1. The problem is to understand and illustrate the Alternate Segment Theorem in geometry. 2. The Alternate Segment Theorem states that the angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment of the circle. 3. Consider a circle with a tangent line touching the circle at point $T$ and a chord $TC$. 4. Let the tangent at $T$ make an angle $\angle ATC$ with the chord $TC$. 5. According to the theorem, this angle $\angle ATC$ is equal to the angle $\angle TCB$ in the alternate segment of the circle. 6. This means $$\angle ATC = \angle TCB$$ where $A$ is a point on the tangent line and $B$ is a point on the circle such that $TCB$ is an angle in the alternate segment. 7. This theorem helps in solving many problems involving tangents and chords in circles by relating these angles. Final answer: The angle between the tangent and chord at the point of contact equals the angle in the alternate segment, i.e., $$\angle ATC = \angle TCB$$.