The given figure shows quadrilateral ABCD. The dimensions of the sides and of the diagonal AC are shown in the figure. What is the measure of ∠DAC?

Comparing ΔABC and ΔACD:

$$\begin{gathered} \frac{AB}{AC}=\frac{3}{6}=\frac{1}{2} \\ \frac{BC}{AD}=\frac{5}{10}=\frac{1}{2} \\ \frac{AC}{CD}=\frac{6}{12}=\frac{1}{2} \\ \therefore \frac{AB}{AC}=\frac{BC}{AD}=\frac{AC}{CD}=\frac{1}{2} \end{gathered}$$

Therefore, by SSS similarity criterion, ΔABC ∼ ΔCAD

∴ ∠BAC = ∠ACD (Corresponding angles of similar triangles are equal)

⇒ ∠ACD = 56° (∠BAC = 56°)

Applying angle sum property of triangles in ΔACD:

∠ACD + ∠DAC + ∠CDA = 180°

⇒ 56° + ∠DAC + 30° = 180°

⇒ ∠DAC = 180° − 86° = 94°

Thus, the measure of ∠DAC is 94°.