MO Z7–I–6 2021

In the triangle ABC, point D lies on the AC side and point E on the BC side. The sizes of the angles ABD, BAE, CAE and CBD are 30°, 60°, 20° and 30°, respectively. Find the size of the AED angle.

Correct answer:

AED =  20 °

Step-by-step explanation:

$$\begin{gathered} ABD=30\text{ ° } \\ BAE=60\text{ ° } \\ CAE=20\text{ ° } \\ CBD=30\text{ ° } \\ \alpha =CAE+BAE=20+60=80\text{ ° } \\ \beta =ABD+CBD=30+30=60\text{ ° } \\ ADB=180-\alpha -ABD=180-80-30=70\text{ ° } \\ \gamma =180-\alpha -\beta =180-80-60=40\text{ ° } \\ AEB=180-BAE-\beta =180-60-60=60\text{ ° } \\ ASB=180-BAE-ABD=180-60-30=90\text{ ° } \\ AED+BDE=90 \\ ADB=180-CAE-(AED+BDE) \\ ADB=180-CAE-90=180-20-90=70\text{ ° } \\ AEC=180-\gamma -CAE=180-40-20=120\text{ ° } \\ CDB=180-\gamma -CBD=180-40-30=110\text{ ° } \\ \mathrm{\mid }AS\mathrm{\mid }=\mathrm{\mid }ES\mathrm{\mid } \\ \mathrm{\Delta }AEB:60\mathrm{°}-60\mathrm{°}-60\mathrm{°} \\ \mathrm{\Delta }ASD\dots DSE \\ AED=CAE=20=20\text{°} \end{gathered}$$

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