1. The problem: Understand the Alternate Segment Theorem in geometry. 2. Statement: 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. Formula and explanation: If a tangent touches a circle at point $T$ and a chord $TC$ is drawn, then the angle between the tangent and chord, say $\angle ATC$, is equal to the angle subtended by the chord in the alternate segment, say $\angle TCB$. 4. Important rules: - The tangent touches the circle at exactly one point. - The chord is a line segment joining two points on the circle. - The alternate segment is the segment of the circle opposite to the angle between the tangent and chord. 5. Intermediate work: - Identify the tangent point $T$. - Draw chord $TC$. - Measure $\angle ATC$ between tangent and chord. - Measure $\angle TCB$ in the alternate segment. 6. Explanation: This theorem helps relate angles formed outside the circle (by tangent and chord) to angles inside the circle (in the alternate segment). It is useful in solving many circle geometry problems involving tangents and chords. Final answer: The angle between the tangent and chord at the point of contact equals the angle in the alternate segment of the circle, i.e., $\angle ATC = \angle TCB$.