2021-II-39

Join $AD$.

In $\mathrm{\Delta }CDE$,

$\begin{array}{rcll}\mathrm{\angle }DCE& =& \mathrm{\angle }CDF-\mathrm{\angle }CED& \text{(ext. }\mathrm{\angle }\text{ of }\mathrm{\Delta }\text{)}\\ \mathrm{\angle }DCE& =& {49}^{\circ }-{31}^{\circ }\\ \mathrm{\angle }DCE& =& {18}^{\circ }\end{array}$

In $\mathrm{\Delta }ACD$,

$\begin{array}{rcll}\mathrm{\angle }ADC& =& {90}^{\circ }& \text{(}\mathrm{\angle }\text{ in semi-circle)}\\ \mathrm{\angle }CAD& =& \mathrm{\angle }CDF& \text{(}\mathrm{\angle }\text{ in alt. segment)}\\ \mathrm{\angle }CAD& =& {49}^{\circ }\\ \mathrm{\angle }ACE& =& {180}^{\circ }-\mathrm{\angle }ADC-\mathrm{\angle }CAD-\mathrm{\angle }DCE& \text{(}\mathrm{\angle }\text{ sum of }\mathrm{\Delta }\text{)}\\ \mathrm{\angle }ACE& =& {180}^{\circ }-{90}^{\circ }-{49}^{\circ }-{18}^{\circ }\\ \mathrm{\angle }ACE& =& {23}^{\circ }\end{array}$

In $\mathrm{\Delta }ABC$,

$\begin{array}{rcll}\mathrm{\angle }BAC& =& \mathrm{\angle }ACE& \text{(alt. }\mathrm{\angle }\text{s, }AB\text{//}EC\text{)}\\ \mathrm{\angle }BAC& =& {23}^{\circ }\\ \mathrm{\angle }ABC& =& {90}^{\circ }& \text{(}\mathrm{\angle }\text{ in semi-circle)}\\ \mathrm{\angle }ACB& =& {180}^{\circ }-\mathrm{\angle }ABC-\mathrm{\angle }BAC& \text{(}\mathrm{\angle }\text{ sum of }\mathrm{\Delta }\text{)}\\ \mathrm{\angle }ACB& =& {180}^{\circ }-{90}^{\circ }-{23}^{\circ }\\ \mathrm{\angle }ACB& =& {57}^{\circ }\end{array}$