MO Z7–I–6 2021

In 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:

∠ABD=30 ∘ ∠BAE=60 ∘ ∠CAE=20 ∘ ∠CBD=30 ∘  α=∠CAE+∠BAE=20+60=80 ∘ β=∠ABD+∠CBD=30+30=60 ∘ ∠ADB=180−α−∠ABD=180−80−30=70 ∘ γ=180−α−β=180−80−60=40 ∘  ∠AEB=180−∠BAE−β=180−60−60=60 ∘ ∠ASB=180−∠BAE−∠ABD=180−60−30=90 ∘  AED+BDE = 90 ADB = 180 − CAE −(AED+BDE) ∠ADB=180−∠CAE−90=180−20−90=70 ∘  ∠AEC=180−γ−∠CAE=180−40−20=120 ∘ ∠CDB=180−γ−∠CBD=180−40−30=110 ∘  ∣AS∣ = ∣ES∣  Δ∠AEB: 60° − 60° − 60° ΔASD … DSE AED=∠AED=∠CAE=20=20∘

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