MHB Find the angle, cyclic quadrilaterals

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The discussion revolves around finding the angles in a cyclic quadrilateral given that angle ABD is 50 degrees. It is established that angle ACD also measures 50 degrees due to the inscribed angle theorem. The participants debate whether lines AC and BD intersect at the circle's center, O, which is crucial for determining angle ACB. Ultimately, if AC intersects BD at O, angle ACB is found to be 40 degrees, but without this intersection, there isn't enough information to conclude the angle's measure. The conversation highlights the importance of the intersection point in applying Thales' theorem to solve the problem.
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Here is a circle with center $O$ :cool:

Its is given that $\angle ABD=50$ & to find the magnitudes of

$\angle ACD$ & $\angle ACB$

Now what I know is (Nerd) $\angle ACD=50$ due to the inscribed angle theorem, Can you help me to find the other angle which I don't know how to find ,stating the reasons
 

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Do $\overline{AC}$ and $\overline{BD}$ intersect at $O$ ?
 
greg1313 said:
Do $\overline{AC}$ and $\overline{BD}$ intersect at $O$ ?

No nothing about the intersection is mentioned in the problem , But it is given that $O$ is the center of the circle
 
I don't think there's enough information. If $\overline{AC}$ and $\overline{BD}$ intersected at $O$ the problem wouldn't make any sense. So, are we missing anything?
 
No that is all what is given in the problem (Sadface)
 
Actually I goofed - :o - if $\overline{AC}$ intersects $\overline{BD}$ at $O$, then $\angle{ACB}=40^\circ$, so I'd assume that is the case - without that I don't think there's enough information.
 
greg1313 said:
Actually I goofed - :o - if $\overline{AC}$ intersects $\overline{BD}$ at $O$, then $\angle{ACB}=40^\circ$, so I'd assume that is the case - without that I don't think there's enough information.

Yes it should be :) You used "Angle at the Center Theorem" , right?
 
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Hey mathlearn! ;)

I believe it suffices if $O$ is on $BD$.
Due to Thales' theorem that implies that the required angle is 40 degrees.

Without it, we indeed do not have sufficient information.
It would mean that the angle at A does not have to be 90 degrees, implying that the required angle does not have to be 40 degrees, but could be anything. (Nerd)

Cheers!
 

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