Euclid's Elements

Label four points $A$ , $B$ , $C$ , and $D$ as shown. Since the yellow rectangle plus blue rectangle forms a square the area of the square is $3.91+12.01 = 16$ and its side length $AC = \sqrt{16} = 4$ . Then the area of the blue rectangle is $AD \times 4 = 12.09 \implies AD = 3.0225$ . We note that

$\begin{aligned} \cos \theta & = \frac {AD}\blue{AB} & \small \blue{\text{Note that }AB = \sqrt{15.32}} \\ & = \frac {3.0225}\blue{\sqrt{15.32}} \\ \theta & \approx 0.688479974 \text{ rad} \\ & \approx \boxed{39.5}^\circ \end{aligned}$