \(\alpha + \beta + \gamma = 180^\circ\).

Proof: the parallel to AC through B. Corresponding and alternate angles show \alpha + \beta + \gamma = 180^\circ.
Proof: the parallel to \(AC\) through \(B\). Corresponding and alternate angles show \(\alpha + \beta + \gamma = 180^\circ\).

Angle sum in a quadrilateral

A diagonal (die Diagonale) cuts the quadrilateral into two triangles. So: \(\alpha + \beta + \gamma + \delta = 2 \cdot 180^\circ = 360^\circ\).

Two triangles: 2 · 180^\circ = 360^\circ.
Two triangles: \(2 \cdot 180^\circ = 360^\circ\).

Missing angle in a triangle: \(\gamma = 180^\circ - 48^\circ - 75^\circ = 57^\circ\). In a quadrilateral you calculate with \(360^\circ\).

Mistake: calculated with 180^\circ in a quadrilateral. The result is a negative angle.
Mistake: calculated with \(180^\circ\) in a quadrilateral. The result is a negative angle.

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

Task

An isosceles triangle has \(40^\circ\) at the top. How big are the two equal angles at the base?

Show solution

\((180^\circ - 40^\circ) : 2 = 70^\circ\)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Winkelsumme im Dreieck (180 Grad) und im Viereck (360 Grad), fehlende Winkel berechnen. 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.