1. **Problem Statement:** We have a circle with center O and points A, B, C, D on the circumference. O lies on chord AB. Given angle $\angle DAB = 44^\circ$, find angles $\angle AOD$, $\angle DOB$, and $\angle DCB$. 2. **Key Theorem:** The angle subtended by an arc at the center of the circle is twice the angle subtended at the circumference by the same arc. That is, if $\angle DAB$ is an inscribed angle subtending arc $DB$, then the central angle $\angle DOB$ subtending the same arc satisfies: $$\angle DOB = 2 \times \angle DAB$$ 3. **Find $\angle DOB$:** Given $\angle DAB = 44^\circ$, $$\angle DOB = 2 \times 44^\circ = 88^\circ$$ 4. **Find $\angle AOD$:** Since O lies on AB, and OA and OD are radii, triangle $AOD$ is isosceles with $OA = OD$. The central angle $\angle AOD$ subtends arc $AD$. The angle at the circumference $\angle ADB$ subtends the same arc $AB$. But we need $\angle AOD$. Note that $\angle AOD$ and $\angle DOB$ are central angles subtending arcs $AD$ and $DB$ respectively, and $\angle AOD + \angle DOB = 180^\circ$ because O lies on AB (a straight line). So, $$\angle AOD = 180^\circ - \angle DOB = 180^\circ - 88^\circ = 92^\circ$$ 5. **Find $\angle DCB$:** $\angle DCB$ is an inscribed angle subtending arc $DB$. By the theorem, $$\angle DCB = \frac{1}{2} \times \angle DOB = \frac{1}{2} \times 88^\circ = 44^\circ$$ **Final answers:** $$\angle AOD = 92^\circ$$ $$\angle DOB = 88^\circ$$ $$\angle DCB = 44^\circ$$