In the figure below, $\overline{EF}$ is a diameter of the circle. What is the measure of $\angle ABC$, in degrees?

Question

In the figure below, $\overline{EF}$ is a diameter of the circle. What is the measure of $\angle ABC$, in degrees?

Would listing angles help me? If so, how?

Answer 1

The arc $\;\overset{\mmlToken{mo}{⏜}}{AC\,} = 180^\circ - \;\overset{\mmlToken{mo}{⏜}}{CF\,} - \;\overset{\mmlToken{mo}{⏜}}{AE\,} = 180^\circ - 80^\circ - 35^\circ = 65^\circ$.

Then by the interior secant angle formula: $ \widehat{ABC} = \frac{1}{2}(\;\overset{\mmlToken{mo}{⏜}}{AC\,} + \;\overset{\mmlToken{mo}{⏜}}{EF\,}) = \frac{1}{2}(65^\circ + 180^\circ) = 122.5^\circ$.

( The above assumes that the given values refer to angles $\widehat{COF} = 80^\circ$ and $\widehat{AOE} = 35^\circ$. )

Answer 2

We have $$\widehat{CEF}=40^\circ,\qquad \widehat{AFE}=35^\circ $$ hence $$\widehat{ABC}=\widehat{EBF}=180^\circ-(40^\circ+35^\circ) = \color{red}{105^\circ}.$$ ( I am guessing that the depicted $80^\circ$ and $35^\circ$ angles are $\widehat{COF}$ and $\widehat{AFE}$ ).

Answer 3

$\angle ABC=180^\circ -(\angle CEF +\angle AFE)=180^\circ-(40^\circ +17.5^\circ)=122.5^\circ$