Consider a diameter $\overline{AF}.$ Since $\overleftrightarrow{AB}$ is tangent to the circle at $A,$ then $\overline{AF}\perp \overleftrightarrow{AB}.$

In addition, since $\overline{AF}$ is a diameter, $\overgroup{ACF}$ is a semicircle and then its measure is $180^{\circ }.$ The Arc Addition Postulate allows setting the following equation. $$\begin{array}{c}m\overgroup{AC}+m\overgroup{FC}=\underset{180^{\circ }}{\underbrace{m\overgroup{ACF}}}\end{array}$$ Since $\angle CAF$ is a central angle, its measure is half the measure of the intercepted arc. That is, $m\angle CAF=\frac{1}{2}m\overgroup{FC}$ or equivalently, $m\overgroup{FC}=2m\angle CAF.$ $$\begin{array}{c}m\overgroup{AC}+2m\angle CAF=180^{\circ }\end{array}$$ Similarly, the Angle Addition Postulate can be used to establish the following equation. $$\begin{array}{c}m\angle 1+m\angle CAF=90^{\circ }\end{array}$$ Next, multiply the equation above by $\text{-}2$ and add it to the previous equation. $$\begin{array}{rl}m\overgroup{AC}+2m\angle CAF& =180^{\circ }\\ {}^{+}\text{-}2m\angle 1-2m\angle CAF& =\text{-}180^{\circ }\\ & \\ m\overgroup{AC}-2m\angle 1& =0\end{array}$$ Solving the last equation by $m\angle 1$ gives the first desired equation.

Once more, the Arc Addition Postulate and the Angle Addition Postulate can be applied to set the following pair of equations. $$\{\begin{array}{ll}m\overgroup{ADF}+m\overgroup{FC}=m\overgroup{ADC}& \text{(I)}\\ m\angle EAF+m\angle CAF=m\angle 2& \text{(II)}\end{array}$$ Recall that $m\overgroup{ADF}=180^{\circ },$ $m\angle EAF=90^{\circ },$ and $m\overgroup{FC}=2m\angle CAF.$ $$\{\begin{array}{ll}180^{\circ }+2m\angle CAF=m\overgroup{ADC}& \text{(I)}\\ 90^{\circ }+m\angle CAF=m\angle 2& \text{(II)}\end{array}$$ Next, multiply Equation $(\text{II})$ by $\text{-}2$ and add it to Equation $(\text{I}).$ $$\begin{array}{rl}180^{\circ }+2m\angle CAF& =m\overgroup{ADC}\\ {}^{+}\text{-}180^{\circ }-2m\angle CAF& =\text{-}2m\angle 2\\ & \\ 0& =m\overgroup{ADC}-2m\angle 2\end{array}$$ Finally, by solving the resulting equation for $m\angle 2,$ the second desired equation is obtained.