To explain the Theorem of Angles in the Alternate Segment, let’s break it down into simple steps:

1. What the Theorem Means: This theorem tells us that if you have a line that just touches (tangent) a circle and a line that cuts through it (chord), the angle formed between the tangent and the chord is the same as the angle made by the chord in the opposite segment of the circle.

2. Drawing a Diagram: Imagine a circle called $O$. We have a tangent line named $AB$ touching the circle at point $A$. We also have a chord called $AC$. Now, draw a line from the center of the circle ($O$) to $A$. Then, continue the line $AC$ until it touches the circle again at point $D$.

3. Looking at the Angles:

   • We can call the angle between the chord and the tangent at point $A$ as $\angle CAB$.
   • The angle made by the points $A$ and $D$ in the opposite segment can be called $\angle ADB$.

4. Using a Special Rule: There’s a rule called the Inscribed Angle Theorem. It says that the angle $\angle ADB$ equals half the size of the arc ($AB$) it touches. Since $\angle CAB$ also relates to this same arc, we can say that $\angle CAB = \angle ADB$.

This theorem is really important for understanding shapes and angles in circles. It helps us solve many problems in geometry!