The diagram shows a circle with centre A. Points G, D, F are on the circle. CE is a tangent to the circle at D. DGE = 51^. AFG = 38^. Find GAF.

We are given:

• G, D, F are points on the circle with centre A.
• $\overline{CE}$ is a tangent to the circle at D.
• $\angle DGE = 51^\circ$.
• $\angle AFG = 38^\circ$.

Since $\overline{AD}$ is a radius and $\overline{CE}$ is a tangent at D, the radius is perpendicular to the tangent at the point of tangency.
Therefore, $\angle ADE = 90^\circ$.

Since A is the centre and F, G are on the circle, $\overline{AF}$ and $\overline{AG}$ are