1. **State the problem:** We need to find the size of angle $\angle DAC$ in a circle with diameter $AC$, where $\angle CAB = 25^\circ$ and $\angle DEC = 100^\circ$. 2. **Recall key properties:** - Since $AC$ is a diameter, $\angle ABC$ (an angle subtended by the diameter) is a right angle ($90^\circ$). - Angles subtended by the same chord are equal. - The sum of angles around point $E$ is $360^\circ$. 3. **Analyze given angles:** - $\angle CAB = 25^\circ$ is given. - $\angle DEC = 100^\circ$ is given. 4. **Find $\angle CEB$:** Since $\angle DEC = 100^\circ$, and $E$ lies on chords $AC$ and $BD$, the angles $\angle DEB$ and $\angle CEB$ are supplementary (they form a straight line along $BD$). So, $$\angle DEB + \angle CEB = 180^\circ$$ $$100^\circ + \angle CEB = 180^\circ$$ $$\angle CEB = 180^\circ - 100^\circ = 80^\circ$$ 5. **Use the fact that $\angle CEB$ and $\angle CAB$ subtend the same chord $CB$:** Angles subtended by the same chord in the circle are equal, so $$\angle CEB = \angle CAB = 25^\circ$$ But from step 4, $\angle CEB = 80^\circ$, so this is a contradiction. Therefore, $\angle CEB$ does not subtend chord $CB$. 6. **Use the cyclic quadrilateral property:** Points $A, B, C, D$ lie on the circle. Opposite angles of a cyclic quadrilateral sum to $180^\circ$. Opposite angles are $\angle BAC$ and $\angle BDC$. Given $\angle CAB = 25^\circ$, so $$\angle BDC = 180^\circ - 25^\circ = 155^\circ$$ 7. **Use the fact that $\angle DEC = 100^\circ$ is part of $\angle BDC$:** Since $E$ lies on $BD$, $\angle BDC$ is split into $\angle BDE$ and $\angle EDC$. Given $\angle DEC = 100^\circ$, and $\angle BDC = 155^\circ$, then $$\angle BDE = 155^\circ - 100^\circ = 55^\circ$$ 8. **Find $\angle DAC$:** Since $AC$ is diameter, $\angle DAC$ is an angle subtended by chord $DC$ at point $A$. Using the property that angles subtended by the same chord are equal, $$\angle DAC = \angle BDE = 55^\circ$$ **Final answer:** $$\boxed{55^\circ}$$