Question

plz do this circle problem fast

Answer 1

How do i find ∠(ODC)∠(ODC)? so i wanted to show my teacher this but his not available yet. Can someone help me to solve? geometry problems which on circle seems hard to me. Thanks!

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It's a difficult one but it doesn't matter.
So, let's start-

First join AO and AC,
Here,
∠AOC=2∠ABC
∠AOC=2∠30°
=60°
Now,
AO=OC(Radius)
thus,
∠CAO=∠OAC=z

In ΔAOC-
∠CAO+∠OAC+∠AOC=180°
z+z+60°=180°
2z=120°
z=60°=∠ACO=∠OAC
Therefore,
ΔAOC is an equilateral Δ.
Hence,
AO=OC=AC

Now,
∠ACB=∠ACO+OCB=60°+20°
=80°
In ΔABC-
∠ABC+∠BAC+∠ACB=180°
30°+∠BAC+80°=180°
∠BAC=180°-110°
=70°
Now,
∠ACD=∠ACO-∠DCO
=60°-20°
=40°

In ΔADC-
∠ADC+∠DAC+∠ACD=180°
70°+40°+∠ADC=180°
∠ADC=70°
∠ADC=∠DAC=70°
thus,
AC=CD=OC
hence,
∠ODC=∠COD=x°
In ΔDOC-
∠ODC+COD+∠DCO=180°
x°+x°+20°=180°
2x=160°
x=80°=∠ODC

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Thank You!