1. **Problem Statement:** We want to understand the theorem that states "The angles formed at the same segment of a circle are equal in magnitude." This means if two angles are subtended by the same chord and lie on the same side of the chord, then these angles are equal. 2. **Theorem Explanation:** Consider a circle with points A, B, C, and D on its circumference. Suppose chord AB is fixed. Then, the angles \(\angle ACB\) and \(\angle ADB\), which are angles subtended by chord AB at points C and D respectively on the same segment, are equal. 3. **Formula/Rule:** If points A, B, C, and D lie on a circle and C and D are on the same segment of chord AB, then: $$\angle ACB = \angle ADB$$ 4. **Example:** - Let chord AB be fixed. - Points C and D lie on the circumference such that both are on the same side of chord AB. - Measure \(\angle ACB\) and \(\angle ADB\). 5. **Proof Sketch:** - Both angles subtend the same chord AB. - By the circle theorem, angles subtended by the same chord in the same segment are equal. 6. **Learner-friendly explanation:** Imagine you have a string stretched between points A and B on a circle. If you pick any point C on the circle on one side of the string and measure the angle formed by points A, C, and B, then pick another point D on the same side of the string and measure the angle formed by A, D, and B, these two angles will be exactly the same. This is because the arc between A and B creates a constant angle for all points on the same segment. 7. **Summary:** The key takeaway is that the angle subtended by a chord at any point on the same segment of the circle is constant. This theorem is very useful in solving many geometry problems involving circles.