Question: User: Line HGF is a tangent to the circle shown below.
Calculate the size of angle a.
Give a reason for your answer.
Not drawn accurately
HGF is a tangent at G, with points D on the left of the circle, E at the top, and G at the bottom; chords DE, DG, and EG are drawn; the angle at D between DE and the line to the interior point is 17\u00b0, the angle at D between the line to the interior point and DG is 58\u00b0, and angle a is at G between EG and the tangent GF.
1. **State the problem:** We have a circle with tangent line HGF at point G. We need to find the size of angle $a$ at point G between chord $EG$ and tangent $GF$.
2. **Identify given information:**
- Angle at $D$ between $DE$ and the interior line is $17^\circ$.
- Angle at $D$ between the interior line and $DG$ is $58^\circ$.
- Tangent $HGF$ touches the circle at $G$.
3. **Calculate angle $EDG$:**
$$\angle EDG = 17^\circ + 58^\circ = 75^\circ$$
4. **Recall the tangent-secant theorem:**
The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
5. **Apply the theorem:**
Angle $a$ between tangent $GF$ and chord $EG$ equals the angle $EDG$ in the alternate segment.
6. **Therefore:**
$$a = 75^\circ$$
7. **Reason:**
Because the angle between a tangent and a chord is equal to the angle in the alternate segment of the circle.
**Final answer:**
$$\boxed{75^\circ}$$